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/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smulCmmap_apply | [173, 1] | [177, 73] | rw [smulCmmap, ←ContinuousMultilinearMap.toFun_eq_coe] | n : ℕ
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid A
inst✝³ : Module 𝕜 A
inst✝² : TopologicalSpace A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
⊢ (smulCmmap 𝕜 A B x xs) z = (x fun x => z 0) • xs fun i => z i.succ | n : ℕ
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid A
inst✝³ : Module 𝕜 A
inst✝² : TopologicalSpace A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
⊢ { toFun := smulCmmapFn x xs, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ }.toFun z =
(x fun x => z 0) • xs fun i => z i.succ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid A
inst✝³ : Module 𝕜 A
inst✝² : TopologicalSpace A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
⊢ (smulCmmap 𝕜 A B x xs) z = (x fun x => z 0) • xs fun i => z i.succ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smulCmmap_apply | [173, 1] | [177, 73] | simp only | n : ℕ
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid A
inst✝³ : Module 𝕜 A
inst✝² : TopologicalSpace A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
⊢ { toFun := smulCmmapFn x xs, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ }.toFun z =
(x fun x => z 0) • xs fun i => z i.succ | n : ℕ
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid A
inst✝³ : Module 𝕜 A
inst✝² : TopologicalSpace A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
⊢ smulCmmapFn x xs z = (x fun x => z 0) • xs fun i => z i.succ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid A
inst✝³ : Module 𝕜 A
inst✝² : TopologicalSpace A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
⊢ { toFun := smulCmmapFn x xs, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ }.toFun z =
(x fun x => z 0) • xs fun i => z i.succ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smulCmmap_apply | [173, 1] | [177, 73] | rfl | n : ℕ
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid A
inst✝³ : Module 𝕜 A
inst✝² : TopologicalSpace A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
⊢ smulCmmapFn x xs z = (x fun x => z 0) • xs fun i => z i.succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid A
inst✝³ : Module 𝕜 A
inst✝² : TopologicalSpace A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
⊢ smulCmmapFn x xs z = (x fun x => z 0) • xs fun i => z i.succ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smulCmmap_norm | [179, 1] | [198, 81] | apply ContinuousMultilinearMap.opNorm_le_bound | n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
⊢ ‖smulCmmap 𝕜 A B x xs‖ ≤ ‖x‖ * ‖xs‖ | case hMp
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
⊢ 0 ≤ ‖x‖ * ‖xs‖
case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
⊢ ∀ (m : Fin (n + 1) → A), ‖(smulCmmap 𝕜 A B x xs) m‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖m i‖ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
⊢ ‖smulCmmap 𝕜 A B x xs‖ ≤ ‖x‖ * ‖xs‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smulCmmap_norm | [179, 1] | [198, 81] | bound | case hMp
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
⊢ 0 ≤ ‖x‖ * ‖xs‖
case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
⊢ ∀ (m : Fin (n + 1) → A), ‖(smulCmmap 𝕜 A B x xs) m‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖m i‖ | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
⊢ ∀ (m : Fin (n + 1) → A), ‖(smulCmmap 𝕜 A B x xs) m‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖m i‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case hMp
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
⊢ 0 ≤ ‖x‖ * ‖xs‖
case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
⊢ ∀ (m : Fin (n + 1) → A), ‖(smulCmmap 𝕜 A B x xs) m‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖m i‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smulCmmap_norm | [179, 1] | [198, 81] | intro z | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
⊢ ∀ (m : Fin (n + 1) → A), ‖(smulCmmap 𝕜 A B x xs) m‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖m i‖ | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
⊢ ‖(smulCmmap 𝕜 A B x xs) z‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
⊢ ∀ (m : Fin (n + 1) → A), ‖(smulCmmap 𝕜 A B x xs) m‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖m i‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smulCmmap_norm | [179, 1] | [198, 81] | rw [smulCmmap_apply] | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
⊢ ‖(smulCmmap 𝕜 A B x xs) z‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
⊢ ‖(smulCmmap 𝕜 A B x xs) z‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smulCmmap_norm | [179, 1] | [198, 81] | have xb := ContinuousMultilinearMap.le_opNorm x fun _ : Fin 1 ↦ z 0 | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * Finset.univ.prod fun i => ‖z 0‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smulCmmap_norm | [179, 1] | [198, 81] | have xsb := ContinuousMultilinearMap.le_opNorm xs fun i : Fin n ↦ z i.succ | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * Finset.univ.prod fun i => ‖z 0‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * Finset.univ.prod fun i => ‖z 0‖
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * Finset.univ.prod fun i => ‖z 0‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smulCmmap_norm | [179, 1] | [198, 81] | simp only [Finset.univ_unique, Fin.default_eq_zero, Finset.prod_const, Finset.card_singleton,
pow_one] at xb xsb | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * Finset.univ.prod fun i => ‖z 0‖
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * Finset.univ.prod fun i => ‖z 0‖
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smulCmmap_norm | [179, 1] | [198, 81] | have e0 := Fin.prod_cons ‖z 0‖ fun i : Fin n ↦ ‖z i.succ‖ | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 : (Finset.univ.prod fun i => Fin.cons ‖z 0‖ (fun i => ‖z i.succ‖) i) = ‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smulCmmap_norm | [179, 1] | [198, 81] | have e1 : ‖z 0‖ = (fun i : Fin (n + 1) ↦ ‖z i‖) 0 := rfl | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 : (Finset.univ.prod fun i => Fin.cons ‖z 0‖ (fun i => ‖z i.succ‖) i) = ‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 : (Finset.univ.prod fun i => Fin.cons ‖z 0‖ (fun i => ‖z i.succ‖) i) = ‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 : (Finset.univ.prod fun i => Fin.cons ‖z 0‖ (fun i => ‖z i.succ‖) i) = ‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smulCmmap_norm | [179, 1] | [198, 81] | have e2 : (fun i : Fin n ↦ ‖z i.succ‖) = Fin.tail fun i : Fin (n + 1) ↦ ‖z i‖ := rfl | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 : (Finset.univ.prod fun i => Fin.cons ‖z 0‖ (fun i => ‖z i.succ‖) i) = ‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 : (Finset.univ.prod fun i => Fin.cons ‖z 0‖ (fun i => ‖z i.succ‖) i) = ‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
e2 : (fun i => ‖z i.succ‖) = Fin.tail fun i => ‖z i‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 : (Finset.univ.prod fun i => Fin.cons ‖z 0‖ (fun i => ‖z i.succ‖) i) = ‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smulCmmap_norm | [179, 1] | [198, 81] | nth_rw 1 [e1] at e0 | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 : (Finset.univ.prod fun i => Fin.cons ‖z 0‖ (fun i => ‖z i.succ‖) i) = ‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
e2 : (fun i => ‖z i.succ‖) = Fin.tail fun i => ‖z i‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 :
(Finset.univ.prod fun i => Fin.cons ((fun i => ‖z i‖) 0) (fun i => ‖z i.succ‖) i) =
‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
e2 : (fun i => ‖z i.succ‖) = Fin.tail fun i => ‖z i‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 : (Finset.univ.prod fun i => Fin.cons ‖z 0‖ (fun i => ‖z i.succ‖) i) = ‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
e2 : (fun i => ‖z i.succ‖) = Fin.tail fun i => ‖z i‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smulCmmap_norm | [179, 1] | [198, 81] | nth_rw 1 [e2] at e0 | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 :
(Finset.univ.prod fun i => Fin.cons ((fun i => ‖z i‖) 0) (fun i => ‖z i.succ‖) i) =
‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
e2 : (fun i => ‖z i.succ‖) = Fin.tail fun i => ‖z i‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 :
(Finset.univ.prod fun i => Fin.cons ((fun i => ‖z i‖) 0) (Fin.tail fun i => ‖z i‖) i) =
‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
e2 : (fun i => ‖z i.succ‖) = Fin.tail fun i => ‖z i‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 :
(Finset.univ.prod fun i => Fin.cons ((fun i => ‖z i‖) 0) (fun i => ‖z i.succ‖) i) =
‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
e2 : (fun i => ‖z i.succ‖) = Fin.tail fun i => ‖z i‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smulCmmap_norm | [179, 1] | [198, 81] | rw [Fin.cons_self_tail (fun i ↦ ‖z i‖)] at e0 | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 :
(Finset.univ.prod fun i => Fin.cons ((fun i => ‖z i‖) 0) (Fin.tail fun i => ‖z i‖) i) =
‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
e2 : (fun i => ‖z i.succ‖) = Fin.tail fun i => ‖z i‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 : (Finset.univ.prod fun i => (fun i => ‖z i‖) i) = ‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
e2 : (fun i => ‖z i.succ‖) = Fin.tail fun i => ‖z i‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 :
(Finset.univ.prod fun i => Fin.cons ((fun i => ‖z i‖) 0) (Fin.tail fun i => ‖z i‖) i) =
‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
e2 : (fun i => ‖z i.succ‖) = Fin.tail fun i => ‖z i‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smulCmmap_norm | [179, 1] | [198, 81] | calc ‖(x fun _ : Fin 1 ↦ z 0) • xs fun i : Fin n ↦ z i.succ‖
_ ≤ ‖x‖ * ‖z 0‖ * (‖xs‖ * Finset.univ.prod fun i : Fin n ↦ ‖z i.succ‖) := by
rw [norm_smul]; bound
_ = ‖x‖ * ‖xs‖ * (‖z 0‖ * Finset.univ.prod fun i : Fin n ↦ ‖z i.succ‖) := by ring
_ = ‖x‖ * ‖xs‖ * Finset.univ.prod fun i : Fin (n + 1) ↦ ‖z i‖ := by rw [←e0] | case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 : (Finset.univ.prod fun i => (fun i => ‖z i‖) i) = ‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
e2 : (fun i => ‖z i.succ‖) = Fin.tail fun i => ‖z i‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hM
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 : (Finset.univ.prod fun i => (fun i => ‖z i‖) i) = ‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
e2 : (fun i => ‖z i.succ‖) = Fin.tail fun i => ‖z i‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smulCmmap_norm | [179, 1] | [198, 81] | rw [norm_smul] | n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 : (Finset.univ.prod fun i => (fun i => ‖z i‖) i) = ‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
e2 : (fun i => ‖z i.succ‖) = Fin.tail fun i => ‖z i‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖z 0‖ * (‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖) | n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 : (Finset.univ.prod fun i => (fun i => ‖z i‖) i) = ‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
e2 : (fun i => ‖z i.succ‖) = Fin.tail fun i => ‖z i‖
⊢ ‖x fun x => z 0‖ * ‖xs fun i => z i.succ‖ ≤ ‖x‖ * ‖z 0‖ * (‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 : (Finset.univ.prod fun i => (fun i => ‖z i‖) i) = ‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
e2 : (fun i => ‖z i.succ‖) = Fin.tail fun i => ‖z i‖
⊢ ‖(x fun x => z 0) • xs fun i => z i.succ‖ ≤ ‖x‖ * ‖z 0‖ * (‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smulCmmap_norm | [179, 1] | [198, 81] | bound | n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 : (Finset.univ.prod fun i => (fun i => ‖z i‖) i) = ‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
e2 : (fun i => ‖z i.succ‖) = Fin.tail fun i => ‖z i‖
⊢ ‖x fun x => z 0‖ * ‖xs fun i => z i.succ‖ ≤ ‖x‖ * ‖z 0‖ * (‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 : (Finset.univ.prod fun i => (fun i => ‖z i‖) i) = ‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
e2 : (fun i => ‖z i.succ‖) = Fin.tail fun i => ‖z i‖
⊢ ‖x fun x => z 0‖ * ‖xs fun i => z i.succ‖ ≤ ‖x‖ * ‖z 0‖ * (‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smulCmmap_norm | [179, 1] | [198, 81] | ring | n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 : (Finset.univ.prod fun i => (fun i => ‖z i‖) i) = ‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
e2 : (fun i => ‖z i.succ‖) = Fin.tail fun i => ‖z i‖
⊢ ‖x‖ * ‖z 0‖ * (‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖) =
‖x‖ * ‖xs‖ * (‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 : (Finset.univ.prod fun i => (fun i => ‖z i‖) i) = ‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
e2 : (fun i => ‖z i.succ‖) = Fin.tail fun i => ‖z i‖
⊢ ‖x‖ * ‖z 0‖ * (‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖) =
‖x‖ * ‖xs‖ * (‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smulCmmap_norm | [179, 1] | [198, 81] | rw [←e0] | n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 : (Finset.univ.prod fun i => (fun i => ‖z i‖) i) = ‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
e2 : (fun i => ‖z i.succ‖) = Fin.tail fun i => ‖z i‖
⊢ ‖x‖ * ‖xs‖ * (‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖) = ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : NormedSpace 𝕜 B
x : ContinuousMultilinearMap 𝕜 (fun x => A) 𝕜
xs : ContinuousMultilinearMap 𝕜 (fun x => A) B
z : Fin (n + 1) → A
xsb : ‖xs fun i => z i.succ‖ ≤ ‖xs‖ * Finset.univ.prod fun i => ‖z i.succ‖
xb : ‖x fun x => z 0‖ ≤ ‖x‖ * ‖z 0‖
e0 : (Finset.univ.prod fun i => (fun i => ‖z i‖) i) = ‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖
e1 : ‖z 0‖ = (fun i => ‖z i‖) 0
e2 : (fun i => ‖z i.succ‖) = Fin.tail fun i => ‖z i‖
⊢ ‖x‖ * ‖xs‖ * (‖z 0‖ * Finset.univ.prod fun i => ‖z i.succ‖) = ‖x‖ * ‖xs‖ * Finset.univ.prod fun i => ‖z i‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_apply | [208, 1] | [226, 94] | induction' n with n h | n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
n k : ℕ
a b : 𝕜
x : E
⊢ ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x | case zero
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
⊢ ((termCmmap 𝕜 0 k x) fun x => (a, b)) = a ^ min k 0 • b ^ (0 - k) • x
case succ
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
⊢ ((termCmmap 𝕜 (n + 1) k x) fun x => (a, b)) = a ^ min k (n + 1) • b ^ (n + 1 - k) • x | Please generate a tactic in lean4 to solve the state.
STATE:
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
n k : ℕ
a b : 𝕜
x : E
⊢ ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_apply | [208, 1] | [226, 94] | simp only [termCmmap, ContinuousMultilinearMap.constOfIsEmpty_apply, min_zero, pow_zero,
zero_tsub, one_smul, Nat.zero_eq] | case zero
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
⊢ ((termCmmap 𝕜 0 k x) fun x => (a, b)) = a ^ min k 0 • b ^ (0 - k) • x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
n : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
⊢ ((termCmmap 𝕜 0 k x) fun x => (a, b)) = a ^ min k 0 • b ^ (0 - k) • x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_apply | [208, 1] | [226, 94] | rw [termCmmap, smulCmmap_apply, h] | case succ
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
⊢ ((termCmmap 𝕜 (n + 1) k x) fun x => (a, b)) = a ^ min k (n + 1) • b ^ (n + 1 - k) • x | case succ
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
⊢ ((if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) fun x => (a, b)) • a ^ min k n • b ^ (n - k) • x =
a ^ min k (n + 1) • b ^ (n + 1 - k) • x | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
⊢ ((termCmmap 𝕜 (n + 1) k x) fun x => (a, b)) = a ^ min k (n + 1) • b ^ (n + 1 - k) • x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_apply | [208, 1] | [226, 94] | by_cases nk : n < k | case succ
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
⊢ ((if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) fun x => (a, b)) • a ^ min k n • b ^ (n - k) • x =
a ^ min k (n + 1) • b ^ (n + 1 - k) • x | case pos
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : n < k
⊢ ((if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) fun x => (a, b)) • a ^ min k n • b ^ (n - k) • x =
a ^ min k (n + 1) • b ^ (n + 1 - k) • x
case neg
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : ¬n < k
⊢ ((if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) fun x => (a, b)) • a ^ min k n • b ^ (n - k) • x =
a ^ min k (n + 1) • b ^ (n + 1 - k) • x | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
⊢ ((if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) fun x => (a, b)) • a ^ min k n • b ^ (n - k) • x =
a ^ min k (n + 1) • b ^ (n + 1 - k) • x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_apply | [208, 1] | [226, 94] | simp [nk] | case pos
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : n < k
⊢ ((if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) fun x => (a, b)) • a ^ min k n • b ^ (n - k) • x =
a ^ min k (n + 1) • b ^ (n + 1 - k) • x | case pos
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : n < k
⊢ ((fstCmmap 𝕜 𝕜 𝕜) fun x => (a, b)) • a ^ min k n • b ^ (n - k) • x = a ^ min k (n + 1) • b ^ (n + 1 - k) • x | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : n < k
⊢ ((if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) fun x => (a, b)) • a ^ min k n • b ^ (n - k) • x =
a ^ min k (n + 1) • b ^ (n + 1 - k) • x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_apply | [208, 1] | [226, 94] | rw [fstCmmap_apply] | case pos
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : n < k
⊢ ((fstCmmap 𝕜 𝕜 𝕜) fun x => (a, b)) • a ^ min k n • b ^ (n - k) • x = a ^ min k (n + 1) • b ^ (n + 1 - k) • x | case pos
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : n < k
⊢ a • a ^ min k n • b ^ (n - k) • x = a ^ min k (n + 1) • b ^ (n + 1 - k) • x | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : n < k
⊢ ((fstCmmap 𝕜 𝕜 𝕜) fun x => (a, b)) • a ^ min k n • b ^ (n - k) • x = a ^ min k (n + 1) • b ^ (n + 1 - k) • x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_apply | [208, 1] | [226, 94] | have nsk : n.succ ≤ k := Nat.succ_le_iff.mpr nk | case pos
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : n < k
⊢ a • a ^ min k n • b ^ (n - k) • x = a ^ min k (n + 1) • b ^ (n + 1 - k) • x | case pos
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : n < k
nsk : n.succ ≤ k
⊢ a • a ^ min k n • b ^ (n - k) • x = a ^ min k (n + 1) • b ^ (n + 1 - k) • x | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : n < k
⊢ a • a ^ min k n • b ^ (n - k) • x = a ^ min k (n + 1) • b ^ (n + 1 - k) • x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_apply | [208, 1] | [226, 94] | rw [min_eq_right nk.le, min_eq_right nsk, Nat.sub_eq_zero_of_le nk.le,
Nat.sub_eq_zero_of_le nsk] | case pos
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : n < k
nsk : n.succ ≤ k
⊢ a • a ^ min k n • b ^ (n - k) • x = a ^ min k (n + 1) • b ^ (n + 1 - k) • x | case pos
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : n < k
nsk : n.succ ≤ k
⊢ a • a ^ n • b ^ 0 • x = a ^ n.succ • b ^ 0 • x | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : n < k
nsk : n.succ ≤ k
⊢ a • a ^ min k n • b ^ (n - k) • x = a ^ min k (n + 1) • b ^ (n + 1 - k) • x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_apply | [208, 1] | [226, 94] | simp only [pow_zero, one_smul, ← smul_assoc, smul_eq_mul, Nat.succ_eq_add_one, pow_succ'] | case pos
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : n < k
nsk : n.succ ≤ k
⊢ a • a ^ n • b ^ 0 • x = a ^ n.succ • b ^ 0 • x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : n < k
nsk : n.succ ≤ k
⊢ a • a ^ n • b ^ 0 • x = a ^ n.succ • b ^ 0 • x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_apply | [208, 1] | [226, 94] | simp [nk] | case neg
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : ¬n < k
⊢ ((if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) fun x => (a, b)) • a ^ min k n • b ^ (n - k) • x =
a ^ min k (n + 1) • b ^ (n + 1 - k) • x | case neg
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : ¬n < k
⊢ ((sndCmmap 𝕜 𝕜 𝕜) fun x => (a, b)) • a ^ min k n • b ^ (n - k) • x = a ^ min k (n + 1) • b ^ (n + 1 - k) • x | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : ¬n < k
⊢ ((if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) fun x => (a, b)) • a ^ min k n • b ^ (n - k) • x =
a ^ min k (n + 1) • b ^ (n + 1 - k) • x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_apply | [208, 1] | [226, 94] | simp at nk | case neg
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : ¬n < k
⊢ ((sndCmmap 𝕜 𝕜 𝕜) fun x => (a, b)) • a ^ min k n • b ^ (n - k) • x = a ^ min k (n + 1) • b ^ (n + 1 - k) • x | case neg
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : k ≤ n
⊢ ((sndCmmap 𝕜 𝕜 𝕜) fun x => (a, b)) • a ^ min k n • b ^ (n - k) • x = a ^ min k (n + 1) • b ^ (n + 1 - k) • x | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : ¬n < k
⊢ ((sndCmmap 𝕜 𝕜 𝕜) fun x => (a, b)) • a ^ min k n • b ^ (n - k) • x = a ^ min k (n + 1) • b ^ (n + 1 - k) • x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_apply | [208, 1] | [226, 94] | rw [sndCmmap_apply] | case neg
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : k ≤ n
⊢ ((sndCmmap 𝕜 𝕜 𝕜) fun x => (a, b)) • a ^ min k n • b ^ (n - k) • x = a ^ min k (n + 1) • b ^ (n + 1 - k) • x | case neg
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : k ≤ n
⊢ b • a ^ min k n • b ^ (n - k) • x = a ^ min k (n + 1) • b ^ (n + 1 - k) • x | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : k ≤ n
⊢ ((sndCmmap 𝕜 𝕜 𝕜) fun x => (a, b)) • a ^ min k n • b ^ (n - k) • x = a ^ min k (n + 1) • b ^ (n + 1 - k) • x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_apply | [208, 1] | [226, 94] | have nsk : k ≤ n.succ := Nat.le_succ_of_le nk | case neg
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : k ≤ n
⊢ b • a ^ min k n • b ^ (n - k) • x = a ^ min k (n + 1) • b ^ (n + 1 - k) • x | case neg
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : k ≤ n
nsk : k ≤ n.succ
⊢ b • a ^ min k n • b ^ (n - k) • x = a ^ min k (n + 1) • b ^ (n + 1 - k) • x | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : k ≤ n
⊢ b • a ^ min k n • b ^ (n - k) • x = a ^ min k (n + 1) • b ^ (n + 1 - k) • x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_apply | [208, 1] | [226, 94] | rw [min_eq_left nk, min_eq_left nsk] | case neg
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : k ≤ n
nsk : k ≤ n.succ
⊢ b • a ^ min k n • b ^ (n - k) • x = a ^ min k (n + 1) • b ^ (n + 1 - k) • x | case neg
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : k ≤ n
nsk : k ≤ n.succ
⊢ b • a ^ k • b ^ (n - k) • x = a ^ k • b ^ (n + 1 - k) • x | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : k ≤ n
nsk : k ≤ n.succ
⊢ b • a ^ min k n • b ^ (n - k) • x = a ^ min k (n + 1) • b ^ (n + 1 - k) • x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_apply | [208, 1] | [226, 94] | rw [smul_comm b _, ← smul_assoc b _ _, smul_eq_mul, ← pow_succ', ← Nat.sub_add_comm nk] | case neg
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : k ≤ n
nsk : k ≤ n.succ
⊢ b • a ^ k • b ^ (n - k) • x = a ^ k • b ^ (n + 1 - k) • x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n✝ : ℕ
𝕜 : Type
inst✝⁵ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁴ : Semiring R
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SMulCommClass 𝕜 𝕜 E
inst✝ : IsScalarTower 𝕜 𝕜 E
k : ℕ
a b : 𝕜
x : E
n : ℕ
h : ((termCmmap 𝕜 n k x) fun x => (a, b)) = a ^ min k n • b ^ (n - k) • x
nk : k ≤ n
nsk : k ≤ n.succ
⊢ b • a ^ k • b ^ (n - k) • x = a ^ k • b ^ (n + 1 - k) • x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_norm | [228, 1] | [237, 88] | induction' n with n nh | n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
n k : ℕ
x : E
⊢ ‖termCmmap 𝕜 n k x‖ ≤ ‖x‖ | case zero
n : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
⊢ ‖termCmmap 𝕜 0 k x‖ ≤ ‖x‖
case succ
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
nh : ‖termCmmap 𝕜 n k x‖ ≤ ‖x‖
⊢ ‖termCmmap 𝕜 (n + 1) k x‖ ≤ ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
n k : ℕ
x : E
⊢ ‖termCmmap 𝕜 n k x‖ ≤ ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_norm | [228, 1] | [237, 88] | simp only [termCmmap, le_refl, ContinuousMultilinearMap.norm_constOfIsEmpty] | case zero
n : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
⊢ ‖termCmmap 𝕜 0 k x‖ ≤ ‖x‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
n : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
⊢ ‖termCmmap 𝕜 0 k x‖ ≤ ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_norm | [228, 1] | [237, 88] | rw [termCmmap] | case succ
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
nh : ‖termCmmap 𝕜 n k x‖ ≤ ‖x‖
⊢ ‖termCmmap 𝕜 (n + 1) k x‖ ≤ ‖x‖ | case succ
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
nh : ‖termCmmap 𝕜 n k x‖ ≤ ‖x‖
⊢ ‖(fun k x => smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) (termCmmap 𝕜 n k x)) k x‖ ≤ ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
nh : ‖termCmmap 𝕜 n k x‖ ≤ ‖x‖
⊢ ‖termCmmap 𝕜 (n + 1) k x‖ ≤ ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_norm | [228, 1] | [237, 88] | simp only | case succ
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
nh : ‖termCmmap 𝕜 n k x‖ ≤ ‖x‖
⊢ ‖(fun k x => smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) (termCmmap 𝕜 n k x)) k x‖ ≤ ‖x‖ | case succ
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
nh : ‖termCmmap 𝕜 n k x‖ ≤ ‖x‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) (termCmmap 𝕜 n k x)‖ ≤ ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
nh : ‖termCmmap 𝕜 n k x‖ ≤ ‖x‖
⊢ ‖(fun k x => smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) (termCmmap 𝕜 n k x)) k x‖ ≤ ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_norm | [228, 1] | [237, 88] | generalize ht : termCmmap 𝕜 n k x = t | case succ
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
nh : ‖termCmmap 𝕜 n k x‖ ≤ ‖x‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) (termCmmap 𝕜 n k x)‖ ≤ ‖x‖ | case succ
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
nh : ‖termCmmap 𝕜 n k x‖ ≤ ‖x‖
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
ht : termCmmap 𝕜 n k x = t
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
nh : ‖termCmmap 𝕜 n k x‖ ≤ ‖x‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) (termCmmap 𝕜 n k x)‖ ≤ ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_norm | [228, 1] | [237, 88] | rw [ht] at nh | case succ
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
nh : ‖termCmmap 𝕜 n k x‖ ≤ ‖x‖
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
ht : termCmmap 𝕜 n k x = t
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖ | case succ
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
nh : ‖termCmmap 𝕜 n k x‖ ≤ ‖x‖
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
ht : termCmmap 𝕜 n k x = t
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_norm | [228, 1] | [237, 88] | have tn := smulCmmap_norm (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t | case succ
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖ | case succ
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
tn :
‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤
‖if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜‖ * ‖t‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_norm | [228, 1] | [237, 88] | by_cases nk : n < k | case succ
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
tn :
‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤
‖if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜‖ * ‖t‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖ | case pos
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
tn :
‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤
‖if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜‖ * ‖t‖
nk : n < k
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖
case neg
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
tn :
‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤
‖if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜‖ * ‖t‖
nk : ¬n < k
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
tn :
‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤
‖if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜‖ * ‖t‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_norm | [228, 1] | [237, 88] | simp [nk] at tn ⊢ | case pos
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
tn :
‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤
‖if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜‖ * ‖t‖
nk : n < k
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖ | case pos
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
nk : n < k
tn : ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (fstCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖fstCmmap 𝕜 𝕜 𝕜‖ * ‖t‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (fstCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
tn :
‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤
‖if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜‖ * ‖t‖
nk : n < k
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_norm | [228, 1] | [237, 88] | rw [fstCmmap_norm] at tn | case pos
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
nk : n < k
tn : ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (fstCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖fstCmmap 𝕜 𝕜 𝕜‖ * ‖t‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (fstCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖ | case pos
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
nk : n < k
tn : ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (fstCmmap 𝕜 𝕜 𝕜) t‖ ≤ 1 * ‖t‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (fstCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
nk : n < k
tn : ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (fstCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖fstCmmap 𝕜 𝕜 𝕜‖ * ‖t‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (fstCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_norm | [228, 1] | [237, 88] | simp at tn | case pos
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
nk : n < k
tn : ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (fstCmmap 𝕜 𝕜 𝕜) t‖ ≤ 1 * ‖t‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (fstCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖ | case pos
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
nk : n < k
tn : ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (fstCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖t‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (fstCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
nk : n < k
tn : ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (fstCmmap 𝕜 𝕜 𝕜) t‖ ≤ 1 * ‖t‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (fstCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_norm | [228, 1] | [237, 88] | exact _root_.trans tn nh | case pos
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
nk : n < k
tn : ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (fstCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖t‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (fstCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
nk : n < k
tn : ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (fstCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖t‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (fstCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_norm | [228, 1] | [237, 88] | simp [nk] at tn ⊢ | case neg
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
tn :
‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤
‖if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜‖ * ‖t‖
nk : ¬n < k
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖ | case neg
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
nk : ¬n < k
tn : ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖sndCmmap 𝕜 𝕜 𝕜‖ * ‖t‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
tn :
‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤
‖if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜‖ * ‖t‖
nk : ¬n < k
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (if n < k then fstCmmap 𝕜 𝕜 𝕜 else sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_norm | [228, 1] | [237, 88] | rw [sndCmmap_norm] at tn | case neg
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
nk : ¬n < k
tn : ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖sndCmmap 𝕜 𝕜 𝕜‖ * ‖t‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖ | case neg
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
nk : ¬n < k
tn : ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ 1 * ‖t‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
nk : ¬n < k
tn : ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖sndCmmap 𝕜 𝕜 𝕜‖ * ‖t‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_norm | [228, 1] | [237, 88] | simp at tn | case neg
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
nk : ¬n < k
tn : ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ 1 * ‖t‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖ | case neg
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
nk : ¬n < k
tn : ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖t‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
nk : ¬n < k
tn : ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ 1 * ‖t‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | termCmmap_norm | [228, 1] | [237, 88] | exact _root_.trans tn nh | case neg
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
nk : ¬n < k
tn : ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖t‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n✝ : ℕ
𝕜✝ : Type
inst✝⁴ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝³ : Semiring R
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
k : ℕ
x : E
n : ℕ
t : ContinuousMultilinearMap 𝕜 (fun x => 𝕜 × 𝕜) E
nh : ‖t‖ ≤ ‖x‖
ht : termCmmap 𝕜 n k x = t
nk : ¬n < k
tn : ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖t‖
⊢ ‖smulCmmap 𝕜 (𝕜 × 𝕜) E (sndCmmap 𝕜 𝕜 𝕜) t‖ ≤ ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | ContinuousLinearMap.apply_eq_zero_of_eq_zero | [258, 1] | [262, 39] | rw [h, ContinuousLinearMap.map_zero] | n : ℕ
𝕜✝ : Type
inst✝⁷ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝⁶ : Semiring R
𝕜 X Y : Type
inst✝⁵ : NormedField 𝕜
inst✝⁴ : TopologicalSpace X
inst✝³ : NormedAddCommGroup X
inst✝² : Module 𝕜 X
inst✝¹ : NormedAddCommGroup Y
inst✝ : Module 𝕜 Y
f : X →L[𝕜] Y
x : X
h : x = 0
⊢ f x = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜✝ : Type
inst✝⁷ : NontriviallyNormedField 𝕜✝
R A B E : Type
inst✝⁶ : Semiring R
𝕜 X Y : Type
inst✝⁵ : NormedField 𝕜
inst✝⁴ : TopologicalSpace X
inst✝³ : NormedAddCommGroup X
inst✝² : Module 𝕜 X
inst✝¹ : NormedAddCommGroup Y
inst✝ : Module 𝕜 Y
f : X →L[𝕜] Y
x : X
h : x = 0
⊢ f x = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | ContinuousLinearMap.smulRight_ne_zero | [264, 1] | [273, 8] | rcases ContinuousLinearMap.exists_ne_zero c0 with ⟨x,cx⟩ | n : ℕ
𝕜 : Type
inst✝¹¹ : NontriviallyNormedField 𝕜
R✝ A✝ B✝ E : Type
inst✝¹⁰ : Semiring R✝
R A B : Type
inst✝⁹ : Ring R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : AddCommMonoid A
inst✝⁶ : TopologicalSpace R
inst✝⁵ : Module R A
inst✝⁴ : TopologicalSpace B
inst✝³ : AddCommMonoid B
inst✝² : Module R B
inst✝¹ : ContinuousSMul R B
inst✝ : NoZeroSMulDivisors R B
c : A →L[R] R
f : B
c0 : c ≠ 0
f0 : f ≠ 0
⊢ c.smulRight f ≠ 0 | case intro
n : ℕ
𝕜 : Type
inst✝¹¹ : NontriviallyNormedField 𝕜
R✝ A✝ B✝ E : Type
inst✝¹⁰ : Semiring R✝
R A B : Type
inst✝⁹ : Ring R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : AddCommMonoid A
inst✝⁶ : TopologicalSpace R
inst✝⁵ : Module R A
inst✝⁴ : TopologicalSpace B
inst✝³ : AddCommMonoid B
inst✝² : Module R B
inst✝¹ : ContinuousSMul R B
inst✝ : NoZeroSMulDivisors R B
c : A →L[R] R
f : B
c0 : c ≠ 0
f0 : f ≠ 0
x : A
cx : c x ≠ 0
⊢ c.smulRight f ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝¹¹ : NontriviallyNormedField 𝕜
R✝ A✝ B✝ E : Type
inst✝¹⁰ : Semiring R✝
R A B : Type
inst✝⁹ : Ring R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : AddCommMonoid A
inst✝⁶ : TopologicalSpace R
inst✝⁵ : Module R A
inst✝⁴ : TopologicalSpace B
inst✝³ : AddCommMonoid B
inst✝² : Module R B
inst✝¹ : ContinuousSMul R B
inst✝ : NoZeroSMulDivisors R B
c : A →L[R] R
f : B
c0 : c ≠ 0
f0 : f ≠ 0
⊢ c.smulRight f ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | ContinuousLinearMap.smulRight_ne_zero | [264, 1] | [273, 8] | simp only [Ne, ContinuousLinearMap.ext_iff, not_forall, ContinuousLinearMap.zero_apply,
ContinuousLinearMap.smulRight_apply, smul_eq_zero, not_or] | case intro
n : ℕ
𝕜 : Type
inst✝¹¹ : NontriviallyNormedField 𝕜
R✝ A✝ B✝ E : Type
inst✝¹⁰ : Semiring R✝
R A B : Type
inst✝⁹ : Ring R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : AddCommMonoid A
inst✝⁶ : TopologicalSpace R
inst✝⁵ : Module R A
inst✝⁴ : TopologicalSpace B
inst✝³ : AddCommMonoid B
inst✝² : Module R B
inst✝¹ : ContinuousSMul R B
inst✝ : NoZeroSMulDivisors R B
c : A →L[R] R
f : B
c0 : c ≠ 0
f0 : f ≠ 0
x : A
cx : c x ≠ 0
⊢ c.smulRight f ≠ 0 | case intro
n : ℕ
𝕜 : Type
inst✝¹¹ : NontriviallyNormedField 𝕜
R✝ A✝ B✝ E : Type
inst✝¹⁰ : Semiring R✝
R A B : Type
inst✝⁹ : Ring R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : AddCommMonoid A
inst✝⁶ : TopologicalSpace R
inst✝⁵ : Module R A
inst✝⁴ : TopologicalSpace B
inst✝³ : AddCommMonoid B
inst✝² : Module R B
inst✝¹ : ContinuousSMul R B
inst✝ : NoZeroSMulDivisors R B
c : A →L[R] R
f : B
c0 : c ≠ 0
f0 : f ≠ 0
x : A
cx : c x ≠ 0
⊢ ∃ x, ¬c x = 0 ∧ ¬f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : ℕ
𝕜 : Type
inst✝¹¹ : NontriviallyNormedField 𝕜
R✝ A✝ B✝ E : Type
inst✝¹⁰ : Semiring R✝
R A B : Type
inst✝⁹ : Ring R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : AddCommMonoid A
inst✝⁶ : TopologicalSpace R
inst✝⁵ : Module R A
inst✝⁴ : TopologicalSpace B
inst✝³ : AddCommMonoid B
inst✝² : Module R B
inst✝¹ : ContinuousSMul R B
inst✝ : NoZeroSMulDivisors R B
c : A →L[R] R
f : B
c0 : c ≠ 0
f0 : f ≠ 0
x : A
cx : c x ≠ 0
⊢ c.smulRight f ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | ContinuousLinearMap.smulRight_ne_zero | [264, 1] | [273, 8] | use x | case intro
n : ℕ
𝕜 : Type
inst✝¹¹ : NontriviallyNormedField 𝕜
R✝ A✝ B✝ E : Type
inst✝¹⁰ : Semiring R✝
R A B : Type
inst✝⁹ : Ring R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : AddCommMonoid A
inst✝⁶ : TopologicalSpace R
inst✝⁵ : Module R A
inst✝⁴ : TopologicalSpace B
inst✝³ : AddCommMonoid B
inst✝² : Module R B
inst✝¹ : ContinuousSMul R B
inst✝ : NoZeroSMulDivisors R B
c : A →L[R] R
f : B
c0 : c ≠ 0
f0 : f ≠ 0
x : A
cx : c x ≠ 0
⊢ ∃ x, ¬c x = 0 ∧ ¬f = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : ℕ
𝕜 : Type
inst✝¹¹ : NontriviallyNormedField 𝕜
R✝ A✝ B✝ E : Type
inst✝¹⁰ : Semiring R✝
R A B : Type
inst✝⁹ : Ring R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : AddCommMonoid A
inst✝⁶ : TopologicalSpace R
inst✝⁵ : Module R A
inst✝⁴ : TopologicalSpace B
inst✝³ : AddCommMonoid B
inst✝² : Module R B
inst✝¹ : ContinuousSMul R B
inst✝ : NoZeroSMulDivisors R B
c : A →L[R] R
f : B
c0 : c ≠ 0
f0 : f ≠ 0
x : A
cx : c x ≠ 0
⊢ ∃ x, ¬c x = 0 ∧ ¬f = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | ContinuousLinearMap.one_ne_zero | [275, 1] | [280, 18] | simp only [Ne, ContinuousLinearMap.ext_iff, not_forall, ContinuousLinearMap.zero_apply,
ContinuousLinearMap.one_apply] | n : ℕ
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
R✝ A✝ B E : Type
inst✝⁵ : Semiring R✝
R A : Type
inst✝⁴ : Ring R
inst✝³ : TopologicalSpace A
inst✝² : AddCommMonoid A
inst✝¹ : Module R A
inst✝ : Nontrivial A
⊢ 1 ≠ 0 | n : ℕ
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
R✝ A✝ B E : Type
inst✝⁵ : Semiring R✝
R A : Type
inst✝⁴ : Ring R
inst✝³ : TopologicalSpace A
inst✝² : AddCommMonoid A
inst✝¹ : Module R A
inst✝ : Nontrivial A
⊢ ∃ x, ¬x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
R✝ A✝ B E : Type
inst✝⁵ : Semiring R✝
R A : Type
inst✝⁴ : Ring R
inst✝³ : TopologicalSpace A
inst✝² : AddCommMonoid A
inst✝¹ : Module R A
inst✝ : Nontrivial A
⊢ 1 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | ContinuousLinearMap.one_ne_zero | [275, 1] | [280, 18] | apply exists_ne | n : ℕ
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
R✝ A✝ B E : Type
inst✝⁵ : Semiring R✝
R A : Type
inst✝⁴ : Ring R
inst✝³ : TopologicalSpace A
inst✝² : AddCommMonoid A
inst✝¹ : Module R A
inst✝ : Nontrivial A
⊢ ∃ x, ¬x = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
R✝ A✝ B E : Type
inst✝⁵ : Semiring R✝
R A : Type
inst✝⁴ : Ring R
inst✝³ : TopologicalSpace A
inst✝² : AddCommMonoid A
inst✝¹ : Module R A
inst✝ : Nontrivial A
⊢ ∃ x, ¬x = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | exist_root_of_unity | [36, 1] | [49, 91] | set n : ℕ+ := ⟨d, lt_of_lt_of_le (by norm_num) d2⟩ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
⊢ ∃ a, a ≠ 1 ∧ a ^ d = 1 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
⊢ ∃ a, a ≠ 1 ∧ a ^ d = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
⊢ ∃ a, a ≠ 1 ∧ a ^ d = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | exist_root_of_unity | [36, 1] | [49, 91] | have two : Nontrivial (rootsOfUnity n ℂ) := by
rw [← Fintype.one_lt_card_iff_nontrivial, Complex.card_rootsOfUnity]
simp only [PNat.mk_coe, n]; exact lt_of_lt_of_le (by norm_num) d2 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
⊢ ∃ a, a ≠ 1 ∧ a ^ d = 1 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
two : Nontrivial ↥(rootsOfUnity n ℂ)
⊢ ∃ a, a ≠ 1 ∧ a ^ d = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
⊢ ∃ a, a ≠ 1 ∧ a ^ d = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | exist_root_of_unity | [36, 1] | [49, 91] | rcases two with ⟨⟨a, am⟩, ⟨b, bm⟩, ab⟩ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
two : Nontrivial ↥(rootsOfUnity n ℂ)
⊢ ∃ a, a ≠ 1 ∧ a ^ d = 1 | case mk.intro.mk.intro.mk
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a : ℂˣ
am : a ∈ rootsOfUnity n ℂ
b : ℂˣ
bm : b ∈ rootsOfUnity n ℂ
ab : ⟨a, am⟩ ≠ ⟨b, bm⟩
⊢ ∃ a, a ≠ 1 ∧ a ^ d = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
two : Nontrivial ↥(rootsOfUnity n ℂ)
⊢ ∃ a, a ≠ 1 ∧ a ^ d = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | exist_root_of_unity | [36, 1] | [49, 91] | simp only [Ne, Subtype.mk_eq_mk, mem_rootsOfUnity, PNat.mk_coe] at am bm ab | case mk.intro.mk.intro.mk
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a : ℂˣ
am : a ∈ rootsOfUnity n ℂ
b : ℂˣ
bm : b ∈ rootsOfUnity n ℂ
ab : ⟨a, am⟩ ≠ ⟨b, bm⟩
⊢ ∃ a, a ≠ 1 ∧ a ^ d = 1 | case mk.intro.mk.intro.mk
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬a = b
⊢ ∃ a, a ≠ 1 ∧ a ^ d = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.mk.intro.mk
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a : ℂˣ
am : a ∈ rootsOfUnity n ℂ
b : ℂˣ
bm : b ∈ rootsOfUnity n ℂ
ab : ⟨a, am⟩ ≠ ⟨b, bm⟩
⊢ ∃ a, a ≠ 1 ∧ a ^ d = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | exist_root_of_unity | [36, 1] | [49, 91] | by_cases a1 : a = 1 | case mk.intro.mk.intro.mk
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬a = b
⊢ ∃ a, a ≠ 1 ∧ a ^ d = 1 | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬a = b
a1 : a = 1
⊢ ∃ a, a ≠ 1 ∧ a ^ d = 1
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬a = b
a1 : ¬a = 1
⊢ ∃ a, a ≠ 1 ∧ a ^ d = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.mk.intro.mk
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬a = b
⊢ ∃ a, a ≠ 1 ∧ a ^ d = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | exist_root_of_unity | [36, 1] | [49, 91] | norm_num | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
⊢ 0 < 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
⊢ 0 < 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | exist_root_of_unity | [36, 1] | [49, 91] | rw [← Fintype.one_lt_card_iff_nontrivial, Complex.card_rootsOfUnity] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
⊢ Nontrivial ↥(rootsOfUnity n ℂ) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
⊢ 1 < ↑n | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
⊢ Nontrivial ↥(rootsOfUnity n ℂ)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | exist_root_of_unity | [36, 1] | [49, 91] | simp only [PNat.mk_coe, n] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
⊢ 1 < ↑n | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
⊢ 1 < d | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
⊢ 1 < ↑n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | exist_root_of_unity | [36, 1] | [49, 91] | exact lt_of_lt_of_le (by norm_num) d2 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
⊢ 1 < d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
⊢ 1 < d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | exist_root_of_unity | [36, 1] | [49, 91] | norm_num | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
⊢ 1 < 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
⊢ 1 < 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | exist_root_of_unity | [36, 1] | [49, 91] | use b | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬a = b
a1 : a = 1
⊢ ∃ a, a ≠ 1 ∧ a ^ d = 1 | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬a = b
a1 : a = 1
⊢ ↑b ≠ 1 ∧ ↑b ^ d = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬a = b
a1 : a = 1
⊢ ∃ a, a ≠ 1 ∧ a ^ d = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | exist_root_of_unity | [36, 1] | [49, 91] | rw [a1] at ab | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬a = b
a1 : a = 1
⊢ ↑b ≠ 1 ∧ ↑b ^ d = 1 | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬1 = b
a1 : a = 1
⊢ ↑b ≠ 1 ∧ ↑b ^ d = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬a = b
a1 : a = 1
⊢ ↑b ≠ 1 ∧ ↑b ^ d = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | exist_root_of_unity | [36, 1] | [49, 91] | constructor | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬1 = b
a1 : a = 1
⊢ ↑b ≠ 1 ∧ ↑b ^ d = 1 | case h.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬1 = b
a1 : a = 1
⊢ ↑b ≠ 1
case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬1 = b
a1 : a = 1
⊢ ↑b ^ d = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬1 = b
a1 : a = 1
⊢ ↑b ≠ 1 ∧ ↑b ^ d = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | exist_root_of_unity | [36, 1] | [49, 91] | simp only [ne_eq, Units.val_eq_one, Ne.symm ab, not_false_eq_true] | case h.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬1 = b
a1 : a = 1
⊢ ↑b ≠ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬1 = b
a1 : a = 1
⊢ ↑b ≠ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | exist_root_of_unity | [36, 1] | [49, 91] | simp only [PNat.mk_coe, n] at bm | case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬1 = b
a1 : a = 1
⊢ ↑b ^ d = 1 | case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ d = 1
ab : ¬1 = b
a1 : a = 1
⊢ ↑b ^ d = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬1 = b
a1 : a = 1
⊢ ↑b ^ d = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | exist_root_of_unity | [36, 1] | [49, 91] | rw [← Units.val_pow_eq_pow_val, bm, Units.val_one] | case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ d = 1
ab : ¬1 = b
a1 : a = 1
⊢ ↑b ^ d = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ d = 1
ab : ¬1 = b
a1 : a = 1
⊢ ↑b ^ d = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | exist_root_of_unity | [36, 1] | [49, 91] | use a | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬a = b
a1 : ¬a = 1
⊢ ∃ a, a ≠ 1 ∧ a ^ d = 1 | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬a = b
a1 : ¬a = 1
⊢ ↑a ≠ 1 ∧ ↑a ^ d = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬a = b
a1 : ¬a = 1
⊢ ∃ a, a ≠ 1 ∧ a ^ d = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | exist_root_of_unity | [36, 1] | [49, 91] | constructor | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬a = b
a1 : ¬a = 1
⊢ ↑a ≠ 1 ∧ ↑a ^ d = 1 | case h.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬a = b
a1 : ¬a = 1
⊢ ↑a ≠ 1
case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬a = b
a1 : ¬a = 1
⊢ ↑a ^ d = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬a = b
a1 : ¬a = 1
⊢ ↑a ≠ 1 ∧ ↑a ^ d = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | exist_root_of_unity | [36, 1] | [49, 91] | simp only [ne_eq, Units.val_eq_one, a1, not_false_eq_true] | case h.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬a = b
a1 : ¬a = 1
⊢ ↑a ≠ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬a = b
a1 : ¬a = 1
⊢ ↑a ≠ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | exist_root_of_unity | [36, 1] | [49, 91] | simp only [PNat.mk_coe, n] at am | case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬a = b
a1 : ¬a = 1
⊢ ↑a ^ d = 1 | case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ d = 1
bm : b ^ ↑n = 1
ab : ¬a = b
a1 : ¬a = 1
⊢ ↑a ^ d = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ ↑n = 1
bm : b ^ ↑n = 1
ab : ¬a = b
a1 : ¬a = 1
⊢ ↑a ^ d = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | exist_root_of_unity | [36, 1] | [49, 91] | rw [← Units.val_pow_eq_pow_val, am, Units.val_one] | case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ d = 1
bm : b ^ ↑n = 1
ab : ¬a = b
a1 : ¬a = 1
⊢ ↑a ^ d = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
d : ℕ
d2 : 2 ≤ d
n : ℕ+ := ⟨d, ⋯⟩
a b : ℂˣ
am : a ^ d = 1
bm : b ^ ↑n = 1
ab : ¬a = b
a1 : ¬a = 1
⊢ ↑a ^ d = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | rcases s.superNear with ⟨t, s⟩ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s : SuperAt f d
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z | case intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s : SuperAt f d
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | have ba : AnalyticAt ℂ (bottcherNear f d) 0 := bottcherNear_analytic_z s _ s.t0 | case intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z | case intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | have nc : mfderiv I I (bottcherNear f d) 0 ≠ 0 := by
rw [mfderiv_eq_fderiv, ← deriv_fderiv, (bottcherNear_monic s).deriv]
exact ContinuousLinearMap.smulRight_ne_zero ContinuousLinearMap.one_ne_zero (by norm_num) | case intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z | case intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | rcases complex_inverse_fun' (ba.holomorphicAt I I) nc with ⟨i, ia, ib, bi⟩ | case intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z | case intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i (bottcherNear f d 0)
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 (bottcherNear f d 0), bottcherNear f d (i x) = x
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | rw [bottcherNear_zero] at bi ia | case intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i (bottcherNear f d 0)
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 (bottcherNear f d 0), bottcherNear f d (i x) = x
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z | case intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i (bottcherNear f d 0)
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 (bottcherNear f d 0), bottcherNear f d (i x) = x
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | have i0 : i 0 = 0 := by nth_rw 1 [← bottcherNear_zero]; rw [ib.self_of_nhds] | case intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z | case intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | have inj : ∀ᶠ p : ℂ × ℂ in 𝓝 (0, 0), i p.1 = i p.2 → p.1 = p.2 := by
refine ia.local_inj ?_
have d0 : mfderiv I I (fun z : ℂ ↦ z) 0 ≠ 0 := id_mderiv_ne_zero
rw [(Filter.EventuallyEq.symm ib).mfderiv_eq] at d0
rw [←Function.comp_def, mfderiv_comp 0 _ ba.differentiableAt.mdifferentiableAt] at d0
simp only [Ne, mderiv_comp_eq_zero_iff, nc, or_false_iff] at d0
rw [bottcherNear_zero] at d0; exact d0
rw [bottcherNear_zero]; exact ia.mdifferentiableAt | case intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z | case intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : ∀ᶠ (p : ℂ × ℂ) in 𝓝 (0, 0), i p.1 = i p.2 → p.1 = p.2
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | rcases exist_root_of_unity s.d2 with ⟨a, a1, ad⟩ | case intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : ∀ᶠ (p : ℂ × ℂ) in 𝓝 (0, 0), i p.1 = i p.2 → p.1 = p.2
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z | case intro.intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : ∀ᶠ (p : ℂ × ℂ) in 𝓝 (0, 0), i p.1 = i p.2 → p.1 = p.2
a : ℂ
a1 : a ≠ 1
ad : a ^ d = 1
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : ∀ᶠ (p : ℂ × ℂ) in 𝓝 (0, 0), i p.1 = i p.2 → p.1 = p.2
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | refine ⟨fun z ↦ i (a * bottcherNear f d z), ?_, ?_, ?_⟩ | case intro.intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : ∀ᶠ (p : ℂ × ℂ) in 𝓝 (0, 0), i p.1 = i p.2 → p.1 = p.2
a : ℂ
a1 : a ≠ 1
ad : a ^ d = 1
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z | case intro.intro.intro.intro.intro.intro.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : ∀ᶠ (p : ℂ × ℂ) in 𝓝 (0, 0), i p.1 = i p.2 → p.1 = p.2
a : ℂ
a1 : a ≠ 1
ad : a ^ d = 1
⊢ AnalyticAt ℂ (fun z => i (a * bottcherNear f d z)) 0
case intro.intro.intro.intro.intro.intro.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : ∀ᶠ (p : ℂ × ℂ) in 𝓝 (0, 0), i p.1 = i p.2 → p.1 = p.2
a : ℂ
a1 : a ≠ 1
ad : a ^ d = 1
⊢ (fun z => i (a * bottcherNear f d z)) 0 = 0
case intro.intro.intro.intro.intro.intro.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : ∀ᶠ (p : ℂ × ℂ) in 𝓝 (0, 0), i p.1 = i p.2 → p.1 = p.2
a : ℂ
a1 : a ≠ 1
ad : a ^ d = 1
⊢ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, (fun z => i (a * bottcherNear f d z)) z ≠ z ∧ f ((fun z => i (a * bottcherNear f d z)) z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : ∀ᶠ (p : ℂ × ℂ) in 𝓝 (0, 0), i p.1 = i p.2 → p.1 = p.2
a : ℂ
a1 : a ≠ 1
ad : a ^ d = 1
⊢ ∃ g, AnalyticAt ℂ g 0 ∧ g 0 = 0 ∧ ∀ᶠ (z : ℂ) in 𝓝[≠] 0, g z ≠ z ∧ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | rw [mfderiv_eq_fderiv, ← deriv_fderiv, (bottcherNear_monic s).deriv] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
⊢ mfderiv I I (bottcherNear f d) 0 ≠ 0 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
⊢ ContinuousLinearMap.smulRight 1 1 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
⊢ mfderiv I I (bottcherNear f d) 0 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | exact ContinuousLinearMap.smulRight_ne_zero ContinuousLinearMap.one_ne_zero (by norm_num) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
⊢ ContinuousLinearMap.smulRight 1 1 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
⊢ ContinuousLinearMap.smulRight 1 1 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | norm_num | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
⊢ 1 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
⊢ 1 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | nth_rw 1 [← bottcherNear_zero] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
⊢ i 0 = 0 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
⊢ i (bottcherNear ?m.13266 ?m.13267 0) = 0
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
⊢ ℂ → ℂ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
⊢ ℕ | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
⊢ i 0 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | rw [ib.self_of_nhds] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
⊢ i (bottcherNear ?m.13266 ?m.13267 0) = 0
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
⊢ ℂ → ℂ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
⊢ ℕ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
⊢ i (bottcherNear ?m.13266 ?m.13267 0) = 0
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
⊢ ℂ → ℂ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
⊢ ℕ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | refine ia.local_inj ?_ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
⊢ ∀ᶠ (p : ℂ × ℂ) in 𝓝 (0, 0), i p.1 = i p.2 → p.1 = p.2 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
⊢ mfderiv I I i 0 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
⊢ ∀ᶠ (p : ℂ × ℂ) in 𝓝 (0, 0), i p.1 = i p.2 → p.1 = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | have d0 : mfderiv I I (fun z : ℂ ↦ z) 0 ≠ 0 := id_mderiv_ne_zero | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
⊢ mfderiv I I i 0 ≠ 0 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : mfderiv I I (fun z => z) 0 ≠ 0
⊢ mfderiv I I i 0 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
⊢ mfderiv I I i 0 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | rw [(Filter.EventuallyEq.symm ib).mfderiv_eq] at d0 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : mfderiv I I (fun z => z) 0 ≠ 0
⊢ mfderiv I I i 0 ≠ 0 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : mfderiv I I (fun x => i (bottcherNear f d x)) 0 ≠ 0
⊢ mfderiv I I i 0 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : mfderiv I I (fun z => z) 0 ≠ 0
⊢ mfderiv I I i 0 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | rw [←Function.comp_def, mfderiv_comp 0 _ ba.differentiableAt.mdifferentiableAt] at d0 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : mfderiv I I (fun x => i (bottcherNear f d x)) 0 ≠ 0
⊢ mfderiv I I i 0 ≠ 0 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : (mfderiv I I i (bottcherNear f d 0)).comp (mfderiv I I (bottcherNear f d) 0) ≠ 0
⊢ mfderiv I I i 0 ≠ 0
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : mfderiv I I (i ∘ fun x => bottcherNear f d x) 0 ≠ 0
⊢ MDifferentiableAt I I i (bottcherNear f d 0) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : mfderiv I I (fun x => i (bottcherNear f d x)) 0 ≠ 0
⊢ mfderiv I I i 0 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | simp only [Ne, mderiv_comp_eq_zero_iff, nc, or_false_iff] at d0 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : (mfderiv I I i (bottcherNear f d 0)).comp (mfderiv I I (bottcherNear f d) 0) ≠ 0
⊢ mfderiv I I i 0 ≠ 0
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : mfderiv I I (i ∘ fun x => bottcherNear f d x) 0 ≠ 0
⊢ MDifferentiableAt I I i (bottcherNear f d 0) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : ¬mfderiv I I i (bottcherNear f d 0) = 0
⊢ mfderiv I I i 0 ≠ 0
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : mfderiv I I (i ∘ fun x => bottcherNear f d x) 0 ≠ 0
⊢ MDifferentiableAt I I i (bottcherNear f d 0) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : (mfderiv I I i (bottcherNear f d 0)).comp (mfderiv I I (bottcherNear f d) 0) ≠ 0
⊢ mfderiv I I i 0 ≠ 0
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : mfderiv I I (i ∘ fun x => bottcherNear f d x) 0 ≠ 0
⊢ MDifferentiableAt I I i (bottcherNear f d 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | rw [bottcherNear_zero] at d0 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : ¬mfderiv I I i (bottcherNear f d 0) = 0
⊢ mfderiv I I i 0 ≠ 0
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : mfderiv I I (i ∘ fun x => bottcherNear f d x) 0 ≠ 0
⊢ MDifferentiableAt I I i (bottcherNear f d 0) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : ¬mfderiv I I i 0 = 0
⊢ mfderiv I I i 0 ≠ 0
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : mfderiv I I (i ∘ fun x => bottcherNear f d x) 0 ≠ 0
⊢ MDifferentiableAt I I i (bottcherNear f d 0) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : ¬mfderiv I I i (bottcherNear f d 0) = 0
⊢ mfderiv I I i 0 ≠ 0
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : mfderiv I I (i ∘ fun x => bottcherNear f d x) 0 ≠ 0
⊢ MDifferentiableAt I I i (bottcherNear f d 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | exact d0 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : ¬mfderiv I I i 0 = 0
⊢ mfderiv I I i 0 ≠ 0
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : mfderiv I I (i ∘ fun x => bottcherNear f d x) 0 ≠ 0
⊢ MDifferentiableAt I I i (bottcherNear f d 0) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : mfderiv I I (i ∘ fun x => bottcherNear f d x) 0 ≠ 0
⊢ MDifferentiableAt I I i (bottcherNear f d 0) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : ¬mfderiv I I i 0 = 0
⊢ mfderiv I I i 0 ≠ 0
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
inst✝³ : AnalyticManifold I S
T : Type
inst✝² : TopologicalSpace T
inst✝¹ : ChartedSpace ℂ T
inst✝ : AnalyticManifold I T
f : ℂ → ℂ
d : ℕ
s✝ : SuperAt f d
t : Set ℂ
s : SuperNear f d t
ba : AnalyticAt ℂ (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ≠ 0
i : ℂ → ℂ
ia : HolomorphicAt I I i 0
ib : ∀ᶠ (x : ℂ) in 𝓝 0, i (bottcherNear f d x) = x
bi : ∀ᶠ (x : ℂ) in 𝓝 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : mfderiv I I (i ∘ fun x => bottcherNear f d x) 0 ≠ 0
⊢ MDifferentiableAt I I i (bottcherNear f d 0)
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.