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/EquivariantMap.lean | Set.image_smul_setₑ | [40, 1] | [44, 30] | apply Set.image_congr | α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝¹ : SMul M α
inst✝ : SMul N β
f : α →ₑ[φ] β
m : M
s : Set α
⊢ (fun x => f.toFun (m • x)) '' s = (fun x => φ m • f.toFun x) '' s | case h
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝¹ : SMul M α
inst✝ : SMul N β
f : α →ₑ[φ] β
m : M
s : Set α
⊢ ∀ a ∈ s, f.toFun (m • a) = φ m • f.toFun a | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝¹ : SMul M α
inst✝ : SMul N β
f : α →ₑ[φ] β
m : M
s : Set α
⊢ (fun x => f.toFun (m • x)) '' s = (fun x => φ m • f.toFun x) '' s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | Set.image_smul_setₑ | [40, 1] | [44, 30] | intro a _ | case h
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝¹ : SMul M α
inst✝ : SMul N β
f : α →ₑ[φ] β
m : M
s : Set α
⊢ ∀ a ∈ s, f.toFun (m • a) = φ m • f.toFun a | case h
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝¹ : SMul M α
inst✝ : SMul N β
f : α →ₑ[φ] β
m : M
s : Set α
a : α
a✝ : a ∈ s
⊢ f.toFun (m • a) = φ m • f.toFun a | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝¹ : SMul M α
inst✝ : SMul N β
f : α →ₑ[φ] β
m : M
s : Set α
⊢ ∀ a ∈ s, f.toFun (m • a) = φ m • f.toFun a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | Set.image_smul_setₑ | [40, 1] | [44, 30] | rw [f.map_smul'] | case h
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝¹ : SMul M α
inst✝ : SMul N β
f : α →ₑ[φ] β
m : M
s : Set α
a : α
a✝ : a ∈ s
⊢ f.toFun (m • a) = φ m • f.toFun a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝¹ : SMul M α
inst✝ : SMul N β
f : α →ₑ[φ] β
m : M
s : Set α
a : α
a✝ : a ∈ s
⊢ f.toFun (m • a) = φ m • f.toFun a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | Set.image_smul_set | [49, 1] | [49, 84] | simp | α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type u_5
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
s : Set α
⊢ ⇑f₁ '' (m • s) = m • ⇑f₁ '' s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type u_5
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
s : Set α
⊢ ⇑f₁ '' (m • s) = m • ⇑f₁ '' s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | Set.preimage_smul_setₑ_le | [53, 1] | [57, 41] | rintro x ⟨y, hy, rfl⟩ | α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.2413
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
t : Set β
⊢ m • ⇑f ⁻¹' t ⊆ ⇑f ⁻¹' (φ m • t) | case intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.2413
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
t : Set β
y : α
hy : y ∈ ⇑f ⁻¹' t
⊢ (fun x => m • x) y ∈ ⇑f ⁻¹' (φ m • t) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.2413
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
t : Set β
⊢ m • ⇑f ⁻¹' t ⊆ ⇑f ⁻¹' (φ m • t)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | Set.preimage_smul_setₑ_le | [53, 1] | [57, 41] | refine ⟨f y, hy, (by rw [map_smulₛₗ])⟩ | case intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.2413
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
t : Set β
y : α
hy : y ∈ ⇑f ⁻¹' t
⊢ (fun x => m • x) y ∈ ⇑f ⁻¹' (φ m • t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.2413
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
t : Set β
y : α
hy : y ∈ ⇑f ⁻¹' t
⊢ (fun x => m • x) y ∈ ⇑f ⁻¹' (φ m • ... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | Set.preimage_smul_setₑ_le | [53, 1] | [57, 41] | rw [map_smulₛₗ] | α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.2413
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
t : Set β
y : α
hy : y ∈ ⇑f ⁻¹' t
⊢ (fun x => φ m • x) (f y) = f ((fun x => m • x) y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.2413
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
t : Set β
y : α
hy : y ∈ ⇑f ⁻¹' t
⊢ (fun x => φ m • x) (f y) = f ((fun x => m • x) y)
TA... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | Set.preimage_smul_setₑ' | [65, 1] | [75, 32] | apply Set.Subset.antisymm | α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun b => φ m • b
t : Set β
⊢ ⇑f ⁻¹' (φ m • t) = m • ⇑f ⁻¹' t | case h₁
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun b => φ m • b
t : Set β
⊢ ⇑f ⁻¹' (φ m • t) ⊆ m • ⇑f ⁻¹' t
case h₂
α : Ty... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun b => φ m • b
t : Se... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | Set.preimage_smul_setₑ' | [65, 1] | [75, 32] | exact preimage_smul_setₑ_le t | case h₂
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun b => φ m • b
t : Set β
⊢ m • ⇑f ⁻¹' t ⊆ ⇑f ⁻¹' (φ m • t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun b => φ m • ... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | Set.preimage_smul_setₑ' | [65, 1] | [75, 32] | rintro x ⟨y, yt, hy⟩ | case h₁
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun b => φ m • b
t : Set β
⊢ ⇑f ⁻¹' (φ m • t) ⊆ m • ⇑f ⁻¹' t | case h₁.intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun b => φ m • b
t : Set β
x : α
y : β
yt : y ∈ t
hy : (fun x =>... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun b => φ m • ... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | Set.preimage_smul_setₑ' | [65, 1] | [75, 32] | dsimp at hy | case h₁.intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun b => φ m • b
t : Set β
x : α
y : β
yt : y ∈ t
hy : (fun x =>... | case h₁.intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun b => φ m • b
t : Set β
x : α
y : β
yt : y ∈ t
hy : φ m • y =... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | Set.preimage_smul_setₑ' | [65, 1] | [75, 32] | obtain ⟨x', hx' : m • x' = x⟩ := hmα.surjective x | case h₁.intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun b => φ m • b
t : Set β
x : α
y : β
yt : y ∈ t
hy : φ m • y =... | case h₁.intro.intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun b => φ m • b
t : Set β
x : α
y : β
yt : y ∈ t
hy : φ m... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | Set.preimage_smul_setₑ' | [65, 1] | [75, 32] | refine ⟨x', ?_, hx'⟩ | case h₁.intro.intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun b => φ m • b
t : Set β
x : α
y : β
yt : y ∈ t
hy : φ m... | case h₁.intro.intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun b => φ m • b
t : Set β
x : α
y : β
yt : y ∈ t
hy : φ m... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.intro.intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijecti... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | Set.preimage_smul_setₑ' | [65, 1] | [75, 32] | simp only [← hx', map_smulₛₗ] at hy | case h₁.intro.intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun b => φ m • b
t : Set β
x : α
y : β
yt : y ∈ t
hy : φ m... | case h₁.intro.intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun b => φ m • b
t : Set β
x : α
y : β
yt : y ∈ t
x' : α
h... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.intro.intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijecti... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | Set.preimage_smul_setₑ' | [65, 1] | [75, 32] | simp only [mem_preimage] | case h₁.intro.intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun b => φ m • b
t : Set β
x : α
y : β
yt : y ∈ t
x' : α
h... | case h₁.intro.intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun b => φ m • b
t : Set β
x : α
y : β
yt : y ∈ t
x' : α
h... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.intro.intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijecti... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | Set.preimage_smul_setₑ' | [65, 1] | [75, 32] | rw [← hmβ.injective hy] | case h₁.intro.intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun b => φ m • b
t : Set β
x : α
y : β
yt : y ∈ t
x' : α
h... | case h₁.intro.intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun b => φ m • b
t : Set β
x : α
y : β
yt : y ∈ t
x' : α
h... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.intro.intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijecti... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | Set.preimage_smul_setₑ' | [65, 1] | [75, 32] | exact yt | case h₁.intro.intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijective fun b => φ m • b
t : Set β
x : α
y : β
yt : y ∈ t
x' : α
h... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.intro.intro.intro
α : Type u_1
β : Type u_2
M : Type u_3
N : Type u_4
φ : M → N
inst✝² : SMul M α
inst✝¹ : SMul N β
f : α →ₑ[φ] β
β₁ : Type ?u.4258
inst✝ : SMul M β₁
f₁ : α →ₑ[id] β₁
m : M
hmα : Function.Bijective fun a => m • a
hmβ : Function.Bijecti... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | isPretransitive.of_surjective_map | [117, 1] | [125, 34] | apply MulAction.IsPretransitive.mk | α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPret... | case exists_smul_eq
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surj... | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | isPretransitive.of_surjective_map | [117, 1] | [125, 34] | intro x y | case exists_smul_eq
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surj... | case exists_smul_eq
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surj... | Please generate a tactic in lean4 to solve the state.
STATE:
case exists_smul_eq
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
ins... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | isPretransitive.of_surjective_map | [117, 1] | [125, 34] | let h_eq := h.exists_smul_eq | case exists_smul_eq
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surj... | case exists_smul_eq
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surj... | Please generate a tactic in lean4 to solve the state.
STATE:
case exists_smul_eq
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
ins... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | isPretransitive.of_surjective_map | [117, 1] | [125, 34] | obtain ⟨x', rfl⟩ := hf x | case exists_smul_eq
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surj... | case exists_smul_eq.intro
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Functio... | Please generate a tactic in lean4 to solve the state.
STATE:
case exists_smul_eq
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
ins... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | isPretransitive.of_surjective_map | [117, 1] | [125, 34] | obtain ⟨y', rfl⟩ := hf y | case exists_smul_eq.intro
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Functio... | case exists_smul_eq.intro.intro
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : F... | Please generate a tactic in lean4 to solve the state.
STATE:
case exists_smul_eq.intro
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | isPretransitive.of_surjective_map | [117, 1] | [125, 34] | obtain ⟨g, rfl⟩ := h_eq x' y' | case exists_smul_eq.intro.intro
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : F... | case exists_smul_eq.intro.intro.intro
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
... | Please generate a tactic in lean4 to solve the state.
STATE:
case exists_smul_eq.intro.intro
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ :... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | isPretransitive.of_surjective_map | [117, 1] | [125, 34] | use φ g | case exists_smul_eq.intro.intro.intro
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
... | case h
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h :... | Please generate a tactic in lean4 to solve the state.
STATE:
case exists_smul_eq.intro.intro.intro
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
in... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | isPretransitive.of_surjective_map | [117, 1] | [125, 34] | simp only [map_smulₛₗ] | case h
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulActio... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | isPretransitive.of_bijective_map_iff | [128, 1] | [142, 31] | constructor | α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Functi... | case mp
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf ... | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | isPretransitive.of_bijective_map_iff | [128, 1] | [142, 31] | apply isPretransitive.of_surjective_map hf.surjective | case mp
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf ... | case mpr
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulActi... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | isPretransitive.of_bijective_map_iff | [128, 1] | [142, 31] | intro hN | case mpr
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf... | case mpr
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAct... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | isPretransitive.of_bijective_map_iff | [128, 1] | [142, 31] | apply IsPretransitive.mk | case mpr
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf... | case mpr.exists_smul_eq
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAct... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | isPretransitive.of_bijective_map_iff | [128, 1] | [142, 31] | intro x y | case mpr.exists_smul_eq
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.... | case mpr.exists_smul_eq
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.exists_smul_eq
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | isPretransitive.of_bijective_map_iff | [128, 1] | [142, 31] | obtain ⟨k, hk⟩ := exists_smul_eq N (f x) (f y) | case mpr.exists_smul_eq
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.... | case mpr.exists_smul_eq.intro
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Fun... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.exists_smul_eq
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | isPretransitive.of_bijective_map_iff | [128, 1] | [142, 31] | obtain ⟨g, rfl⟩ := hφ k | case mpr.exists_smul_eq.intro
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Fun... | case mpr.exists_smul_eq.intro.intro
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.exists_smul_eq.intro
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : G... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | isPretransitive.of_bijective_map_iff | [128, 1] | [142, 31] | use g | case mpr.exists_smul_eq.intro.intro
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ... | case h
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf :... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.exists_smul_eq.intro.intro
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | isPretransitive.of_bijective_map_iff | [128, 1] | [142, 31] | apply hf.injective | case h
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf :... | case h.a
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulActio... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/EquivariantMap.lean | isPretransitive.of_bijective_map_iff | [128, 1] | [142, 31] | simp only [hk, map_smulₛₗ] | case h.a
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
α✝ : Type u_1
β✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
φ✝ : M✝ → N✝
inst✝⁵ : SMul M✝ α✝
inst✝⁴ : SMul N✝ β✝
f✝ : α✝ →ₑ[φ✝] β✝
M : Type u_8
inst✝³ : Group M
α : Type u_6
inst✝² : MulAction M α
N : Type u_7
β : Type u_5
inst✝¹ : Group N
inst✝ : MulAct... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.isMultiplyPreprimitive_zero | [54, 1] | [58, 9] | constructor | M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
⊢ IsMultiplyPreprimitive M α 0 | case left
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
⊢ IsMultiplyPretransitive M α 0
case right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
⊢ ∀ (s : Set α), Set.encard s + 1 = ↑0 → IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s) | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
⊢ IsMultiplyPreprimitive M α 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.isMultiplyPreprimitive_zero | [54, 1] | [58, 9] | apply MulAction.is_zero_pretransitive | case left
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
⊢ IsMultiplyPretransitive M α 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
⊢ IsMultiplyPretransitive M α 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.isMultiplyPreprimitive_zero | [54, 1] | [58, 9] | intro s | case right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
⊢ ∀ (s : Set α), Set.encard s + 1 = ↑0 → IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s) | case right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s : Set α
⊢ Set.encard s + 1 = ↑0 → IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
⊢ ∀ (s : Set α), Set.encard s + 1 = ↑0 → IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.isMultiplyPreprimitive_zero | [54, 1] | [58, 9] | simp | case right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s : Set α
⊢ Set.encard s + 1 = ↑0 → IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s : Set α
⊢ Set.encard s + 1 = ↑0 → IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.isPreprimitive_iff_is_one_preprimitive | [62, 1] | [86, 57] | constructor | M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
⊢ IsPreprimitive M α ↔ IsMultiplyPreprimitive M α 1 | case mp
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
⊢ IsPreprimitive M α → IsMultiplyPreprimitive M α 1
case mpr
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
⊢ IsMultiplyPreprimitive M α 1 → IsPreprimitive M α | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
⊢ IsPreprimitive M α ↔ IsMultiplyPreprimitive M α 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.isPreprimitive_iff_is_one_preprimitive | [62, 1] | [86, 57] | intro h | case mp
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
⊢ IsPreprimitive M α → IsMultiplyPreprimitive M α 1 | case mp
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
⊢ IsMultiplyPreprimitive M α 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
⊢ IsPreprimitive M α → IsMultiplyPreprimitive M α 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.isPreprimitive_iff_is_one_preprimitive | [62, 1] | [86, 57] | constructor | case mp
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
⊢ IsMultiplyPreprimitive M α 1 | case mp.left
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
⊢ IsMultiplyPretransitive M α 1
case mp.right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
⊢ ∀ (s : Set α), Set.encard s + 1 = ↑1 → IsPreprimitive ↥(fixingSubgroup M s) ↥(SubM... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
⊢ IsMultiplyPreprimitive M α 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.isPreprimitive_iff_is_one_preprimitive | [62, 1] | [86, 57] | rw [← isPretransitive_iff_is_one_pretransitive] | case mp.left
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
⊢ IsMultiplyPretransitive M α 1 | case mp.left
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
⊢ IsPretransitive M α | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.left
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
⊢ IsMultiplyPretransitive M α 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.isPreprimitive_iff_is_one_preprimitive | [62, 1] | [86, 57] | exact h.toIsPretransitive | case mp.left
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
⊢ IsPretransitive M α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.left
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
⊢ IsPretransitive M α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.isPreprimitive_iff_is_one_preprimitive | [62, 1] | [86, 57] | intro s hs | case mp.right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
⊢ ∀ (s : Set α), Set.encard s + 1 = ↑1 → IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s) | case mp.right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
s : Set α
hs : Set.encard s + 1 = ↑1
⊢ IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
⊢ ∀ (s : Set α), Set.encard s + 1 = ↑1 → IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.isPreprimitive_iff_is_one_preprimitive | [62, 1] | [86, 57] | simp only [Nat.cast_one] at hs | case mp.right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
s : Set α
hs : Set.encard s + 1 = ↑1
⊢ IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s) | case mp.right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
s : Set α
hs : Set.encard s + 1 = 1
⊢ IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
s : Set α
hs : Set.encard s + 1 = ↑1
⊢ IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.isPreprimitive_iff_is_one_preprimitive | [62, 1] | [86, 57] | suffices s = ∅ by
rw [this]
rw [isPreprimitive_of_bijective_map_iff
(SubMulAction.of_fixingSubgroupEmpty_mapScalars_surjective M α)
(SubMulAction.ofFixingSubgroupEmpty_equivariantMap_bijective M α)]
exact h | case mp.right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
s : Set α
hs : Set.encard s + 1 = 1
⊢ IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s) | case mp.right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
s : Set α
hs : Set.encard s + 1 = 1
⊢ s = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
s : Set α
hs : Set.encard s + 1 = 1
⊢ IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.isPreprimitive_iff_is_one_preprimitive | [62, 1] | [86, 57] | rw [← Set.encard_eq_zero, ← Nat.cast_zero, ← WithTop.add_one_eq_coe_succ_iff] | case mp.right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
s : Set α
hs : Set.encard s + 1 = 1
⊢ s = ∅ | case mp.right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
s : Set α
hs : Set.encard s + 1 = 1
⊢ Set.encard s + 1 = ↑(Nat.succ 0) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
s : Set α
hs : Set.encard s + 1 = 1
⊢ s = ∅
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.isPreprimitive_iff_is_one_preprimitive | [62, 1] | [86, 57] | exact hs | case mp.right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
s : Set α
hs : Set.encard s + 1 = 1
⊢ Set.encard s + 1 = ↑(Nat.succ 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
s : Set α
hs : Set.encard s + 1 = 1
⊢ Set.encard s + 1 = ↑(Nat.succ 0)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.isPreprimitive_iff_is_one_preprimitive | [62, 1] | [86, 57] | rw [this] | M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
s : Set α
hs : Set.encard s + 1 = 1
this : s = ∅
⊢ IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s) | M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
s : Set α
hs : Set.encard s + 1 = 1
this : s = ∅
⊢ IsPreprimitive ↥(fixingSubgroup M ∅) ↥(SubMulAction.ofFixingSubgroup M ∅) | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
s : Set α
hs : Set.encard s + 1 = 1
this : s = ∅
⊢ IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.isPreprimitive_iff_is_one_preprimitive | [62, 1] | [86, 57] | rw [isPreprimitive_of_bijective_map_iff
(SubMulAction.of_fixingSubgroupEmpty_mapScalars_surjective M α)
(SubMulAction.ofFixingSubgroupEmpty_equivariantMap_bijective M α)] | M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
s : Set α
hs : Set.encard s + 1 = 1
this : s = ∅
⊢ IsPreprimitive ↥(fixingSubgroup M ∅) ↥(SubMulAction.ofFixingSubgroup M ∅) | M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
s : Set α
hs : Set.encard s + 1 = 1
this : s = ∅
⊢ IsPreprimitive M α | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
s : Set α
hs : Set.encard s + 1 = 1
this : s = ∅
⊢ IsPreprimitive ↥(fixingSubgroup M ∅) ↥(SubMulAction.ofFixingSubgroup M ∅)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.isPreprimitive_iff_is_one_preprimitive | [62, 1] | [86, 57] | exact h | M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
s : Set α
hs : Set.encard s + 1 = 1
this : s = ∅
⊢ IsPreprimitive M α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
s : Set α
hs : Set.encard s + 1 = 1
this : s = ∅
⊢ IsPreprimitive M α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.isPreprimitive_iff_is_one_preprimitive | [62, 1] | [86, 57] | rintro ⟨_, h1'⟩ | case mpr
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
⊢ IsMultiplyPreprimitive M α 1 → IsPreprimitive M α | case mpr.intro
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
left✝ : IsMultiplyPretransitive M α 1
h1' : ∀ (s : Set α), Set.encard s + 1 = ↑1 → IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s)
⊢ IsPreprimitive M α | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
⊢ IsMultiplyPreprimitive M α 1 → IsPreprimitive M α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.isPreprimitive_iff_is_one_preprimitive | [62, 1] | [86, 57] | apply isPreprimitive_of_surjective_map
(Function.Bijective.surjective
(SubMulAction.ofFixingSubgroupEmpty_equivariantMap_bijective M α)) | case mpr.intro
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
left✝ : IsMultiplyPretransitive M α 1
h1' : ∀ (s : Set α), Set.encard s + 1 = ↑1 → IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s)
⊢ IsPreprimitive M α | case mpr.intro
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
left✝ : IsMultiplyPretransitive M α 1
h1' : ∀ (s : Set α), Set.encard s + 1 = ↑1 → IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s)
⊢ IsPreprimitive ↥(fixingSubgroup M ∅) ↥(SubMulAction.ofFixingSubgroup M ∅) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
left✝ : IsMultiplyPretransitive M α 1
h1' : ∀ (s : Set α), Set.encard s + 1 = ↑1 → IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s)
⊢ IsPreprimitive M α
T... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.isPreprimitive_iff_is_one_preprimitive | [62, 1] | [86, 57] | apply h1' | case mpr.intro
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
left✝ : IsMultiplyPretransitive M α 1
h1' : ∀ (s : Set α), Set.encard s + 1 = ↑1 → IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s)
⊢ IsPreprimitive ↥(fixingSubgroup M ∅) ↥(SubMulAction.ofFixingSubgroup M ∅) | case mpr.intro.x
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
left✝ : IsMultiplyPretransitive M α 1
h1' : ∀ (s : Set α), Set.encard s + 1 = ↑1 → IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s)
⊢ Set.encard ∅ + 1 = ↑1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
left✝ : IsMultiplyPretransitive M α 1
h1' : ∀ (s : Set α), Set.encard s + 1 = ↑1 → IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s)
⊢ IsPreprimitive ↥(fix... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.isPreprimitive_iff_is_one_preprimitive | [62, 1] | [86, 57] | simp only [Set.encard_empty, zero_add, Nat.cast_one] | case mpr.intro.x
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
left✝ : IsMultiplyPretransitive M α 1
h1' : ∀ (s : Set α), Set.encard s + 1 = ↑1 → IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s)
⊢ Set.encard ∅ + 1 = ↑1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.x
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
left✝ : IsMultiplyPretransitive M α 1
h1' : ∀ (s : Set α), Set.encard s + 1 = ↑1 → IsPreprimitive ↥(fixingSubgroup M s) ↥(SubMulAction.ofFixingSubgroup M s)
⊢ Set.encard ∅ + 1 =... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | let h_eq := h.exists_smul_eq | M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
⊢ IsMultiplyPreprimitive M α (Nat.succ n) ↔
IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n | M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
⊢ IsMultiplyPreprimitive M α (Nat.succ n) ↔
IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
⊢ IsMultiplyPreprimitive M α (Nat.succ n) ↔
IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | constructor | M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
⊢ IsMultiplyPreprimitive M α (Nat.succ n) ↔
IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n | case mp
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
⊢ IsMultiplyPreprimitive M α (Nat.succ n) →
IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
ca... | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
⊢ IsMultiplyPreprimitive M α (Nat.succ n) ↔
IsMultiplyPreprimitive (↥(stab... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | intro hn | case mp
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
⊢ IsMultiplyPreprimitive M α (Nat.succ n) →
IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n | case mp
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
⊢ IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
⊢ IsMultiplyPreprimitive M α (Nat.succ n) →
IsMultiplyPreprimitive... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | cases' Nat.lt_or_ge n 1 with h0 h1 | case mp
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
⊢ IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n | case mp.inl
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h0 : n < 1
⊢ IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabili... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
⊢ IsMultiplyPreprimitive... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | constructor | case mp.inr
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
⊢ IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabili... | case mp.inr.left
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
⊢ IsMultiplyPretransitive (↥(stabilizer M a)) (↥(SubMulAction.ofS... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
⊢ IsMulti... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | exact (stabilizer.isMultiplyPretransitive M α h).mp hn.left | case mp.inr.left
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
⊢ IsMultiplyPretransitive (↥(stabilizer M a)) (↥(SubMulAction.ofS... | case mp.inr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
⊢ ∀ (s : Set ↥(SubMulAction.ofStabilizer M a)),
Set.encard s... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr.left
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
⊢ Is... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | intro s hs | case mp.inr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
⊢ ∀ (s : Set ↥(SubMulAction.ofStabilizer M a)),
Set.encard s... | case mp.inr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s : Set ↥(SubMulAction.ofStabilizer M a)
hs : Set.encard s + 1 =... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
⊢ ∀... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | apply isPreprimitive_of_surjective_map
(SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_bijective M a s).surjective | case mp.inr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s : Set ↥(SubMulAction.ofStabilizer M a)
hs : Set.encard s + 1 =... | case mp.inr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s : Set ↥(SubMulAction.ofStabilizer M a)
hs : Set.encard s + 1 =... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s :... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | apply hn.right | case mp.inr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s : Set ↥(SubMulAction.ofStabilizer M a)
hs : Set.encard s + 1 =... | case mp.inr.right.x
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s : Set ↥(SubMulAction.ofStabilizer M a)
hs : Set.encard s + 1... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s :... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | rw [Set.encard_insert_of_not_mem, Subtype.coe_injective.encard_image, hs, Nat.cast_succ] | case mp.inr.right.x
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s : Set ↥(SubMulAction.ofStabilizer M a)
hs : Set.encard s + 1... | case mp.inr.right.x
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s : Set ↥(SubMulAction.ofStabilizer M a)
hs : Set.encard s + 1... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr.right.x
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | rw [Nat.lt_one_iff] at h0 | case mp.inl
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h0 : n < 1
⊢ IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabili... | case mp.inl
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h0 : n = 0
⊢ IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabili... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inl
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h0 : n < 1
⊢ IsMulti... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | rw [h0] | case mp.inl
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h0 : n = 0
⊢ IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabili... | case mp.inl
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h0 : n = 0
⊢ IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabili... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inl
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h0 : n = 0
⊢ IsMulti... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | apply isMultiplyPreprimitive_zero | case mp.inl
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h0 : n = 0
⊢ IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabili... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inl
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h0 : n = 0
⊢ IsMulti... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | intro ha | case mp.inr.right.x
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s : Set ↥(SubMulAction.ofStabilizer M a)
hs : Set.encard s + 1... | case mp.inr.right.x
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s : Set ↥(SubMulAction.ofStabilizer M a)
hs : Set.encard s + 1... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr.right.x
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | simp only [Set.mem_image, Subtype.exists, exists_and_right, exists_eq_right] at ha | case mp.inr.right.x
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s : Set ↥(SubMulAction.ofStabilizer M a)
hs : Set.encard s + 1... | case mp.inr.right.x
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s : Set ↥(SubMulAction.ofStabilizer M a)
hs : Set.encard s + 1... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr.right.x
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | obtain ⟨b, _⟩ := ha | case mp.inr.right.x
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s : Set ↥(SubMulAction.ofStabilizer M a)
hs : Set.encard s + 1... | case mp.inr.right.x.intro
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s : Set ↥(SubMulAction.ofStabilizer M a)
hs : Set.encard... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr.right.x
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | apply b | case mp.inr.right.x.intro
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s : Set ↥(SubMulAction.ofStabilizer M a)
hs : Set.encard... | case mp.inr.right.x.intro
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s : Set ↥(SubMulAction.ofStabilizer M a)
hs : Set.encard... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr.right.x.intro
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | simp only [Set.mem_singleton_iff] | case mp.inr.right.x.intro
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n ≥ 1
s : Set ↥(SubMulAction.ofStabilizer M a)
hs : Set.encard... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr.right.x.intro
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn✝ : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn : IsMultiplyPreprimitive M α (Nat.succ n)
h1 : n... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | intro hn_0 | case mpr
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
⊢ IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n →
IsMultiplyPreprimitive M α (Nat.succ n) | case mpr
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
⊢ IsMultiplyPreprimitive M α (Nat.succ n) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
⊢ IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabil... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | constructor | case mpr
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
⊢ IsMultiplyPreprimitive M α (Nat.succ n) | case mpr.left
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
⊢ IsMultiplyPretransitive M α (Nat.suc... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofS... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | rw [stabilizer.isMultiplyPretransitive M α h] | case mpr.left
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
⊢ IsMultiplyPretransitive M α (Nat.suc... | case mpr.left
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
⊢ IsMultiplyPretransitive (↥(stabilize... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.left
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulActio... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | exact hn_0.left | case mpr.left
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
⊢ IsMultiplyPretransitive (↥(stabilize... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.left
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulActio... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | suffices
∀ (s : Set α) (_ : s.encard + 1 = n.succ) (_ : a ∈ s),
IsPreprimitive (fixingSubgroup M s) (SubMulAction.ofFixingSubgroup M s) by
intro s hs
have : ∃ b : α, b ∈ s := by
rw [← Set.nonempty_def, Set.nonempty_iff_ne_empty]
intro h
apply not_lt.mpr hn
rw [h, Set.encard_empty, zero_add, ← ... | case mpr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
⊢ ∀ (s : Set α),
Set.encard s + 1... | case mpr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
⊢ ∀ (s : Set α),
Set.encard s + 1... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulActi... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | intro s hs has | case mpr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
⊢ ∀ (s : Set α),
Set.encard s + 1... | case mpr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
s : Set α
hs : Set.encard s + 1 = ↑(N... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulActi... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | rw [WithTop.add_one_eq_coe_succ_iff] at hs | case mpr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
s : Set α
hs : Set.encard s + 1 = ↑(N... | case mpr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
s : Set α
hs : Set.encard s = ↑n
has ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulActi... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | let t : Set (SubMulAction.ofStabilizer M a) := Subtype.val ⁻¹' s | case mpr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
s : Set α
hs : Set.encard s = ↑n
has ... | case mpr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
s : Set α
hs : Set.encard s = ↑n
has ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulActi... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | rw [hst] | case mpr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
s : Set α
hs : Set.encard s = ↑n
has ... | case mpr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
s : Set α
hs : Set.encard s = ↑n
has ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulActi... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | rw [isPreprimitive_of_bijective_map_iff
(SubMulAction.scalarMap_ofFixingSubgroupOfStabilizer_bijective M a t).surjective
(SubMulAction.equivariantMap_ofFixingSubgroup_to_ofStabilizer_bijective M a t)] | case mpr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
s : Set α
hs : Set.encard s = ↑n
has ... | case mpr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
s : Set α
hs : Set.encard s = ↑n
has ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.right
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulActi... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | intro s hs | M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this :
∀ (s : Set α),
Set.encard s + 1 = ↑(Nat... | M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this :
∀ (s : Set α),
Set.encard s + 1 = ↑(Nat... | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | have : ∃ b : α, b ∈ s := by
rw [← Set.nonempty_def, Set.nonempty_iff_ne_empty]
intro h
apply not_lt.mpr hn
rw [h, Set.encard_empty, zero_add, ← Nat.cast_one, Nat.cast_inj, Nat.succ_inj'] at hs
simp only [← hs, zero_lt_one] | M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this :
∀ (s : Set α),
Set.encard s + 1 = ↑(Nat... | M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this✝ :
∀ (s : Set α),
Set.encard s + 1 = ↑(Na... | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | obtain ⟨b, hb⟩ := this | M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this✝ :
∀ (s : Set α),
Set.encard s + 1 = ↑(Na... | case intro
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this :
∀ (s : Set α),
Set.encard s ... | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | obtain ⟨g, hg : g • b = a⟩ := h_eq b a | case intro
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this :
∀ (s : Set α),
Set.encard s ... | case intro.intro
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this :
∀ (s : Set α),
Set.enc... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.o... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | apply isPreprimitive_of_surjective_map
(SubMulAction.conjMap_ofFixingSubgroup_bijective M (inv_smul_smul g s)).surjective | case intro.intro
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this :
∀ (s : Set α),
Set.enc... | case intro.intro
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this :
∀ (s : Set α),
Set.enc... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAc... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | refine' this (g • s) _ ⟨b, hb, hg⟩ | case intro.intro
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this :
∀ (s : Set α),
Set.enc... | case intro.intro
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this :
∀ (s : Set α),
Set.enc... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAc... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | rw [smul_set_encard_eq, hs] | case intro.intro
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this :
∀ (s : Set α),
Set.enc... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAc... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | rw [← Set.nonempty_def, Set.nonempty_iff_ne_empty] | M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this :
∀ (s : Set α),
Set.encard s + 1 = ↑(Nat... | M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this :
∀ (s : Set α),
Set.encard s + 1 = ↑(Nat... | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | intro h | M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this :
∀ (s : Set α),
Set.encard s + 1 = ↑(Nat... | M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h✝ : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this :
∀ (s : Set α),
Set.encard s + 1 = ↑(Na... | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | apply not_lt.mpr hn | M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h✝ : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this :
∀ (s : Set α),
Set.encard s + 1 = ↑(Na... | M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h✝ : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this :
∀ (s : Set α),
Set.encard s + 1 = ↑(Na... | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h✝ : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilize... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | rw [h, Set.encard_empty, zero_add, ← Nat.cast_one, Nat.cast_inj, Nat.succ_inj'] at hs | M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h✝ : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this :
∀ (s : Set α),
Set.encard s + 1 = ↑(Na... | M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h✝ : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this :
∀ (s : Set α),
Set.encard s + 1 = ↑(Na... | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h✝ : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilize... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | simp only [← hs, zero_lt_one] | M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h✝ : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
this :
∀ (s : Set α),
Set.encard s + 1 = ↑(Na... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h✝ : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilize... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | ext x | M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
s : Set α
hs : Set.encard s = ↑n
has : a ∈ s
t : Set... | case h
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
s : Set α
hs : Set.encard s = ↑n
has : a ∈ s
... | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | constructor | case h
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
s : Set α
hs : Set.encard s = ↑n
has : a ∈ s
... | case h.mp
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
s : Set α
hs : Set.encard s = ↑n
has : a ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofSta... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | intro hxs | case h.mp
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
s : Set α
hs : Set.encard s = ↑n
has : a ∈... | case h.mp
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
s : Set α
hs : Set.encard s = ↑n
has : a ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.of... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/MultiplePrimitivity.lean | MulAction.stabilizer.isMultiplyPreprimitive | [97, 1] | [173, 67] | by_cases hxa : x = a | case h.mp
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
s : Set α
hs : Set.encard s = ↑n
has : a ∈... | case pos
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.ofStabilizer M a)) n
s : Set α
hs : Set.encard s = ↑n
has : a ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
M : Type u_2
α : Type u_1
inst✝¹ : Group M
inst✝ : MulAction M α
n : ℕ
hn : 1 ≤ n
h : IsPretransitive M α
a : α
h_eq : ∀ (x y : α), ∃ g, g • x = y := IsPretransitive.exists_smul_eq
hn_0 : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(SubMulAction.of... |
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