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
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stringlengths 3
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| state_after
stringlengths 6
2.09M
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stringlengths 73
2.09M
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https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticAt_log | [65, 1] | [69, 35] | filter_upwards [Complex.isOpen_slitPlane.eventually_mem m] | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
m : c ∈ Complex.slitPlane
⊢ ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ log z | case h
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
m : c ∈ Complex.slitPlane
⊢ ∀ a ∈ Complex.slitPlane, DifferentiableAt ℂ log a | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
m : c ∈ Complex.slitPlane
⊢ ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ log z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticAt_log | [65, 1] | [69, 35] | intro z m | case h
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
m : c ∈ Complex.slitPlane
⊢ ∀ a ∈ Complex.slitPlane, DifferentiableAt ℂ log a | case h
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
m✝ : c ∈ Complex.slitPlane
z : ℂ
m : z ∈ Complex.slitPlane
⊢ DifferentiableAt ℂ log z | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
m : c ∈ Complex.slitPlane
⊢ ∀ a ∈ Complex.slitPlane, DifferentiableAt ℂ log a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticAt_log | [65, 1] | [69, 35] | exact differentiableAt_id.clog m | case h
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
m✝ : c ∈ Complex.slitPlane
z : ℂ
m : z ∈ Complex.slitPlane
⊢ DifferentiableAt ℂ log z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
m✝ : c ∈ Complex.slitPlane
z : ℂ
m : z ∈ Complex.slitPlane
⊢ DifferentiableAt ℂ log z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | AnalyticAt.cpow | [82, 1] | [89, 48] | have fc : f c ≠ 0 := Complex.slitPlane_ne_zero m | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f g : E → ℂ
c : E
fa : AnalyticAt ℂ f c
ga : AnalyticAt ℂ g c
m : f c ∈ Complex.slitPlane
⊢ AnalyticAt ℂ (fun z => f z ^ g z) c | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f g : E → ℂ
c : E
fa : AnalyticAt ℂ f c
ga : AnalyticAt ℂ g c
m : f c ∈ Complex.slitPlane
fc : f c ≠ 0
⊢ AnalyticAt ℂ (fun z => f z ^ g z) c | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f g : E → ℂ
c : E
fa : AnalyticAt ℂ f c
ga : AnalyticAt ℂ g c
m : f c ∈ Complex.slitPlane
⊢ AnalyticAt ℂ (fun z => f z ^ g z) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | AnalyticAt.cpow | [82, 1] | [89, 48] | have e : (fun z ↦ f z ^ g z) =ᶠ[𝓝 c] fun z ↦ Complex.exp (Complex.log (f z) * g z) := by
refine (fa.continuousAt.eventually_ne fc).mp (Filter.eventually_of_forall ?_)
intro z fz; simp only [fz, Complex.cpow_def, if_false] | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f g : E → ℂ
c : E
fa : AnalyticAt ℂ f c
ga : AnalyticAt ℂ g c
m : f c ∈ Complex.slitPlane
fc : f c ≠ 0
⊢ AnalyticAt ℂ (fun z => f z ^ g z) c | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f g : E → ℂ
c : E
fa : AnalyticAt ℂ f c
ga : AnalyticAt ℂ g c
m : f c ∈ Complex.slitPlane
fc : f c ≠ 0
e : (𝓝 c).EventuallyEq (fun z => f z ^ g z) fun z => ((f z).log * g z).exp
⊢ AnalyticAt ℂ (fun z => f z ^ g z) c | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f g : E → ℂ
c : E
fa : AnalyticAt ℂ f c
ga : AnalyticAt ℂ g c
m : f c ∈ Complex.slitPlane
fc : f c ≠ 0
⊢ AnalyticAt ℂ (fun z => f z ^ g z) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | AnalyticAt.cpow | [82, 1] | [89, 48] | rw [analyticAt_congr e] | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f g : E → ℂ
c : E
fa : AnalyticAt ℂ f c
ga : AnalyticAt ℂ g c
m : f c ∈ Complex.slitPlane
fc : f c ≠ 0
e : (𝓝 c).EventuallyEq (fun z => f z ^ g z) fun z => ((f z).log * g z).exp
⊢ AnalyticAt ℂ (fun z => f z ^ g z) c | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f g : E → ℂ
c : E
fa : AnalyticAt ℂ f c
ga : AnalyticAt ℂ g c
m : f c ∈ Complex.slitPlane
fc : f c ≠ 0
e : (𝓝 c).EventuallyEq (fun z => f z ^ g z) fun z => ((f z).log * g z).exp
⊢ AnalyticAt ℂ (fun z => ((f z).log * g z).exp) c | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f g : E → ℂ
c : E
fa : AnalyticAt ℂ f c
ga : AnalyticAt ℂ g c
m : f c ∈ Complex.slitPlane
fc : f c ≠ 0
e : (𝓝 c).EventuallyEq (fun z => f z ^ g z) fun z => ((f z).log * g z).exp
⊢ AnalyticAt ℂ (fun z => f z ^ g z) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | AnalyticAt.cpow | [82, 1] | [89, 48] | exact AnalyticAt.exp.comp ((fa.log m).mul ga) | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f g : E → ℂ
c : E
fa : AnalyticAt ℂ f c
ga : AnalyticAt ℂ g c
m : f c ∈ Complex.slitPlane
fc : f c ≠ 0
e : (𝓝 c).EventuallyEq (fun z => f z ^ g z) fun z => ((f z).log * g z).exp
⊢ AnalyticAt ℂ (fun z => ((f z).log * g z).exp) c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f g : E → ℂ
c : E
fa : AnalyticAt ℂ f c
ga : AnalyticAt ℂ g c
m : f c ∈ Complex.slitPlane
fc : f c ≠ 0
e : (𝓝 c).EventuallyEq (fun z => f z ^ g z) fun z => ((f z).log * g z).exp
⊢ AnalyticAt ℂ (fun z => ((f z).log * g z).exp) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | AnalyticAt.cpow | [82, 1] | [89, 48] | refine (fa.continuousAt.eventually_ne fc).mp (Filter.eventually_of_forall ?_) | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f g : E → ℂ
c : E
fa : AnalyticAt ℂ f c
ga : AnalyticAt ℂ g c
m : f c ∈ Complex.slitPlane
fc : f c ≠ 0
⊢ (𝓝 c).EventuallyEq (fun z => f z ^ g z) fun z => ((f z).log * g z).exp | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f g : E → ℂ
c : E
fa : AnalyticAt ℂ f c
ga : AnalyticAt ℂ g c
m : f c ∈ Complex.slitPlane
fc : f c ≠ 0
⊢ ∀ (x : E), f x ≠ 0 → (fun z => f z ^ g z) x = (fun z => ((f z).log * g z).exp) x | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f g : E → ℂ
c : E
fa : AnalyticAt ℂ f c
ga : AnalyticAt ℂ g c
m : f c ∈ Complex.slitPlane
fc : f c ≠ 0
⊢ (𝓝 c).EventuallyEq (fun z => f z ^ g z) fun z => ((f z).log * g z).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | AnalyticAt.cpow | [82, 1] | [89, 48] | intro z fz | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f g : E → ℂ
c : E
fa : AnalyticAt ℂ f c
ga : AnalyticAt ℂ g c
m : f c ∈ Complex.slitPlane
fc : f c ≠ 0
⊢ ∀ (x : E), f x ≠ 0 → (fun z => f z ^ g z) x = (fun z => ((f z).log * g z).exp) x | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f g : E → ℂ
c : E
fa : AnalyticAt ℂ f c
ga : AnalyticAt ℂ g c
m : f c ∈ Complex.slitPlane
fc : f c ≠ 0
z : E
fz : f z ≠ 0
⊢ (fun z => f z ^ g z) z = (fun z => ((f z).log * g z).exp) z | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f g : E → ℂ
c : E
fa : AnalyticAt ℂ f c
ga : AnalyticAt ℂ g c
m : f c ∈ Complex.slitPlane
fc : f c ≠ 0
⊢ ∀ (x : E), f x ≠ 0 → (fun z => f z ^ g z) x = (fun z => ((f z).log * g z).exp) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | AnalyticAt.cpow | [82, 1] | [89, 48] | simp only [fz, Complex.cpow_def, if_false] | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f g : E → ℂ
c : E
fa : AnalyticAt ℂ f c
ga : AnalyticAt ℂ g c
m : f c ∈ Complex.slitPlane
fc : f c ≠ 0
z : E
fz : f z ≠ 0
⊢ (fun z => f z ^ g z) z = (fun z => ((f z).log * g z).exp) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f g : E → ℂ
c : E
fa : AnalyticAt ℂ f c
ga : AnalyticAt ℂ g c
m : f c ∈ Complex.slitPlane
fc : f c ≠ 0
z : E
fz : f z ≠ 0
⊢ (fun z => f z ^ g z) z = (fun z => ((f z).log * g z).exp) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | ContinuousMultilinearMap.toFun_eq_coe | [30, 1] | [33, 41] | rw [MultilinearMap.toFun_eq_coe] | n : ℕ
𝕜 : Type
inst✝⁸ : NontriviallyNormedField 𝕜
R✝ A✝ B✝ E : Type
inst✝⁷ : Semiring R✝
R A B : Type
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid A
inst✝⁴ : Module R A
inst✝³ : TopologicalSpace A
inst✝² : AddCommMonoid B
inst✝¹ : Module R B
inst✝ : TopologicalSpace B
f : ContinuousMultilinearMap R (fun x => A) B
⊢ f.toFun = ⇑f | n : ℕ
𝕜 : Type
inst✝⁸ : NontriviallyNormedField 𝕜
R✝ A✝ B✝ E : Type
inst✝⁷ : Semiring R✝
R A B : Type
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid A
inst✝⁴ : Module R A
inst✝³ : TopologicalSpace A
inst✝² : AddCommMonoid B
inst✝¹ : Module R B
inst✝ : TopologicalSpace B
f : ContinuousMultilinearMap R (fun x => A) B
⊢ ⇑f.toMultilinearMap = ⇑f | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝⁸ : NontriviallyNormedField 𝕜
R✝ A✝ B✝ E : Type
inst✝⁷ : Semiring R✝
R A B : Type
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid A
inst✝⁴ : Module R A
inst✝³ : TopologicalSpace A
inst✝² : AddCommMonoid B
inst✝¹ : Module R B
inst✝ : TopologicalSpace B
f : ContinuousMultilinearMap R (fun x => A) B
⊢ f.toFun = ⇑f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | ContinuousMultilinearMap.toFun_eq_coe | [30, 1] | [33, 41] | simp | n : ℕ
𝕜 : Type
inst✝⁸ : NontriviallyNormedField 𝕜
R✝ A✝ B✝ E : Type
inst✝⁷ : Semiring R✝
R A B : Type
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid A
inst✝⁴ : Module R A
inst✝³ : TopologicalSpace A
inst✝² : AddCommMonoid B
inst✝¹ : Module R B
inst✝ : TopologicalSpace B
f : ContinuousMultilinearMap R (fun x => A) B
⊢ ⇑f.toMultilinearMap = ⇑f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝⁸ : NontriviallyNormedField 𝕜
R✝ A✝ B✝ E : Type
inst✝⁷ : Semiring R✝
R A B : Type
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid A
inst✝⁴ : Module R A
inst✝³ : TopologicalSpace A
inst✝² : AddCommMonoid B
inst✝¹ : Module R B
inst✝ : TopologicalSpace B
f : ContinuousMultilinearMap R (fun x => A) B
⊢ ⇑f.toMultilinearMap = ⇑f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_apply | [47, 1] | [50, 34] | simp only [fstCmmap, ContinuousMultilinearMap.ofSubsingleton_apply_apply,
ContinuousLinearMap.coe_fst'] | n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid A
inst✝⁴ : Module R A
inst✝³ : TopologicalSpace A
inst✝² : AddCommMonoid B
inst✝¹ : Module R B
inst✝ : TopologicalSpace B
a : A
b : B
⊢ ((fstCmmap R A B) fun x => (a, b)) = a | 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
inst✝⁵ : AddCommMonoid A
inst✝⁴ : Module R A
inst✝³ : TopologicalSpace A
inst✝² : AddCommMonoid B
inst✝¹ : Module R B
inst✝ : TopologicalSpace B
a : A
b : B
⊢ ((fstCmmap R A B) fun x => (a, b)) = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_apply | [52, 1] | [55, 34] | simp only [sndCmmap, ContinuousMultilinearMap.ofSubsingleton_apply_apply,
ContinuousLinearMap.coe_snd'] | n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid A
inst✝⁴ : Module R A
inst✝³ : TopologicalSpace A
inst✝² : AddCommMonoid B
inst✝¹ : Module R B
inst✝ : TopologicalSpace B
a : A
b : B
⊢ ((sndCmmap R A B) fun x => (a, b)) = b | 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
inst✝⁵ : AddCommMonoid A
inst✝⁴ : Module R A
inst✝³ : TopologicalSpace A
inst✝² : AddCommMonoid B
inst✝¹ : Module R B
inst✝ : TopologicalSpace B
a : A
b : B
⊢ ((sndCmmap R A B) fun x => (a, b)) = b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_norm | [57, 1] | [68, 20] | apply le_antisymm | n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fstCmmap 𝕜 A B‖ = 1 | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fstCmmap 𝕜 A B‖ ≤ 1
case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 1 ≤ ‖fstCmmap 𝕜 A B‖ | 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
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fstCmmap 𝕜 A B‖ = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_norm | [57, 1] | [68, 20] | refine (fstCmmap 𝕜 A B).op_norm_le_bound (M := 1) (by norm_num) ?_ | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fstCmmap 𝕜 A B‖ ≤ 1 | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ∀ (m : Fin 1 → A × B), ‖(fstCmmap 𝕜 A B) m‖ ≤ 1 * Finset.univ.prod fun i => ‖m i‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fstCmmap 𝕜 A B‖ ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_norm | [57, 1] | [68, 20] | intro z | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ∀ (m : Fin 1 → A × B), ‖(fstCmmap 𝕜 A B) m‖ ≤ 1 * Finset.univ.prod fun i => ‖m i‖ | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ ‖(fstCmmap 𝕜 A B) z‖ ≤ 1 * Finset.univ.prod fun i => ‖z i‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ∀ (m : Fin 1 → A × B), ‖(fstCmmap 𝕜 A B) m‖ ≤ 1 * Finset.univ.prod fun i => ‖m i‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_norm | [57, 1] | [68, 20] | simp only [Finset.univ_unique, Fin.default_eq_zero, Finset.prod_singleton, one_mul] | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ ‖(fstCmmap 𝕜 A B) z‖ ≤ 1 * Finset.univ.prod fun i => ‖z i‖ | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ ‖(fstCmmap 𝕜 A B) z‖ ≤ ‖z 0‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ ‖(fstCmmap 𝕜 A B) z‖ ≤ 1 * Finset.univ.prod fun i => ‖z i‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_norm | [57, 1] | [68, 20] | have e : z = (fun _ ↦ ((z 0).1, (z 0).2)) := by apply funext; intro i; rw [Fin.eq_zero i] | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ ‖(fstCmmap 𝕜 A B) z‖ ≤ ‖z 0‖ | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(fstCmmap 𝕜 A B) z‖ ≤ ‖z 0‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ ‖(fstCmmap 𝕜 A B) z‖ ≤ ‖z 0‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_norm | [57, 1] | [68, 20] | rw [e] | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(fstCmmap 𝕜 A B) z‖ ≤ ‖z 0‖ | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(fstCmmap 𝕜 A B) fun x => ((z 0).1, (z 0).2)‖ ≤ ‖(fun x => ((z 0).1, (z 0).2)) 0‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(fstCmmap 𝕜 A B) z‖ ≤ ‖z 0‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_norm | [57, 1] | [68, 20] | rw [fstCmmap_apply] | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(fstCmmap 𝕜 A B) fun x => ((z 0).1, (z 0).2)‖ ≤ ‖(fun x => ((z 0).1, (z 0).2)) 0‖ | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(z 0).1‖ ≤ ‖(fun x => ((z 0).1, (z 0).2)) 0‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(fstCmmap 𝕜 A B) fun x => ((z 0).1, (z 0).2)‖ ≤ ‖(fun x => ((z 0).1, (z 0).2)) 0‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_norm | [57, 1] | [68, 20] | simp | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(z 0).1‖ ≤ ‖(fun x => ((z 0).1, (z 0).2)) 0‖ | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(z 0).1‖ ≤ ‖z 0‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(z 0).1‖ ≤ ‖(fun x => ((z 0).1, (z 0).2)) 0‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_norm | [57, 1] | [68, 20] | exact norm_fst_le (z 0) | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(z 0).1‖ ≤ ‖z 0‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(z 0).1‖ ≤ ‖z 0‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_norm | [57, 1] | [68, 20] | norm_num | n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1 | 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
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_norm | [57, 1] | [68, 20] | apply funext | n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ z = fun x => ((z 0).1, (z 0).2) | case h
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ ∀ (x : Fin 1), z x = ((z 0).1, (z 0).2) | 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
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ z = fun x => ((z 0).1, (z 0).2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_norm | [57, 1] | [68, 20] | intro i | case h
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ ∀ (x : Fin 1), z x = ((z 0).1, (z 0).2) | case h
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
i : Fin 1
⊢ z i = ((z 0).1, (z 0).2) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ ∀ (x : Fin 1), z x = ((z 0).1, (z 0).2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_norm | [57, 1] | [68, 20] | rw [Fin.eq_zero i] | case h
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
i : Fin 1
⊢ z i = ((z 0).1, (z 0).2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
i : Fin 1
⊢ z i = ((z 0).1, (z 0).2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_norm | [57, 1] | [68, 20] | have lo := (fstCmmap 𝕜 A B).unit_le_op_norm (fun _ ↦ (1, 1)) ?_ | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 1 ≤ ‖fstCmmap 𝕜 A B‖ | case a.refine_2
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
lo : ‖(fstCmmap 𝕜 A B) fun x => (1, 1)‖ ≤ ‖fstCmmap 𝕜 A B‖
⊢ 1 ≤ ‖fstCmmap 𝕜 A B‖
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fun x => (1, 1)‖ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 1 ≤ ‖fstCmmap 𝕜 A B‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_norm | [57, 1] | [68, 20] | rw [fstCmmap_apply, norm_one] at lo | case a.refine_2
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
lo : ‖(fstCmmap 𝕜 A B) fun x => (1, 1)‖ ≤ ‖fstCmmap 𝕜 A B‖
⊢ 1 ≤ ‖fstCmmap 𝕜 A B‖
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fun x => (1, 1)‖ ≤ 1 | case a.refine_2
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
lo : 1 ≤ ‖fstCmmap 𝕜 A B‖
⊢ 1 ≤ ‖fstCmmap 𝕜 A B‖
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fun x => (1, 1)‖ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_2
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
lo : ‖(fstCmmap 𝕜 A B) fun x => (1, 1)‖ ≤ ‖fstCmmap 𝕜 A B‖
⊢ 1 ≤ ‖fstCmmap 𝕜 A B‖
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fun x => (1, 1)‖ ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_norm | [57, 1] | [68, 20] | assumption | case a.refine_2
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
lo : 1 ≤ ‖fstCmmap 𝕜 A B‖
⊢ 1 ≤ ‖fstCmmap 𝕜 A B‖
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fun x => (1, 1)‖ ≤ 1 | case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fun x => (1, 1)‖ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_2
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
lo : 1 ≤ ‖fstCmmap 𝕜 A B‖
⊢ 1 ≤ ‖fstCmmap 𝕜 A B‖
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fun x => (1, 1)‖ ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_norm | [57, 1] | [68, 20] | rw [pi_norm_le_iff_of_nonneg] | case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fun x => (1, 1)‖ ≤ 1 | case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ∀ (i : Fin 1), ‖(1, 1)‖ ≤ 1
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fun x => (1, 1)‖ ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_norm | [57, 1] | [68, 20] | intro i | case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ∀ (i : Fin 1), ‖(1, 1)‖ ≤ 1
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1 | case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
i : Fin 1
⊢ ‖(1, 1)‖ ≤ 1
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ∀ (i : Fin 1), ‖(1, 1)‖ ≤ 1
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_norm | [57, 1] | [68, 20] | simp only [Prod.norm_def, norm_one, max_eq_right] | case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
i : Fin 1
⊢ ‖(1, 1)‖ ≤ 1
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1 | case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
i : Fin 1
⊢ max 1 1 ≤ 1
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
i : Fin 1
⊢ ‖(1, 1)‖ ≤ 1
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_norm | [57, 1] | [68, 20] | repeat norm_num | case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
i : Fin 1
⊢ max 1 1 ≤ 1
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
i : Fin 1
⊢ max 1 1 ≤ 1
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | fstCmmap_norm | [57, 1] | [68, 20] | norm_num | case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_norm | [70, 1] | [81, 20] | apply le_antisymm | n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖sndCmmap 𝕜 A B‖ = 1 | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖sndCmmap 𝕜 A B‖ ≤ 1
case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 1 ≤ ‖sndCmmap 𝕜 A B‖ | 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
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖sndCmmap 𝕜 A B‖ = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_norm | [70, 1] | [81, 20] | apply (sndCmmap 𝕜 A B).op_norm_le_bound (M := 1) (by norm_num) | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖sndCmmap 𝕜 A B‖ ≤ 1 | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ∀ (m : Fin 1 → A × B), ‖(sndCmmap 𝕜 A B) m‖ ≤ 1 * Finset.univ.prod fun i => ‖m i‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖sndCmmap 𝕜 A B‖ ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_norm | [70, 1] | [81, 20] | intro z | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ∀ (m : Fin 1 → A × B), ‖(sndCmmap 𝕜 A B) m‖ ≤ 1 * Finset.univ.prod fun i => ‖m i‖ | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ ‖(sndCmmap 𝕜 A B) z‖ ≤ 1 * Finset.univ.prod fun i => ‖z i‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ∀ (m : Fin 1 → A × B), ‖(sndCmmap 𝕜 A B) m‖ ≤ 1 * Finset.univ.prod fun i => ‖m i‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_norm | [70, 1] | [81, 20] | simp only [Finset.univ_unique, Fin.default_eq_zero, Finset.prod_singleton, one_mul] | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ ‖(sndCmmap 𝕜 A B) z‖ ≤ 1 * Finset.univ.prod fun i => ‖z i‖ | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ ‖(sndCmmap 𝕜 A B) z‖ ≤ ‖z 0‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ ‖(sndCmmap 𝕜 A B) z‖ ≤ 1 * Finset.univ.prod fun i => ‖z i‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_norm | [70, 1] | [81, 20] | have e : z = (fun _ ↦ ((z 0).1, (z 0).2)) := by apply funext; intro i; rw [Fin.eq_zero i] | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ ‖(sndCmmap 𝕜 A B) z‖ ≤ ‖z 0‖ | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(sndCmmap 𝕜 A B) z‖ ≤ ‖z 0‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ ‖(sndCmmap 𝕜 A B) z‖ ≤ ‖z 0‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_norm | [70, 1] | [81, 20] | rw [e] | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(sndCmmap 𝕜 A B) z‖ ≤ ‖z 0‖ | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(sndCmmap 𝕜 A B) fun x => ((z 0).1, (z 0).2)‖ ≤ ‖(fun x => ((z 0).1, (z 0).2)) 0‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(sndCmmap 𝕜 A B) z‖ ≤ ‖z 0‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_norm | [70, 1] | [81, 20] | rw [sndCmmap_apply] | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(sndCmmap 𝕜 A B) fun x => ((z 0).1, (z 0).2)‖ ≤ ‖(fun x => ((z 0).1, (z 0).2)) 0‖ | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(z 0).2‖ ≤ ‖(fun x => ((z 0).1, (z 0).2)) 0‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(sndCmmap 𝕜 A B) fun x => ((z 0).1, (z 0).2)‖ ≤ ‖(fun x => ((z 0).1, (z 0).2)) 0‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_norm | [70, 1] | [81, 20] | simp | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(z 0).2‖ ≤ ‖(fun x => ((z 0).1, (z 0).2)) 0‖ | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(z 0).2‖ ≤ ‖z 0‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(z 0).2‖ ≤ ‖(fun x => ((z 0).1, (z 0).2)) 0‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_norm | [70, 1] | [81, 20] | exact norm_snd_le (z 0) | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(z 0).2‖ ≤ ‖z 0‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
e : z = fun x => ((z 0).1, (z 0).2)
⊢ ‖(z 0).2‖ ≤ ‖z 0‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_norm | [70, 1] | [81, 20] | norm_num | n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1 | 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
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_norm | [70, 1] | [81, 20] | apply funext | n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ z = fun x => ((z 0).1, (z 0).2) | case h
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ ∀ (x : Fin 1), z x = ((z 0).1, (z 0).2) | 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
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ z = fun x => ((z 0).1, (z 0).2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_norm | [70, 1] | [81, 20] | intro i | case h
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ ∀ (x : Fin 1), z x = ((z 0).1, (z 0).2) | case h
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
i : Fin 1
⊢ z i = ((z 0).1, (z 0).2) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
⊢ ∀ (x : Fin 1), z x = ((z 0).1, (z 0).2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_norm | [70, 1] | [81, 20] | rw [Fin.eq_zero i] | case h
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
i : Fin 1
⊢ z i = ((z 0).1, (z 0).2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
z : Fin 1 → A × B
i : Fin 1
⊢ z i = ((z 0).1, (z 0).2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_norm | [70, 1] | [81, 20] | have lo := (sndCmmap 𝕜 A B).unit_le_op_norm (fun _ ↦ (1, 1)) ?_ | case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 1 ≤ ‖sndCmmap 𝕜 A B‖ | case a.refine_2
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
lo : ‖(sndCmmap 𝕜 A B) fun x => (1, 1)‖ ≤ ‖sndCmmap 𝕜 A B‖
⊢ 1 ≤ ‖sndCmmap 𝕜 A B‖
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fun x => (1, 1)‖ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 1 ≤ ‖sndCmmap 𝕜 A B‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_norm | [70, 1] | [81, 20] | rw [sndCmmap_apply, norm_one] at lo | case a.refine_2
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
lo : ‖(sndCmmap 𝕜 A B) fun x => (1, 1)‖ ≤ ‖sndCmmap 𝕜 A B‖
⊢ 1 ≤ ‖sndCmmap 𝕜 A B‖
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fun x => (1, 1)‖ ≤ 1 | case a.refine_2
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
lo : 1 ≤ ‖sndCmmap 𝕜 A B‖
⊢ 1 ≤ ‖sndCmmap 𝕜 A B‖
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fun x => (1, 1)‖ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_2
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
lo : ‖(sndCmmap 𝕜 A B) fun x => (1, 1)‖ ≤ ‖sndCmmap 𝕜 A B‖
⊢ 1 ≤ ‖sndCmmap 𝕜 A B‖
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fun x => (1, 1)‖ ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_norm | [70, 1] | [81, 20] | assumption | case a.refine_2
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
lo : 1 ≤ ‖sndCmmap 𝕜 A B‖
⊢ 1 ≤ ‖sndCmmap 𝕜 A B‖
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fun x => (1, 1)‖ ≤ 1 | case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fun x => (1, 1)‖ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_2
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
lo : 1 ≤ ‖sndCmmap 𝕜 A B‖
⊢ 1 ≤ ‖sndCmmap 𝕜 A B‖
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fun x => (1, 1)‖ ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_norm | [70, 1] | [81, 20] | rw [pi_norm_le_iff_of_nonneg] | case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fun x => (1, 1)‖ ≤ 1 | case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ∀ (i : Fin 1), ‖(1, 1)‖ ≤ 1
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ‖fun x => (1, 1)‖ ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_norm | [70, 1] | [81, 20] | intro i | case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ∀ (i : Fin 1), ‖(1, 1)‖ ≤ 1
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1 | case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
i : Fin 1
⊢ ‖(1, 1)‖ ≤ 1
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ ∀ (i : Fin 1), ‖(1, 1)‖ ≤ 1
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_norm | [70, 1] | [81, 20] | simp only [Prod.norm_def, norm_one, max_eq_right] | case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
i : Fin 1
⊢ ‖(1, 1)‖ ≤ 1
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1 | case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
i : Fin 1
⊢ max 1 1 ≤ 1
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
i : Fin 1
⊢ ‖(1, 1)‖ ≤ 1
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_norm | [70, 1] | [81, 20] | repeat norm_num | case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
i : Fin 1
⊢ max 1 1 ≤ 1
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
i : Fin 1
⊢ max 1 1 ≤ 1
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | sndCmmap_norm | [70, 1] | [81, 20] | norm_num | case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_1
n : ℕ
𝕜 : Type
inst✝⁷ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝⁶ : Semiring R
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : NormOneClass A
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : NormOneClass B
⊢ 0 ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_0_0 | [84, 1] | [88, 44] | apply funext | n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
z : Fin (n + 1) → A
x : A
⊢ Function.update (fun x => z 0) 0 x = fun x_1 => x | case h
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
z : Fin (n + 1) → A
x : A
⊢ ∀ (x_1 : Fin 1), Function.update (fun x => z 0) 0 x x_1 = x | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
z : Fin (n + 1) → A
x : A
⊢ Function.update (fun x => z 0) 0 x = fun x_1 => x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_0_0 | [84, 1] | [88, 44] | intro i | case h
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
z : Fin (n + 1) → A
x : A
⊢ ∀ (x_1 : Fin 1), Function.update (fun x => z 0) 0 x x_1 = x | case h
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
z : Fin (n + 1) → A
x : A
i : Fin 1
⊢ Function.update (fun x => z 0) 0 x i = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
z : Fin (n + 1) → A
x : A
⊢ ∀ (x_1 : Fin 1), Function.update (fun x => z 0) 0 x x_1 = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_0_0 | [84, 1] | [88, 44] | have i0 : i = 0 := by simp only [eq_iff_true_of_subsingleton] | case h
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
z : Fin (n + 1) → A
x : A
i : Fin 1
⊢ Function.update (fun x => z 0) 0 x i = x | case h
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
z : Fin (n + 1) → A
x : A
i : Fin 1
i0 : i = 0
⊢ Function.update (fun x => z 0) 0 x i = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
z : Fin (n + 1) → A
x : A
i : Fin 1
⊢ Function.update (fun x => z 0) 0 x i = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_0_0 | [84, 1] | [88, 44] | rw [i0] | case h
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
z : Fin (n + 1) → A
x : A
i : Fin 1
i0 : i = 0
⊢ Function.update (fun x => z 0) 0 x i = x | case h
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
z : Fin (n + 1) → A
x : A
i : Fin 1
i0 : i = 0
⊢ Function.update (fun x => z 0) 0 x 0 = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
z : Fin (n + 1) → A
x : A
i : Fin 1
i0 : i = 0
⊢ Function.update (fun x => z 0) 0 x i = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_0_0 | [84, 1] | [88, 44] | simp only [Function.update_same] | case h
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
z : Fin (n + 1) → A
x : A
i : Fin 1
i0 : i = 0
⊢ Function.update (fun x => z 0) 0 x 0 = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
z : Fin (n + 1) → A
x : A
i : Fin 1
i0 : i = 0
⊢ Function.update (fun x => z 0) 0 x 0 = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_0_0 | [84, 1] | [88, 44] | simp only [eq_iff_true_of_subsingleton] | n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
z : Fin (n + 1) → A
x : A
i : Fin 1
⊢ i = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
z : Fin (n + 1) → A
x : A
i : Fin 1
⊢ i = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_0_succ | [90, 1] | [94, 31] | rw [Function.update_apply] | n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin n
⊢ Function.update f 0 x i.succ = f i.succ | n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin n
⊢ (if i.succ = 0 then x else f i.succ) = f i.succ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin n
⊢ Function.update f 0 x i.succ = f i.succ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_0_succ | [90, 1] | [94, 31] | simp only [ite_eq_right_iff] | n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin n
⊢ (if i.succ = 0 then x else f i.succ) = f i.succ | n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin n
⊢ i.succ = 0 → x = f i.succ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin n
⊢ (if i.succ = 0 then x else f i.succ) = f i.succ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_0_succ | [90, 1] | [94, 31] | have i0 := Fin.succ_ne_zero i | n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin n
⊢ i.succ = 0 → x = f i.succ | n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin n
i0 : i.succ ≠ 0
⊢ i.succ = 0 → x = f i.succ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin n
⊢ i.succ = 0 → x = f i.succ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_0_succ | [90, 1] | [94, 31] | intro h | n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin n
i0 : i.succ ≠ 0
⊢ i.succ = 0 → x = f i.succ | n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin n
i0 : i.succ ≠ 0
h : i.succ = 0
⊢ x = f i.succ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin n
i0 : i.succ ≠ 0
⊢ i.succ = 0 → x = f i.succ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_0_succ | [90, 1] | [94, 31] | exfalso | n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin n
i0 : i.succ ≠ 0
h : i.succ = 0
⊢ x = f i.succ | n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin n
i0 : i.succ ≠ 0
h : i.succ = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin n
i0 : i.succ ≠ 0
h : i.succ = 0
⊢ x = f i.succ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_0_succ | [90, 1] | [94, 31] | exact i0 h | n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin n
i0 : i.succ ≠ 0
h : i.succ = 0
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin n
i0 : i.succ ≠ 0
h : i.succ = 0
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_nz_0 | [96, 1] | [97, 97] | rw [Function.update_noteq i0.symm] | n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
⊢ Function.update f i x 0 = f 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
⊢ Function.update f i x 0 = f 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_nz_succ | [99, 1] | [110, 58] | apply funext | n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
⊢ (fun j => Function.update f i x j.succ) = Function.update (fun j => f j.succ) (i.pred i0) x | case h
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
⊢ ∀ (x_1 : Fin n), Function.update f i x x_1.succ = Function.update (fun j => f j.succ) (i.pred i0) x x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
⊢ (fun j => Function.update f i x j.succ) = Function.update (fun j => f j.succ) (i.pred i0) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_nz_succ | [99, 1] | [110, 58] | intro k | case h
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
⊢ ∀ (x_1 : Fin n), Function.update f i x x_1.succ = Function.update (fun j => f j.succ) (i.pred i0) x x_1 | case h
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
⊢ Function.update f i x k.succ = Function.update (fun j => f j.succ) (i.pred i0) x k | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
⊢ ∀ (x_1 : Fin n), Function.update f i x x_1.succ = Function.update (fun j => f j.succ) (i.pred i0) x x_1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_nz_succ | [99, 1] | [110, 58] | by_cases ki : k.succ = i | case h
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
⊢ Function.update f i x k.succ = Function.update (fun j => f j.succ) (i.pred i0) x k | case pos
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : k.succ = i
⊢ Function.update f i x k.succ = Function.update (fun j => f j.succ) (i.pred i0) x k
case neg
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : ¬k.succ = i
⊢ Function.update f i x k.succ = Function.update (fun j => f j.succ) (i.pred i0) x k | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
⊢ Function.update f i x k.succ = Function.update (fun j => f j.succ) (i.pred i0) x k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_nz_succ | [99, 1] | [110, 58] | have ki' : k = i.pred i0 := by simp_rw [← ki, Fin.pred_succ] | case pos
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : k.succ = i
⊢ Function.update f i x k.succ = Function.update (fun j => f j.succ) (i.pred i0) x k | case pos
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : k.succ = i
ki' : k = i.pred i0
⊢ Function.update f i x k.succ = Function.update (fun j => f j.succ) (i.pred i0) x k | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : k.succ = i
⊢ Function.update f i x k.succ = Function.update (fun j => f j.succ) (i.pred i0) x k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_nz_succ | [99, 1] | [110, 58] | rw [ki, ki'] | case pos
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : k.succ = i
ki' : k = i.pred i0
⊢ Function.update f i x k.succ = Function.update (fun j => f j.succ) (i.pred i0) x k | case pos
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : k.succ = i
ki' : k = i.pred i0
⊢ Function.update f i x i = Function.update (fun j => f j.succ) (i.pred i0) x (i.pred i0) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : k.succ = i
ki' : k = i.pred i0
⊢ Function.update f i x k.succ = Function.update (fun j => f j.succ) (i.pred i0) x k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_nz_succ | [99, 1] | [110, 58] | rw [Function.update_same] | case pos
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : k.succ = i
ki' : k = i.pred i0
⊢ Function.update f i x i = Function.update (fun j => f j.succ) (i.pred i0) x (i.pred i0) | case pos
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : k.succ = i
ki' : k = i.pred i0
⊢ x = Function.update (fun j => f j.succ) (i.pred i0) x (i.pred i0) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : k.succ = i
ki' : k = i.pred i0
⊢ Function.update f i x i = Function.update (fun j => f j.succ) (i.pred i0) x (i.pred i0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_nz_succ | [99, 1] | [110, 58] | rw [Function.update_same] | case pos
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : k.succ = i
ki' : k = i.pred i0
⊢ x = Function.update (fun j => f j.succ) (i.pred i0) x (i.pred i0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : k.succ = i
ki' : k = i.pred i0
⊢ x = Function.update (fun j => f j.succ) (i.pred i0) x (i.pred i0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_nz_succ | [99, 1] | [110, 58] | simp_rw [← ki, Fin.pred_succ] | n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : k.succ = i
⊢ k = i.pred i0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : k.succ = i
⊢ k = i.pred i0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_nz_succ | [99, 1] | [110, 58] | rw [Function.update_noteq ki] | case neg
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : ¬k.succ = i
⊢ Function.update f i x k.succ = Function.update (fun j => f j.succ) (i.pred i0) x k | case neg
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : ¬k.succ = i
⊢ f k.succ = Function.update (fun j => f j.succ) (i.pred i0) x k | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : ¬k.succ = i
⊢ Function.update f i x k.succ = Function.update (fun j => f j.succ) (i.pred i0) x k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_nz_succ | [99, 1] | [110, 58] | rw [Function.update_noteq _] | case neg
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : ¬k.succ = i
⊢ f k.succ = Function.update (fun j => f j.succ) (i.pred i0) x k | n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : ¬k.succ = i
⊢ k ≠ i.pred i0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : ¬k.succ = i
⊢ f k.succ = Function.update (fun j => f j.succ) (i.pred i0) x k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_nz_succ | [99, 1] | [110, 58] | by_contra h | n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : ¬k.succ = i
⊢ k ≠ i.pred i0 | n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : ¬k.succ = i
h : k = i.pred i0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : ¬k.succ = i
⊢ k ≠ i.pred i0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | update_nz_succ | [99, 1] | [110, 58] | simp only [h, Fin.succ_pred, not_true_eq_false] at ki | n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : ¬k.succ = i
h : k = i.pred i0
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
𝕜 : Type
inst✝¹ : NontriviallyNormedField 𝕜
R A B E : Type
inst✝ : Semiring R
d : DecidableEq (Fin (n + 1))
f : Fin (n + 1) → A
x : A
i : Fin (n + 1)
i0 : i ≠ 0
k : Fin n
ki : ¬k.succ = i
h : k = i.pred i0
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smul_cmmap_add | [119, 1] | [135, 19] | intro d z i u v | 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
⊢ ∀ (d : DecidableEq (Fin (n + 1))) (z : Fin (n + 1) → A) (i : Fin (n + 1)) (u v : A),
smulCmmapFn x xs (Function.update z i (u + v)) =
smulCmmapFn x xs (Function.update z i u) + smulCmmapFn x xs (Function.update z i v) | 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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
u v : A
⊢ smulCmmapFn x xs (Function.update z i (u + v)) =
smulCmmapFn x xs (Function.update z i u) + smulCmmapFn x xs (Function.update z i v) | 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
⊢ ∀ (d : DecidableEq (Fin (n + 1))) (z : Fin (n + 1) → A) (i : Fin (n + 1)) (u v : A),
smulCmmapFn x xs (Function.update z i (u + v)) =
smulCmmapFn x xs (Function.update z i u) + smulCmmapFn x xs (Function.update z i v)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smul_cmmap_add | [119, 1] | [135, 19] | by_cases i0 : i = 0 | 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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
u v : A
⊢ smulCmmapFn x xs (Function.update z i (u + v)) =
smulCmmapFn x xs (Function.update z i u) + smulCmmapFn x xs (Function.update z i v) | case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
u v : A
i0 : i = 0
⊢ smulCmmapFn x xs (Function.update z i (u + v)) =
smulCmmapFn x xs (Function.update z i u) + smulCmmapFn x xs (Function.update z i v)
case neg
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
u v : A
i0 : ¬i = 0
⊢ smulCmmapFn x xs (Function.update z i (u + v)) =
smulCmmapFn x xs (Function.update z i u) + smulCmmapFn x xs (Function.update z i v) | 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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
u v : A
⊢ smulCmmapFn x xs (Function.update z i (u + v)) =
smulCmmapFn x xs (Function.update z i u) + smulCmmapFn x xs (Function.update z i v)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smul_cmmap_add | [119, 1] | [135, 19] | rw [i0] | case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
u v : A
i0 : i = 0
⊢ smulCmmapFn x xs (Function.update z i (u + v)) =
smulCmmapFn x xs (Function.update z i u) + smulCmmapFn x xs (Function.update z i v) | case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
u v : A
i0 : i = 0
⊢ smulCmmapFn x xs (Function.update z 0 (u + v)) =
smulCmmapFn x xs (Function.update z 0 u) + smulCmmapFn x xs (Function.update z 0 v) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
u v : A
i0 : i = 0
⊢ smulCmmapFn x xs (Function.update z i (u + v)) =
smulCmmapFn x xs (Function.update z i u) + smulCmmapFn x xs (Function.update z i v)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smul_cmmap_add | [119, 1] | [135, 19] | have uv := x.map_add (fun _ ↦ z 0) 0 u v | case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
u v : A
i0 : i = 0
⊢ smulCmmapFn x xs (Function.update z 0 (u + v)) =
smulCmmapFn x xs (Function.update z 0 u) + smulCmmapFn x xs (Function.update z 0 v) | case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
u v : A
i0 : i = 0
uv :
x (Function.update (fun x => z 0) 0 (u + v)) =
x (Function.update (fun x => z 0) 0 u) + x (Function.update (fun x => z 0) 0 v)
⊢ smulCmmapFn x xs (Function.update z 0 (u + v)) =
smulCmmapFn x xs (Function.update z 0 u) + smulCmmapFn x xs (Function.update z 0 v) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
u v : A
i0 : i = 0
⊢ smulCmmapFn x xs (Function.update z 0 (u + v)) =
smulCmmapFn x xs (Function.update z 0 u) + smulCmmapFn x xs (Function.update z 0 v)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smul_cmmap_add | [119, 1] | [135, 19] | simp only [update_0_0 z _] at uv | case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
u v : A
i0 : i = 0
uv :
x (Function.update (fun x => z 0) 0 (u + v)) =
x (Function.update (fun x => z 0) 0 u) + x (Function.update (fun x => z 0) 0 v)
⊢ smulCmmapFn x xs (Function.update z 0 (u + v)) =
smulCmmapFn x xs (Function.update z 0 u) + smulCmmapFn x xs (Function.update z 0 v) | case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
u v : A
i0 : i = 0
uv : (x fun x => u + v) = (x fun x => u) + x fun x => v
⊢ smulCmmapFn x xs (Function.update z 0 (u + v)) =
smulCmmapFn x xs (Function.update z 0 u) + smulCmmapFn x xs (Function.update z 0 v) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
u v : A
i0 : i = 0
uv :
x (Function.update (fun x => z 0) 0 (u + v)) =
x (Function.update (fun x => z 0) 0 u) + x (Function.update (fun x => z 0) 0 v)
⊢ smulCmmapFn x xs (Function.update z 0 (u + v)) =
smulCmmapFn x xs (Function.update z 0 u) + smulCmmapFn x xs (Function.update z 0 v)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smul_cmmap_add | [119, 1] | [135, 19] | simp only [Function.update_same, MultilinearMap.toFun_eq_coe, ContinuousMultilinearMap.coe_coe,
ne_eq, uv, add_smul, smulCmmapFn, update_0_succ] | case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
u v : A
i0 : i = 0
uv : (x fun x => u + v) = (x fun x => u) + x fun x => v
⊢ smulCmmapFn x xs (Function.update z 0 (u + v)) =
smulCmmapFn x xs (Function.update z 0 u) + smulCmmapFn x xs (Function.update z 0 v) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
u v : A
i0 : i = 0
uv : (x fun x => u + v) = (x fun x => u) + x fun x => v
⊢ smulCmmapFn x xs (Function.update z 0 (u + v)) =
smulCmmapFn x xs (Function.update z 0 u) + smulCmmapFn x xs (Function.update z 0 v)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smul_cmmap_add | [119, 1] | [135, 19] | simp only [smul_add, ne_eq, update_nz_0 d z i0, MultilinearMap.toFun_eq_coe,
ContinuousMultilinearMap.coe_coe, update_nz_succ d z _ i0, MultilinearMap.map_add, smul_add,
smulCmmapFn] | case neg
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
u v : A
i0 : ¬i = 0
⊢ smulCmmapFn x xs (Function.update z i (u + v)) =
smulCmmapFn x xs (Function.update z i u) + smulCmmapFn x xs (Function.update z i v) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
u v : A
i0 : ¬i = 0
⊢ smulCmmapFn x xs (Function.update z i (u + v)) =
smulCmmapFn x xs (Function.update z i u) + smulCmmapFn x xs (Function.update z i v)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smul_cmmap_smul | [138, 1] | [154, 83] | intro d z i s u | 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
⊢ ∀ (d : DecidableEq (Fin (n + 1))) (z : Fin (n + 1) → A) (i : Fin (n + 1)) (s : 𝕜) (u : A),
smulCmmapFn x xs (Function.update z i (s • u)) = s • smulCmmapFn x xs (Function.update z i u) | 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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
⊢ smulCmmapFn x xs (Function.update z i (s • u)) = s • smulCmmapFn x xs (Function.update z i u) | 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
⊢ ∀ (d : DecidableEq (Fin (n + 1))) (z : Fin (n + 1) → A) (i : Fin (n + 1)) (s : 𝕜) (u : A),
smulCmmapFn x xs (Function.update z i (s • u)) = s • smulCmmapFn x xs (Function.update z i u)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smul_cmmap_smul | [138, 1] | [154, 83] | rw [smulCmmapFn] | 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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
⊢ smulCmmapFn x xs (Function.update z i (s • u)) = s • smulCmmapFn x xs (Function.update z i u) | 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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
⊢ ((x.toFun fun x => Function.update z i (s • u) 0) • xs.toFun fun i_1 => Function.update z i (s • u) i_1.succ) =
s • smulCmmapFn x xs (Function.update z i u) | 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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
⊢ smulCmmapFn x xs (Function.update z i (s • u)) = s • smulCmmapFn x xs (Function.update z i u)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smul_cmmap_smul | [138, 1] | [154, 83] | by_cases i0 : i = 0 | 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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
⊢ ((x.toFun fun x => Function.update z i (s • u) 0) • xs.toFun fun i_1 => Function.update z i (s • u) i_1.succ) =
s • smulCmmapFn x xs (Function.update z i u) | case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
i0 : i = 0
⊢ ((x.toFun fun x => Function.update z i (s • u) 0) • xs.toFun fun i_1 => Function.update z i (s • u) i_1.succ) =
s • smulCmmapFn x xs (Function.update z i u)
case neg
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
i0 : ¬i = 0
⊢ ((x.toFun fun x => Function.update z i (s • u) 0) • xs.toFun fun i_1 => Function.update z i (s • u) i_1.succ) =
s • smulCmmapFn x xs (Function.update z i u) | 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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
⊢ ((x.toFun fun x => Function.update z i (s • u) 0) • xs.toFun fun i_1 => Function.update z i (s • u) i_1.succ) =
s • smulCmmapFn x xs (Function.update z i u)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smul_cmmap_smul | [138, 1] | [154, 83] | rw [i0] | case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
i0 : i = 0
⊢ ((x.toFun fun x => Function.update z i (s • u) 0) • xs.toFun fun i_1 => Function.update z i (s • u) i_1.succ) =
s • smulCmmapFn x xs (Function.update z i u) | case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
i0 : i = 0
⊢ ((x.toFun fun x => Function.update z 0 (s • u) 0) • xs.toFun fun i => Function.update z 0 (s • u) i.succ) =
s • smulCmmapFn x xs (Function.update z 0 u) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
i0 : i = 0
⊢ ((x.toFun fun x => Function.update z i (s • u) 0) • xs.toFun fun i_1 => Function.update z i (s • u) i_1.succ) =
s • smulCmmapFn x xs (Function.update z i u)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smul_cmmap_smul | [138, 1] | [154, 83] | have su := x.map_smul (fun _ ↦ z 0) 0 s u | case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
i0 : i = 0
⊢ ((x.toFun fun x => Function.update z 0 (s • u) 0) • xs.toFun fun i => Function.update z 0 (s • u) i.succ) =
s • smulCmmapFn x xs (Function.update z 0 u) | case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
i0 : i = 0
su : x (Function.update (fun x => z 0) 0 (s • u)) = s • x (Function.update (fun x => z 0) 0 u)
⊢ ((x.toFun fun x => Function.update z 0 (s • u) 0) • xs.toFun fun i => Function.update z 0 (s • u) i.succ) =
s • smulCmmapFn x xs (Function.update z 0 u) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
i0 : i = 0
⊢ ((x.toFun fun x => Function.update z 0 (s • u) 0) • xs.toFun fun i => Function.update z 0 (s • u) i.succ) =
s • smulCmmapFn x xs (Function.update z 0 u)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smul_cmmap_smul | [138, 1] | [154, 83] | rw [update_0_0 z _, update_0_0 z _] at su | case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
i0 : i = 0
su : x (Function.update (fun x => z 0) 0 (s • u)) = s • x (Function.update (fun x => z 0) 0 u)
⊢ ((x.toFun fun x => Function.update z 0 (s • u) 0) • xs.toFun fun i => Function.update z 0 (s • u) i.succ) =
s • smulCmmapFn x xs (Function.update z 0 u) | case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
i0 : i = 0
su : (x fun x => s • u) = s • x fun x => u
⊢ ((x.toFun fun x => Function.update z 0 (s • u) 0) • xs.toFun fun i => Function.update z 0 (s • u) i.succ) =
s • smulCmmapFn x xs (Function.update z 0 u) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
i0 : i = 0
su : x (Function.update (fun x => z 0) 0 (s • u)) = s • x (Function.update (fun x => z 0) 0 u)
⊢ ((x.toFun fun x => Function.update z 0 (s • u) 0) • xs.toFun fun i => Function.update z 0 (s • u) i.succ) =
s • smulCmmapFn x xs (Function.update z 0 u)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smul_cmmap_smul | [138, 1] | [154, 83] | simp only [Function.update_same, MultilinearMap.toFun_eq_coe, ContinuousMultilinearMap.coe_coe,
su, smul_eq_mul, ne_eq, update_0_succ d z _ _, smulCmmapFn, ←smul_assoc] | case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
i0 : i = 0
su : (x fun x => s • u) = s • x fun x => u
⊢ ((x.toFun fun x => Function.update z 0 (s • u) 0) • xs.toFun fun i => Function.update z 0 (s • u) i.succ) =
s • smulCmmapFn x xs (Function.update z 0 u) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
i0 : i = 0
su : (x fun x => s • u) = s • x fun x => u
⊢ ((x.toFun fun x => Function.update z 0 (s • u) 0) • xs.toFun fun i => Function.update z 0 (s • u) i.succ) =
s • smulCmmapFn x xs (Function.update z 0 u)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smul_cmmap_smul | [138, 1] | [154, 83] | have su := xs.map_smul (fun j ↦ z j.succ) (i.pred i0) s u | case neg
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
i0 : ¬i = 0
⊢ ((x.toFun fun x => Function.update z i (s • u) 0) • xs.toFun fun i_1 => Function.update z i (s • u) i_1.succ) =
s • smulCmmapFn x xs (Function.update z i u) | case neg
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
i0 : ¬i = 0
su :
xs (Function.update (fun j => z j.succ) (i.pred i0) (s • u)) =
s • xs (Function.update (fun j => z j.succ) (i.pred i0) u)
⊢ ((x.toFun fun x => Function.update z i (s • u) 0) • xs.toFun fun i_1 => Function.update z i (s • u) i_1.succ) =
s • smulCmmapFn x xs (Function.update z i u) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
i0 : ¬i = 0
⊢ ((x.toFun fun x => Function.update z i (s • u) 0) • xs.toFun fun i_1 => Function.update z i (s • u) i_1.succ) =
s • smulCmmapFn x xs (Function.update z i u)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Multilinear.lean | smul_cmmap_smul | [138, 1] | [154, 83] | simp only [ne_eq, MultilinearMap.toFun_eq_coe, ContinuousMultilinearMap.coe_coe,
update_nz_0 d z i0, update_nz_succ d z _ i0, su, smul_comm _ s, smulCmmapFn] | case neg
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
i0 : ¬i = 0
su :
xs (Function.update (fun j => z j.succ) (i.pred i0) (s • u)) =
s • xs (Function.update (fun j => z j.succ) (i.pred i0) u)
⊢ ((x.toFun fun x => Function.update z i (s • u) 0) • xs.toFun fun i_1 => Function.update z i (s • u) i_1.succ) =
s • smulCmmapFn x xs (Function.update z i u) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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
d : DecidableEq (Fin (n + 1))
z : Fin (n + 1) → A
i : Fin (n + 1)
s : 𝕜
u : A
i0 : ¬i = 0
su :
xs (Function.update (fun j => z j.succ) (i.pred i0) (s • u)) =
s • xs (Function.update (fun j => z j.succ) (i.pred i0) u)
⊢ ((x.toFun fun x => Function.update z i (s • u) 0) • xs.toFun fun i_1 => Function.update z i (s • u) i_1.succ) =
s • smulCmmapFn x xs (Function.update z i u)
TACTIC:
|
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:
|
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