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 | smul_cmmap_cont | [157, 1] | [160, 43] | apply Continuous.smul | 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
⊢ Continuous (smulCmmapFn x xs) | case hf
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
⊢ Continuous fun x_1 => x.toFun fun x => x_1 0
case hg
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
⊢ Continuous fun x => xs.toFun fun i => x 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
⊢ Continuous (smulCmmapFn x xs)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smul_cmmap_cont | [157, 1] | [160, 43] | repeat continuity | case hf
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
⊢ Continuous fun x_1 => x.toFun fun x => x_1 0
case hg
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
⊢ Continuous fun x => xs.toFun fun i => x i.succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
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
⊢ Continuous fun x_1 => x.toFun fun x => x_1 0
case hg
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
⊢ Continuous fun x => xs.toFun fun i => x i.succ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smul_cmmap_cont | [157, 1] | [160, 43] | continuity | case hg
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
⊢ Continuous fun x => xs.toFun fun i => x i.succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
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
⊢ Continuous fun x => xs.toFun fun i => x i.succ
TACTIC:
|
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/Multibrot/Iterates.lean | iter_large_z3 | [42, 1] | [46, 45] | rw [(by norm_num : (2:ℝ) = 3-1)] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 2 ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ (3 - 1) ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 2 ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_large_z3 | [42, 1] | [46, 45] | exact iter_large d 3 (by norm_num) z3 cz n | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ (3 - 1) ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ (3 - 1) ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_large_z3 | [42, 1] | [46, 45] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 2 = 3 - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 2 = 3 - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_large_z3 | [42, 1] | [46, 45] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 2 ≤ 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 2 ≤ 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_large_z4 | [48, 1] | [52, 45] | rw [(by norm_num : (3:ℝ) = 4-1)] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 3 ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ (4 - 1) ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 3 ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_large_z4 | [48, 1] | [52, 45] | exact iter_large d 4 (by norm_num) z4 cz n | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ (4 - 1) ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ (4 - 1) ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_large_z4 | [48, 1] | [52, 45] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 3 = 4 - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 3 = 4 - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_large_z4 | [48, 1] | [52, 45] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 2 ≤ 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 2 ≤ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | le_self_iter | [54, 1] | [58, 92] | refine le_trans ?_ (iter_large_z3 d z3 cz n) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ Complex.abs z ≤ Complex.abs ((f' d c)^[n] z) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ Complex.abs z ≤ 2 ^ n * Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ Complex.abs z ≤ Complex.abs ((f' d c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | le_self_iter | [54, 1] | [58, 92] | exact le_mul_of_one_le_left (Complex.abs.nonneg _) (one_le_pow_of_one_le (by norm_num) _) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ Complex.abs z ≤ 2 ^ n * Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ Complex.abs z ≤ 2 ^ n * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | le_self_iter | [54, 1] | [58, 92] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 1 ≤ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 1 ≤ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | tendsto_iter_atInf | [61, 1] | [65, 100] | simp only [tendsto_atInf_iff_norm_tendsto_atTop, Complex.norm_eq_abs] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun n => (f' d c)^[n] z) atTop atInf | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun x => Complex.abs ((f' d c)^[x] z)) atTop atTop | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun n => (f' d c)^[n] z) atTop atInf
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | tendsto_iter_atInf | [61, 1] | [65, 100] | refine Filter.tendsto_atTop_mono (iter_large_z3 d z3 cz) ?_ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun x => Complex.abs ((f' d c)^[x] z)) atTop atTop | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun n => 2 ^ n * Complex.abs z) atTop atTop | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun x => Complex.abs ((f' d c)^[x] z)) atTop atTop
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | tendsto_iter_atInf | [61, 1] | [65, 100] | exact Filter.Tendsto.atTop_mul_const (by linarith) (tendsto_pow_atTop_atTop_of_one_lt one_lt_two) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun n => 2 ^ n * Complex.abs z) atTop atTop | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun n => 2 ^ n * Complex.abs z) atTop atTop
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | tendsto_iter_atInf | [61, 1] | [65, 100] | linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0 < Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0 < Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | rw [← Complex.abs.ne_zero_iff] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ z ^ d + c ≠ 0 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ Complex.abs (z ^ d + c) ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ z ^ d + c ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | apply ne_of_gt | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ Complex.abs (z ^ d + c) ≠ 0 | case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ 0 < Complex.abs (z ^ d + c) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ Complex.abs (z ^ d + c) ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | have z1 : 1 ≤ abs z := le_trans (by norm_num) z3 | case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ 0 < Complex.abs (z ^ d + c) | case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ 0 < Complex.abs (z ^ d + c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ 0 < Complex.abs (z ^ d + c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | calc abs (z ^ d + c)
_ ≥ abs (z ^ d) - abs c := by bound
_ = abs z ^ d - abs c := by rw [Complex.abs.map_pow]
_ ≥ abs z ^ 2 - abs z := by bound
_ = abs z * (abs z - 1) := by ring
_ ≥ 3 * (3 - 1) := by bound
_ > 0 := by norm_num | case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ 0 < Complex.abs (z ^ d + c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ 0 < Complex.abs (z ^ d + c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | norm_num | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ 1 ≤ 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ 1 ≤ 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | bound | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ Complex.abs (z ^ d + c) ≥ Complex.abs (z ^ d) - Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ Complex.abs (z ^ d + c) ≥ Complex.abs (z ^ d) - Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | rw [Complex.abs.map_pow] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ Complex.abs (z ^ d) - Complex.abs c = Complex.abs z ^ d - Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ Complex.abs (z ^ d) - Complex.abs c = Complex.abs z ^ d - Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | bound | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ Complex.abs z ^ d - Complex.abs c ≥ Complex.abs z ^ 2 - Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ Complex.abs z ^ d - Complex.abs c ≥ Complex.abs z ^ 2 - Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | ring | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ Complex.abs z ^ 2 - Complex.abs z = Complex.abs z * (Complex.abs z - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ Complex.abs z ^ 2 - Complex.abs z = Complex.abs z * (Complex.abs z - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | bound | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ Complex.abs z * (Complex.abs z - 1) ≥ 3 * (3 - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ Complex.abs z * (Complex.abs z - 1) ≥ 3 * (3 - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | norm_num | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ 3 * (3 - 1) > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ 3 * (3 - 1) > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | le_log_abs_z | [90, 1] | [95, 24] | rw [Real.le_log_iff_exp_le (by linarith)] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ 1.0986 ≤ (Complex.abs z).log | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ exp 1.0986 ≤ Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ 1.0986 ≤ (Complex.abs z).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | le_log_abs_z | [90, 1] | [95, 24] | refine le_trans ?_ z3 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ exp 1.0986 ≤ Complex.abs z | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ exp 1.0986 ≤ 3 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ exp 1.0986 ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | le_log_abs_z | [90, 1] | [95, 24] | norm_num | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ exp 1.0986 ≤ 3 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ (5493 / 5000).exp ≤ 3 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ exp 1.0986 ≤ 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | le_log_abs_z | [90, 1] | [95, 24] | exact (exp_div_lt).le | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ (5493 / 5000).exp ≤ 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ (5493 / 5000).exp ≤ 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | le_log_abs_z | [90, 1] | [95, 24] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ 0 < Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ 0 < Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | have z0 : 0 < abs z := lt_of_lt_of_le (by norm_num) z3 | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | have z0' : z ≠ 0 := by exact nnnorm_pos.mp z0 | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | have i1 : 1 / abs z ≤ 1 := by rw [one_div_le z0]; exact le_trans (by norm_num) z3; norm_num | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | have s1 : 1 - 1 / abs z < 1 := by rw [tsub_lt_iff_tsub_lt]; norm_num; exact z0'; exact i1; rfl | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
s1 : 1 - 1 / Complex.abs z < 1
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | have l1 := le_log_abs_z z3 | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
s1 : 1 - 1 / Complex.abs z < 1
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
s1 : 1 - 1 / Complex.abs z < 1
l1 : 1.0986 ≤ (Complex.abs z).log
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
s1 : 1 - 1 / Complex.abs z < 1
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | exact div_nonneg (neg_nonneg.mpr (Real.log_nonpos (sub_nonneg.mpr i1) s1.le)) (by positivity) | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
s1 : 1 - 1 / Complex.abs z < 1
l1 : 1.0986 ≤ (Complex.abs z).log
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
s1 : 1 - 1 / Complex.abs z < 1
l1 : 1.0986 ≤ (Complex.abs z).log
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | norm_num | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ 0 < 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ 0 < 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | exact nnnorm_pos.mp z0 | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
⊢ z ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
⊢ z ≠ 0
TACTIC:
|
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