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stringlengths 6
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|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_mfderiv_inf_ne_zero | [184, 1] | [195, 40] | use 1 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∃ x, ¬(ContinuousLinearMap.smulRight 1 1) x = 0 x | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ¬(ContinuousLinearMap.smulRight 1 1) 1 = 0 1 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∃ x, ¬(ContinuousLinearMap.smulRight 1 1) x = 0 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_mfderiv_inf_ne_zero | [184, 1] | [195, 40] | simp only [ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply,
Algebra.id.smul_eq_mul, mul_one, ContinuousLinearMap.zero_apply] | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ¬(ContinuousLinearMap.smulRight 1 1) 1 = 0 1 | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ¬1 = 0 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ¬(ContinuousLinearMap.smulRight 1 1) 1 = 0 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_mfderiv_inf_ne_zero | [184, 1] | [195, 40] | convert one_ne_zero | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ¬1 = 0 1 | case h.convert_4
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ NeZero 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ¬1 = 0 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_mfderiv_inf_ne_zero | [184, 1] | [195, 40] | exact NeZero.one | case h.convert_4
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ NeZero 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.convert_4
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ NeZero 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | tangentSpace_norm_eq_complex_norm | [52, 1] | [53, 29] | rw [← Complex.norm_eq_abs] | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
x : TangentSpace I z
⊢ ‖x‖ = Complex.abs x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
x : TangentSpace I z
⊢ ‖x‖ = Complex.abs x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_eq_zero_iff | [100, 1] | [112, 48] | constructor | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
⊢ f u = 0 ↔ f = 0 ∨ u = 0 | case mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
⊢ f u = 0 → f = 0 ∨ u = 0
case mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
⊢ f = 0 ∨ u = 0 → f u = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
⊢ f u = 0 ↔ f = 0 ∨ u = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_eq_zero_iff | [100, 1] | [112, 48] | rw [or_iff_not_imp_right] | case mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
⊢ f u = 0 → f = 0 ∨ u = 0 | case mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
⊢ f u = 0 → ¬u = 0 → f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
⊢ f u = 0 → f = 0 ∨ u = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_eq_zero_iff | [100, 1] | [112, 48] | intro f0 u0 | case mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
⊢ f u = 0 → ¬u = 0 → f = 0 | case mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
f0 : f u = 0
u0 : ¬u = 0
⊢ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
⊢ f u = 0 → ¬u = 0 → f = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_eq_zero_iff | [100, 1] | [112, 48] | apply ContinuousLinearMap.ext | case mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
f0 : f u = 0
u0 : ¬u = 0
⊢ f = 0 | case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
f0 : f u = 0
u0 : ¬u = 0
⊢ ∀ (x : TangentSpace I z), f x = 0 x | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
f0 : f u = 0
u0 : ¬u = 0
⊢ f = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_eq_zero_iff | [100, 1] | [112, 48] | intro v | case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
f0 : f u = 0
u0 : ¬u = 0
⊢ ∀ (x : TangentSpace I z), f x = 0 x | case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
f0 : f u = 0
u0 : ¬u = 0
v : TangentSpace I z
⊢ f v = 0 v | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
f0 : f u = 0
u0 : ¬u = 0
⊢ ∀ (x : TangentSpace I z), f x = 0 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_eq_zero_iff | [100, 1] | [112, 48] | simp only [TangentSpace] at f u v u0 | case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
f0 : f u = 0
u0 : ¬u = 0
v : TangentSpace I z
⊢ f v = 0 v | case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : ℂ →L[ℂ] ℂ
u : ℂ
f0 : f u = 0
u0 : ¬u = 0
v : ℂ
⊢ f v = 0 v | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
f0 : f u = 0
u0 : ¬u = 0
v : TangentSpace I z
⊢ f v = 0 v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_eq_zero_iff | [100, 1] | [112, 48] | have e : v = (v * u⁻¹) • u := by simp only [smul_eq_mul, mul_assoc, inv_mul_cancel u0, mul_one] | case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : ℂ →L[ℂ] ℂ
u : ℂ
f0 : f u = 0
u0 : ¬u = 0
v : ℂ
⊢ f v = 0 v | case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : ℂ →L[ℂ] ℂ
u : ℂ
f0 : f u = 0
u0 : ¬u = 0
v : ℂ
e : v = (v * u⁻¹) • u
⊢ f v = 0 v | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : ℂ →L[ℂ] ℂ
u : ℂ
f0 : f u = 0
u0 : ¬u = 0
v : ℂ
⊢ f v = 0 v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_eq_zero_iff | [100, 1] | [112, 48] | rw [ContinuousLinearMap.zero_apply, e] | case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : ℂ →L[ℂ] ℂ
u : ℂ
f0 : f u = 0
u0 : ¬u = 0
v : ℂ
e : v = (v * u⁻¹) • u
⊢ f v = 0 v | case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : ℂ →L[ℂ] ℂ
u : ℂ
f0 : f u = 0
u0 : ¬u = 0
v : ℂ
e : v = (v * u⁻¹) • u
⊢ f ((v * u⁻¹) • u) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : ℂ →L[ℂ] ℂ
u : ℂ
f0 : f u = 0
u0 : ¬u = 0
v : ℂ
e : v = (v * u⁻¹) • u
⊢ f v = 0 v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_eq_zero_iff | [100, 1] | [112, 48] | refine Eq.trans (f.map_smul _ _) ?_ | case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : ℂ →L[ℂ] ℂ
u : ℂ
f0 : f u = 0
u0 : ¬u = 0
v : ℂ
e : v = (v * u⁻¹) • u
⊢ f ((v * u⁻¹) • u) = 0 | case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : ℂ →L[ℂ] ℂ
u : ℂ
f0 : f u = 0
u0 : ¬u = 0
v : ℂ
e : v = (v * u⁻¹) • u
⊢ (v * u⁻¹) • f u = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : ℂ →L[ℂ] ℂ
u : ℂ
f0 : f u = 0
u0 : ¬u = 0
v : ℂ
e : v = (v * u⁻¹) • u
⊢ f ((v * u⁻¹) • u) = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_eq_zero_iff | [100, 1] | [112, 48] | rw [smul_eq_zero] | case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : ℂ →L[ℂ] ℂ
u : ℂ
f0 : f u = 0
u0 : ¬u = 0
v : ℂ
e : v = (v * u⁻¹) • u
⊢ (v * u⁻¹) • f u = 0 | case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : ℂ →L[ℂ] ℂ
u : ℂ
f0 : f u = 0
u0 : ¬u = 0
v : ℂ
e : v = (v * u⁻¹) • u
⊢ v * u⁻¹ = 0 ∨ f u = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : ℂ →L[ℂ] ℂ
u : ℂ
f0 : f u = 0
u0 : ¬u = 0
v : ℂ
e : v = (v * u⁻¹) • u
⊢ (v * u⁻¹) • f u = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_eq_zero_iff | [100, 1] | [112, 48] | right | case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : ℂ →L[ℂ] ℂ
u : ℂ
f0 : f u = 0
u0 : ¬u = 0
v : ℂ
e : v = (v * u⁻¹) • u
⊢ v * u⁻¹ = 0 ∨ f u = 0 | case mp.h.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : ℂ →L[ℂ] ℂ
u : ℂ
f0 : f u = 0
u0 : ¬u = 0
v : ℂ
e : v = (v * u⁻¹) • u
⊢ f u = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : ℂ →L[ℂ] ℂ
u : ℂ
f0 : f u = 0
u0 : ¬u = 0
v : ℂ
e : v = (v * u⁻¹) • u
⊢ v * u⁻¹ = 0 ∨ f u = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_eq_zero_iff | [100, 1] | [112, 48] | exact f0 | case mp.h.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : ℂ →L[ℂ] ℂ
u : ℂ
f0 : f u = 0
u0 : ¬u = 0
v : ℂ
e : v = (v * u⁻¹) • u
⊢ f u = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.h.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : ℂ →L[ℂ] ℂ
u : ℂ
f0 : f u = 0
u0 : ¬u = 0
v : ℂ
e : v = (v * u⁻¹) • u
⊢ f u = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_eq_zero_iff | [100, 1] | [112, 48] | simp only [smul_eq_mul, mul_assoc, inv_mul_cancel u0, mul_one] | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : ℂ →L[ℂ] ℂ
u : ℂ
f0 : f u = 0
u0 : ¬u = 0
v : ℂ
⊢ v = (v * u⁻¹) • u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : ℂ →L[ℂ] ℂ
u : ℂ
f0 : f u = 0
u0 : ¬u = 0
v : ℂ
⊢ v = (v * u⁻¹) • u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_eq_zero_iff | [100, 1] | [112, 48] | intro h | case mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
⊢ f = 0 ∨ u = 0 → f u = 0 | case mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
h : f = 0 ∨ u = 0
⊢ f u = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
⊢ f = 0 ∨ u = 0 → f u = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_eq_zero_iff | [100, 1] | [112, 48] | cases' h with h h | case mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
h : f = 0 ∨ u = 0
⊢ f u = 0 | case mpr.inl
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
h : f = 0
⊢ f u = 0
case mpr.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
h : u = 0
⊢ f u = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
h : f = 0 ∨ u = 0
⊢ f u = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_eq_zero_iff | [100, 1] | [112, 48] | simp only [h, ContinuousLinearMap.zero_apply] | case mpr.inl
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
h : f = 0
⊢ f u = 0
case mpr.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
h : u = 0
⊢ f u = 0 | case mpr.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
h : u = 0
⊢ f u = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inl
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
h : f = 0
⊢ f u = 0
case mpr.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
h : u = 0
⊢ f u = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_eq_zero_iff | [100, 1] | [112, 48] | simp only [h, ContinuousLinearMap.map_zero] | case mpr.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
h : u = 0
⊢ f u = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
h : u = 0
⊢ f u = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_eq_zero_iff' | [115, 1] | [117, 51] | simp only [mderiv_eq_zero_iff, u0, or_false_iff] | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
u0 : u ≠ 0
⊢ f u = 0 ↔ f = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
u0 : u ≠ 0
⊢ f u = 0 ↔ f = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_ne_zero_iff | [120, 1] | [122, 63] | simp only [← not_or] | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
⊢ f u ≠ 0 ↔ f ≠ 0 ∧ u ≠ 0 | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
⊢ f u ≠ 0 ↔ ¬(f = 0 ∨ u = 0) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
⊢ f u ≠ 0 ↔ f ≠ 0 ∧ u ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_ne_zero_iff | [120, 1] | [122, 63] | exact Iff.not (mderiv_eq_zero_iff _ _) | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
⊢ f u ≠ 0 ↔ ¬(f = 0 ∨ u = 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
⊢ f u ≠ 0 ↔ ¬(f = 0 ∨ u = 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_ne_zero_iff' | [125, 1] | [127, 73] | simp only [ne_eq, mderiv_ne_zero_iff, u0, not_false_eq_true, and_true] | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
u0 : u ≠ 0
⊢ f u ≠ 0 ↔ f ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
u : TangentSpace I z
u0 : u ≠ 0
⊢ f u ≠ 0 ↔ f ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_comp_eq_zero_iff | [130, 1] | [139, 87] | rcases exists_ne (0 : TangentSpace I x) with ⟨t, t0⟩ | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
⊢ f.comp g = 0 ↔ f = 0 ∨ g = 0 | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
⊢ f.comp g = 0 ↔ f = 0 ∨ g = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
⊢ f.comp g = 0 ↔ f = 0 ∨ g = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_comp_eq_zero_iff | [130, 1] | [139, 87] | constructor | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
⊢ f.comp g = 0 ↔ f = 0 ∨ g = 0 | case intro.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
⊢ f.comp g = 0 → f = 0 ∨ g = 0
case intro.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
⊢ f = 0 ∨ g = 0 → f.comp g = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
⊢ f.comp g = 0 ↔ f = 0 ∨ g = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_comp_eq_zero_iff | [130, 1] | [139, 87] | intro h | case intro.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
⊢ f.comp g = 0 → f = 0 ∨ g = 0 | case intro.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f.comp g = 0
⊢ f = 0 ∨ g = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
⊢ f.comp g = 0 → f = 0 ∨ g = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_comp_eq_zero_iff | [130, 1] | [139, 87] | simp only [← mderiv_eq_zero_iff' t0, ContinuousLinearMap.comp_apply] at h | case intro.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f.comp g = 0
⊢ f = 0 ∨ g = 0 | case intro.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
⊢ f = 0 ∨ g = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f.comp g = 0
⊢ f = 0 ∨ g = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_comp_eq_zero_iff | [130, 1] | [139, 87] | by_cases g0 : g t = 0 | case intro.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
⊢ f = 0 ∨ g = 0 | case pos
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : g t = 0
⊢ f = 0 ∨ g = 0
case neg
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : ¬g t = 0
⊢ f = 0 ∨ g = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
⊢ f = 0 ∨ g = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_comp_eq_zero_iff | [130, 1] | [139, 87] | right | case pos
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : g t = 0
⊢ f = 0 ∨ g = 0
case neg
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : ¬g t = 0
⊢ f = 0 ∨ g = 0 | case pos.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : g t = 0
⊢ g = 0
case neg
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : ¬g t = 0
⊢ f = 0 ∨ g = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : g t = 0
⊢ f = 0 ∨ g = 0
case neg
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : ¬g t = 0
⊢ f = 0 ∨ g = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_comp_eq_zero_iff | [130, 1] | [139, 87] | rw [mderiv_eq_zero_iff' t0] at g0 | case pos.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : g t = 0
⊢ g = 0
case neg
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : ¬g t = 0
⊢ f = 0 ∨ g = 0 | case pos.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : g = 0
⊢ g = 0
case neg
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : ¬g t = 0
⊢ f = 0 ∨ g = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : g t = 0
⊢ g = 0
case neg
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : ¬g t = 0
⊢ f = 0 ∨ g = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_comp_eq_zero_iff | [130, 1] | [139, 87] | exact g0 | case pos.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : g = 0
⊢ g = 0
case neg
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : ¬g t = 0
⊢ f = 0 ∨ g = 0 | case neg
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : ¬g t = 0
⊢ f = 0 ∨ g = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : g = 0
⊢ g = 0
case neg
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : ¬g t = 0
⊢ f = 0 ∨ g = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_comp_eq_zero_iff | [130, 1] | [139, 87] | left | case neg
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : ¬g t = 0
⊢ f = 0 ∨ g = 0 | case neg.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : ¬g t = 0
⊢ f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : ¬g t = 0
⊢ f = 0 ∨ g = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_comp_eq_zero_iff | [130, 1] | [139, 87] | rwa [← mderiv_eq_zero_iff' g0] | case neg.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : ¬g t = 0
⊢ f = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f (g t) = 0
g0 : ¬g t = 0
⊢ f = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_comp_eq_zero_iff | [130, 1] | [139, 87] | intro h | case intro.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
⊢ f = 0 ∨ g = 0 → f.comp g = 0 | case intro.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f = 0 ∨ g = 0
⊢ f.comp g = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
⊢ f = 0 ∨ g = 0 → f.comp g = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_comp_eq_zero_iff | [130, 1] | [139, 87] | cases' h with h h | case intro.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f = 0 ∨ g = 0
⊢ f.comp g = 0 | case intro.mpr.inl
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f = 0
⊢ f.comp g = 0
case intro.mpr.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : g = 0
⊢ f.comp g = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f = 0 ∨ g = 0
⊢ f.comp g = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_comp_eq_zero_iff | [130, 1] | [139, 87] | simp only [h, g.zero_comp] | case intro.mpr.inl
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f = 0
⊢ f.comp g = 0
case intro.mpr.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : g = 0
⊢ f.comp g = 0 | case intro.mpr.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : g = 0
⊢ f.comp g = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mpr.inl
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : f = 0
⊢ f.comp g = 0
case intro.mpr.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : g = 0
⊢ f.comp g = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_comp_eq_zero_iff | [130, 1] | [139, 87] | simp only [h, f.comp_zero] | case intro.mpr.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : g = 0
⊢ f.comp g = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mpr.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
t : TangentSpace I x
t0 : t ≠ 0
h : g = 0
⊢ f.comp g = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_comp_ne_zero | [142, 1] | [144, 91] | intro f0 g0 | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
⊢ f ≠ 0 → g ≠ 0 → f.comp g ≠ 0 | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
f0 : f ≠ 0
g0 : g ≠ 0
⊢ f.comp g ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
⊢ f ≠ 0 → g ≠ 0 → f.comp g ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_comp_ne_zero | [142, 1] | [144, 91] | simp only [Ne, mderiv_comp_eq_zero_iff, f0, g0, or_self_iff, not_false_iff] | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
f0 : f ≠ 0
g0 : g ≠ 0
⊢ f.comp g ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
x : S
y : T
z : U
f : TangentSpace I y →L[ℂ] TangentSpace I z
g : TangentSpace I x →L[ℂ] TangentSpace I y
f0 : f ≠ 0
g0 : g ≠ 0
⊢ f.comp g ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | has_mfderiv_at_of_mderiv_ne_zero | [147, 1] | [150, 83] | contrapose d0 | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
x : S
d0 : mfderiv I I f x ≠ 0
⊢ MDifferentiableAt I I f x | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
x : S
d0 : ¬MDifferentiableAt I I f x
⊢ ¬mfderiv I I f x ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
x : S
d0 : mfderiv I I f x ≠ 0
⊢ MDifferentiableAt I I f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | has_mfderiv_at_of_mderiv_ne_zero | [147, 1] | [150, 83] | simp only [mfderiv, d0, if_false, Ne, eq_self_iff_true, not_true, not_false_iff] | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
x : S
d0 : ¬MDifferentiableAt I I f x
⊢ ¬mfderiv I I f x ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
x : S
d0 : ¬MDifferentiableAt I I f x
⊢ ¬mfderiv I I f x ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_comp_ne_zero' | [153, 1] | [158, 38] | intro df dg | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : T → U
g : S → T
x : S
⊢ mfderiv I I f (g x) ≠ 0 → mfderiv I I g x ≠ 0 → mfderiv I I (fun x => f (g x)) x ≠ 0 | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : T → U
g : S → T
x : S
df : mfderiv I I f (g x) ≠ 0
dg : mfderiv I I g x ≠ 0
⊢ mfderiv I I (fun x => f (g x)) x ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : T → U
g : S → T
x : S
⊢ mfderiv I I f (g x) ≠ 0 → mfderiv I I g x ≠ 0 → mfderiv I I (fun x => f (g x)) x ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_comp_ne_zero' | [153, 1] | [158, 38] | have e : (fun x ↦ f (g x)) = f ∘ g := rfl | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : T → U
g : S → T
x : S
df : mfderiv I I f (g x) ≠ 0
dg : mfderiv I I g x ≠ 0
⊢ mfderiv I I (fun x => f (g x)) x ≠ 0 | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : T → U
g : S → T
x : S
df : mfderiv I I f (g x) ≠ 0
dg : mfderiv I I g x ≠ 0
e : (fun x => f (g x)) = f ∘ g
⊢ mfderiv I I (fun x => f (g x)) x ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : T → U
g : S → T
x : S
df : mfderiv I I f (g x) ≠ 0
dg : mfderiv I I g x ≠ 0
⊢ mfderiv I I (fun x => f (g x)) x ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_comp_ne_zero' | [153, 1] | [158, 38] | rw [e, mfderiv_comp x (has_mfderiv_at_of_mderiv_ne_zero df) (has_mfderiv_at_of_mderiv_ne_zero dg)] | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : T → U
g : S → T
x : S
df : mfderiv I I f (g x) ≠ 0
dg : mfderiv I I g x ≠ 0
e : (fun x => f (g x)) = f ∘ g
⊢ mfderiv I I (fun x => f (g x)) x ≠ 0 | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : T → U
g : S → T
x : S
df : mfderiv I I f (g x) ≠ 0
dg : mfderiv I I g x ≠ 0
e : (fun x => f (g x)) = f ∘ g
⊢ (mfderiv I I f (g x)).comp (mfderiv I I g x) ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : T → U
g : S → T
x : S
df : mfderiv I I f (g x) ≠ 0
dg : mfderiv I I g x ≠ 0
e : (fun x => f (g x)) = f ∘ g
⊢ mfderiv I I (fun x => f (g x)) x ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderiv_comp_ne_zero' | [153, 1] | [158, 38] | exact mderiv_comp_ne_zero _ _ df dg | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : T → U
g : S → T
x : S
df : mfderiv I I f (g x) ≠ 0
dg : mfderiv I I g x ≠ 0
e : (fun x => f (g x)) = f ∘ g
⊢ (mfderiv I I f (g x)).comp (mfderiv I I g x) ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : T → U
g : S → T
x : S
df : mfderiv I I f (g x) ≠ 0
dg : mfderiv I I g x ≠ 0
e : (fun x => f (g x)) = f ∘ g
⊢ (mfderiv I I f (g x)).comp (mfderiv I I g x) ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderivEquiv_eq | [194, 1] | [195, 78] | apply ContinuousLinearMap.ext | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
f0 : f ≠ 0
⊢ ↑(mderivEquiv f f0) = f | case h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
f0 : f ≠ 0
⊢ ∀ (x : TangentSpace I z), ↑(mderivEquiv f f0) x = f x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
f0 : f ≠ 0
⊢ ↑(mderivEquiv f f0) = f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderivEquiv_eq | [194, 1] | [195, 78] | intro x | case h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
f0 : f ≠ 0
⊢ ∀ (x : TangentSpace I z), ↑(mderivEquiv f f0) x = f x | case h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
f0 : f ≠ 0
x : TangentSpace I z
⊢ ↑(mderivEquiv f f0) x = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
f0 : f ≠ 0
⊢ ∀ (x : TangentSpace I z), ↑(mderivEquiv f f0) x = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mderivEquiv_eq | [194, 1] | [195, 78] | rfl | case h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
f0 : f ≠ 0
x : TangentSpace I z
⊢ ↑(mderivEquiv f f0) x = f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
f0 : f ≠ 0
x : TangentSpace I z
⊢ ↑(mderivEquiv f f0) x = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | extChartAt_mderiv_ne_zero' | [198, 1] | [205, 11] | rcases exists_ne (0 : TangentSpace I z) with ⟨t, t0⟩ | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
⊢ mfderiv I I (↑(extChartAt I z)) w ≠ 0 | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
t : TangentSpace I z
t0 : t ≠ 0
⊢ mfderiv I I (↑(extChartAt I z)) w ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
⊢ mfderiv I I (↑(extChartAt I z)) w ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | extChartAt_mderiv_ne_zero' | [198, 1] | [205, 11] | rw [← mderiv_ne_zero_iff' t0] | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
t : TangentSpace I z
t0 : t ≠ 0
⊢ mfderiv I I (↑(extChartAt I z)) w ≠ 0 | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
t : TangentSpace I z
t0 : t ≠ 0
⊢ (mfderiv I I (↑(extChartAt I z)) w) t ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
t : TangentSpace I z
t0 : t ≠ 0
⊢ mfderiv I I (↑(extChartAt I z)) w ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | extChartAt_mderiv_ne_zero' | [198, 1] | [205, 11] | contrapose t0 | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
t : TangentSpace I z
t0 : t ≠ 0
⊢ (mfderiv I I (↑(extChartAt I z)) w) t ≠ 0 | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
t : TangentSpace I z
t0 : ¬(mfderiv I I (↑(extChartAt I z)) w) t ≠ 0
⊢ ¬t ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
t : TangentSpace I z
t0 : t ≠ 0
⊢ (mfderiv I I (↑(extChartAt I z)) w) t ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | extChartAt_mderiv_ne_zero' | [198, 1] | [205, 11] | simp only [not_not] at t0 ⊢ | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
t : TangentSpace I z
t0 : ¬(mfderiv I I (↑(extChartAt I z)) w) t ≠ 0
⊢ ¬t ≠ 0 | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
t : TangentSpace I z
t0 : (mfderiv I I (↑(extChartAt I z)) w) t = 0
⊢ t = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
t : TangentSpace I z
t0 : ¬(mfderiv I I (↑(extChartAt I z)) w) t ≠ 0
⊢ ¬t ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | extChartAt_mderiv_ne_zero' | [198, 1] | [205, 11] | have h := ContinuousLinearMap.ext_iff.mp (extChartAt_mderiv_left_inverse m) t | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
t : TangentSpace I z
t0 : (mfderiv I I (↑(extChartAt I z)) w) t = 0
⊢ t = 0 | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
t : TangentSpace I z
t0 : (mfderiv I I (↑(extChartAt I z)) w) t = 0
h :
((mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w)).comp (mfderiv I I (↑(extChartAt I z)) w)) t =
(ContinuousLinearMap.id ℂ (TangentSpace I w)) t
⊢ t = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
t : TangentSpace I z
t0 : (mfderiv I I (↑(extChartAt I z)) w) t = 0
⊢ t = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | extChartAt_mderiv_ne_zero' | [198, 1] | [205, 11] | simp only [ContinuousLinearMap.comp_apply, ContinuousLinearMap.map_zero] at h | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
t : TangentSpace I z
t0 : (mfderiv I I (↑(extChartAt I z)) w) t = 0
h :
((mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w)).comp (mfderiv I I (↑(extChartAt I z)) w)) t =
(ContinuousLinearMap.id ℂ (TangentSpace I w)) t
⊢ t = 0 | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
t : TangentSpace I z
t0 : (mfderiv I I (↑(extChartAt I z)) w) t = 0
h :
(mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w)) ((mfderiv I I (↑(extChartAt I z)) w) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I w)) t
⊢ t = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
t : TangentSpace I z
t0 : (mfderiv I I (↑(extChartAt I z)) w) t = 0
h :
((mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w)).comp (mfderiv I I (↑(extChartAt I z)) w)) t =
(ContinuousLinearMap.id ℂ (TangentSpace I w)) t
⊢ t = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | extChartAt_mderiv_ne_zero' | [198, 1] | [205, 11] | rw [←h.trans (ContinuousLinearMap.id_apply _), ContinuousLinearMap.apply_eq_zero_of_eq_zero] | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
t : TangentSpace I z
t0 : (mfderiv I I (↑(extChartAt I z)) w) t = 0
h :
(mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w)) ((mfderiv I I (↑(extChartAt I z)) w) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I w)) t
⊢ t = 0 | case intro.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
t : TangentSpace I z
t0 : (mfderiv I I (↑(extChartAt I z)) w) t = 0
h :
(mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w)) ((mfderiv I I (↑(extChartAt I z)) w) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I w)) t
⊢ (mfderiv I I (↑(extChartAt I z)) w) t = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
t : TangentSpace I z
t0 : (mfderiv I I (↑(extChartAt I z)) w) t = 0
h :
(mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w)) ((mfderiv I I (↑(extChartAt I z)) w) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I w)) t
⊢ t = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | extChartAt_mderiv_ne_zero' | [198, 1] | [205, 11] | exact t0 | case intro.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
t : TangentSpace I z
t0 : (mfderiv I I (↑(extChartAt I z)) w) t = 0
h :
(mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w)) ((mfderiv I I (↑(extChartAt I z)) w) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I w)) t
⊢ (mfderiv I I (↑(extChartAt I z)) w) t = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z w : S
m : w ∈ (extChartAt I z).source
t : TangentSpace I z
t0 : (mfderiv I I (↑(extChartAt I z)) w) t = 0
h :
(mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w)) ((mfderiv I I (↑(extChartAt I z)) w) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I w)) t
⊢ (mfderiv I I (↑(extChartAt I z)) w) t = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | extChartAt_symm_mderiv_ne_zero' | [208, 1] | [215, 11] | rcases exists_ne (0 : TangentSpace I (extChartAt I z z)) with ⟨t, t0⟩ | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
⊢ mfderiv I I (↑(extChartAt I z).symm) w ≠ 0 | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
t : TangentSpace I (↑(extChartAt I z) z)
t0 : t ≠ 0
⊢ mfderiv I I (↑(extChartAt I z).symm) w ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
⊢ mfderiv I I (↑(extChartAt I z).symm) w ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | extChartAt_symm_mderiv_ne_zero' | [208, 1] | [215, 11] | rw [← mderiv_ne_zero_iff' t0] | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
t : TangentSpace I (↑(extChartAt I z) z)
t0 : t ≠ 0
⊢ mfderiv I I (↑(extChartAt I z).symm) w ≠ 0 | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
t : TangentSpace I (↑(extChartAt I z) z)
t0 : t ≠ 0
⊢ (mfderiv I I (↑(extChartAt I z).symm) w) t ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
t : TangentSpace I (↑(extChartAt I z) z)
t0 : t ≠ 0
⊢ mfderiv I I (↑(extChartAt I z).symm) w ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | extChartAt_symm_mderiv_ne_zero' | [208, 1] | [215, 11] | contrapose t0 | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
t : TangentSpace I (↑(extChartAt I z) z)
t0 : t ≠ 0
⊢ (mfderiv I I (↑(extChartAt I z).symm) w) t ≠ 0 | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
t : TangentSpace I (↑(extChartAt I z) z)
t0 : ¬(mfderiv I I (↑(extChartAt I z).symm) w) t ≠ 0
⊢ ¬t ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
t : TangentSpace I (↑(extChartAt I z) z)
t0 : t ≠ 0
⊢ (mfderiv I I (↑(extChartAt I z).symm) w) t ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | extChartAt_symm_mderiv_ne_zero' | [208, 1] | [215, 11] | simp only [not_not] at t0 ⊢ | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
t : TangentSpace I (↑(extChartAt I z) z)
t0 : ¬(mfderiv I I (↑(extChartAt I z).symm) w) t ≠ 0
⊢ ¬t ≠ 0 | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
t : TangentSpace I (↑(extChartAt I z) z)
t0 : (mfderiv I I (↑(extChartAt I z).symm) w) t = 0
⊢ t = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
t : TangentSpace I (↑(extChartAt I z) z)
t0 : ¬(mfderiv I I (↑(extChartAt I z).symm) w) t ≠ 0
⊢ ¬t ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | extChartAt_symm_mderiv_ne_zero' | [208, 1] | [215, 11] | have h := ContinuousLinearMap.ext_iff.mp (extChartAt_mderiv_right_inverse m) t | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
t : TangentSpace I (↑(extChartAt I z) z)
t0 : (mfderiv I I (↑(extChartAt I z).symm) w) t = 0
⊢ t = 0 | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
t : TangentSpace I (↑(extChartAt I z) z)
t0 : (mfderiv I I (↑(extChartAt I z).symm) w) t = 0
h :
((mfderiv I I (↑(extChartAt I z)) (↑(extChartAt I z).symm w)).comp (mfderiv I I (↑(extChartAt I z).symm) w)) t =
(ContinuousLinearMap.id ℂ (TangentSpace I w)) t
⊢ t = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
t : TangentSpace I (↑(extChartAt I z) z)
t0 : (mfderiv I I (↑(extChartAt I z).symm) w) t = 0
⊢ t = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | extChartAt_symm_mderiv_ne_zero' | [208, 1] | [215, 11] | simp only [ContinuousLinearMap.comp_apply, ContinuousLinearMap.map_zero] at h | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
t : TangentSpace I (↑(extChartAt I z) z)
t0 : (mfderiv I I (↑(extChartAt I z).symm) w) t = 0
h :
((mfderiv I I (↑(extChartAt I z)) (↑(extChartAt I z).symm w)).comp (mfderiv I I (↑(extChartAt I z).symm) w)) t =
(ContinuousLinearMap.id ℂ (TangentSpace I w)) t
⊢ t = 0 | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
t : TangentSpace I (↑(extChartAt I z) z)
t0 : (mfderiv I I (↑(extChartAt I z).symm) w) t = 0
h :
(mfderiv I I (↑(extChartAt I z)) (↑(extChartAt I z).symm w)) ((mfderiv I I (↑(extChartAt I z).symm) w) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I w)) t
⊢ t = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
t : TangentSpace I (↑(extChartAt I z) z)
t0 : (mfderiv I I (↑(extChartAt I z).symm) w) t = 0
h :
((mfderiv I I (↑(extChartAt I z)) (↑(extChartAt I z).symm w)).comp (mfderiv I I (↑(extChartAt I z).symm) w)) t =
(ContinuousLinearMap.id ℂ (TangentSpace I w)) t
⊢ t = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | extChartAt_symm_mderiv_ne_zero' | [208, 1] | [215, 11] | rw [←h.trans (ContinuousLinearMap.id_apply _), ContinuousLinearMap.apply_eq_zero_of_eq_zero] | case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
t : TangentSpace I (↑(extChartAt I z) z)
t0 : (mfderiv I I (↑(extChartAt I z).symm) w) t = 0
h :
(mfderiv I I (↑(extChartAt I z)) (↑(extChartAt I z).symm w)) ((mfderiv I I (↑(extChartAt I z).symm) w) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I w)) t
⊢ t = 0 | case intro.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
t : TangentSpace I (↑(extChartAt I z) z)
t0 : (mfderiv I I (↑(extChartAt I z).symm) w) t = 0
h :
(mfderiv I I (↑(extChartAt I z)) (↑(extChartAt I z).symm w)) ((mfderiv I I (↑(extChartAt I z).symm) w) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I w)) t
⊢ (mfderiv I I (↑(extChartAt I z).symm) w) t = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
t : TangentSpace I (↑(extChartAt I z) z)
t0 : (mfderiv I I (↑(extChartAt I z).symm) w) t = 0
h :
(mfderiv I I (↑(extChartAt I z)) (↑(extChartAt I z).symm w)) ((mfderiv I I (↑(extChartAt I z).symm) w) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I w)) t
⊢ t = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | extChartAt_symm_mderiv_ne_zero' | [208, 1] | [215, 11] | exact t0 | case intro.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
t : TangentSpace I (↑(extChartAt I z) z)
t0 : (mfderiv I I (↑(extChartAt I z).symm) w) t = 0
h :
(mfderiv I I (↑(extChartAt I z)) (↑(extChartAt I z).symm w)) ((mfderiv I I (↑(extChartAt I z).symm) w) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I w)) t
⊢ (mfderiv I I (↑(extChartAt I z).symm) w) t = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : ℂ
m : w ∈ (extChartAt I z).target
t : TangentSpace I (↑(extChartAt I z) z)
t0 : (mfderiv I I (↑(extChartAt I z).symm) w) t = 0
h :
(mfderiv I I (↑(extChartAt I z)) (↑(extChartAt I z).symm w)) ((mfderiv I I (↑(extChartAt I z).symm) w) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I w)) t
⊢ (mfderiv I I (↑(extChartAt I z).symm) w) t = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | id_mderiv_ne_zero | [227, 1] | [237, 21] | have d : MDifferentiableAt I I (fun z ↦ z) z := mdifferentiableAt_id I | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
⊢ mfderiv I I (fun z => z) z ≠ 0 | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
d : MDifferentiableAt I I (fun z => z) z
⊢ mfderiv I I (fun z => z) z ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
⊢ mfderiv I I (fun z => z) z ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | id_mderiv_ne_zero | [227, 1] | [237, 21] | simp only [mfderiv, d, if_true, writtenInExtChartAt, Function.comp,
ModelWithCorners.Boundaryless.range_eq_univ, fderivWithin_univ] | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
d : MDifferentiableAt I I (fun z => z) z
⊢ mfderiv I I (fun z => z) z ≠ 0 | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
d : MDifferentiableAt I I (fun z => z) z
⊢ fderiv ℂ (fun x => ↑(extChartAt I z) (↑(extChartAt I z).symm x)) (↑(extChartAt I z) z) ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
d : MDifferentiableAt I I (fun z => z) z
⊢ mfderiv I I (fun z => z) z ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | id_mderiv_ne_zero | [227, 1] | [237, 21] | have e : (fun w ↦ extChartAt I z ((extChartAt I z).symm w)) =ᶠ[𝓝 (extChartAt I z z)] id := by
apply ((isOpen_extChartAt_target I z).eventually_mem (mem_extChartAt_target I z)).mp
refine eventually_of_forall fun w m ↦ ?_
simp only [id, PartialEquiv.right_inv _ m] | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
d : MDifferentiableAt I I (fun z => z) z
⊢ fderiv ℂ (fun x => ↑(extChartAt I z) (↑(extChartAt I z).symm x)) (↑(extChartAt I z) z) ≠ 0 | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
d : MDifferentiableAt I I (fun z => z) z
e : (𝓝 (↑(extChartAt I z) z)).EventuallyEq (fun w => ↑(extChartAt I z) (↑(extChartAt I z).symm w)) id
⊢ fderiv ℂ (fun x => ↑(extChartAt I z) (↑(extChartAt I z).symm x)) (↑(extChartAt I z) z) ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
d : MDifferentiableAt I I (fun z => z) z
⊢ fderiv ℂ (fun x => ↑(extChartAt I z) (↑(extChartAt I z).symm x)) (↑(extChartAt I z) z) ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | id_mderiv_ne_zero | [227, 1] | [237, 21] | simp only [e.fderiv_eq, fderiv_id, Ne, ContinuousLinearMap.ext_iff, not_forall,
ContinuousLinearMap.zero_apply, ContinuousLinearMap.id_apply] | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
d : MDifferentiableAt I I (fun z => z) z
e : (𝓝 (↑(extChartAt I z) z)).EventuallyEq (fun w => ↑(extChartAt I z) (↑(extChartAt I z).symm w)) id
⊢ fderiv ℂ (fun x => ↑(extChartAt I z) (↑(extChartAt I z).symm x)) (↑(extChartAt I z) z) ≠ 0 | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
d : MDifferentiableAt I I (fun z => z) z
e : (𝓝 (↑(extChartAt I z) z)).EventuallyEq (fun w => ↑(extChartAt I z) (↑(extChartAt I z).symm w)) id
⊢ ∃ x, ¬x = 0 x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
d : MDifferentiableAt I I (fun z => z) z
e : (𝓝 (↑(extChartAt I z) z)).EventuallyEq (fun w => ↑(extChartAt I z) (↑(extChartAt I z).symm w)) id
⊢ fderiv ℂ (fun x => ↑(extChartAt I z) (↑(extChartAt I z).symm x)) (↑(extChartAt I z) z) ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | id_mderiv_ne_zero | [227, 1] | [237, 21] | use 1, one_ne_zero | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
d : MDifferentiableAt I I (fun z => z) z
e : (𝓝 (↑(extChartAt I z) z)).EventuallyEq (fun w => ↑(extChartAt I z) (↑(extChartAt I z).symm w)) id
⊢ ∃ x, ¬x = 0 x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
d : MDifferentiableAt I I (fun z => z) z
e : (𝓝 (↑(extChartAt I z) z)).EventuallyEq (fun w => ↑(extChartAt I z) (↑(extChartAt I z).symm w)) id
⊢ ∃ x, ¬x = 0 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | id_mderiv_ne_zero | [227, 1] | [237, 21] | apply ((isOpen_extChartAt_target I z).eventually_mem (mem_extChartAt_target I z)).mp | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
d : MDifferentiableAt I I (fun z => z) z
⊢ (𝓝 (↑(extChartAt I z) z)).EventuallyEq (fun w => ↑(extChartAt I z) (↑(extChartAt I z).symm w)) id | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
d : MDifferentiableAt I I (fun z => z) z
⊢ ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I z) z),
x ∈ (extChartAt I z).target → (fun w => ↑(extChartAt I z) (↑(extChartAt I z).symm w)) x = id x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
d : MDifferentiableAt I I (fun z => z) z
⊢ (𝓝 (↑(extChartAt I z) z)).EventuallyEq (fun w => ↑(extChartAt I z) (↑(extChartAt I z).symm w)) id
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | id_mderiv_ne_zero | [227, 1] | [237, 21] | refine eventually_of_forall fun w m ↦ ?_ | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
d : MDifferentiableAt I I (fun z => z) z
⊢ ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I z) z),
x ∈ (extChartAt I z).target → (fun w => ↑(extChartAt I z) (↑(extChartAt I z).symm w)) x = id x | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
d : MDifferentiableAt I I (fun z => z) z
w : ℂ
m : w ∈ (extChartAt I z).target
⊢ (fun w => ↑(extChartAt I z) (↑(extChartAt I z).symm w)) w = id w | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
d : MDifferentiableAt I I (fun z => z) z
⊢ ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt I z) z),
x ∈ (extChartAt I z).target → (fun w => ↑(extChartAt I z) (↑(extChartAt I z).symm w)) x = id x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | id_mderiv_ne_zero | [227, 1] | [237, 21] | simp only [id, PartialEquiv.right_inv _ m] | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
d : MDifferentiableAt I I (fun z => z) z
w : ℂ
m : w ∈ (extChartAt I z).target
⊢ (fun w => ↑(extChartAt I z) (↑(extChartAt I z).symm w)) w = id w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
d : MDifferentiableAt I I (fun z => z) z
w : ℂ
m : w ∈ (extChartAt I z).target
⊢ (fun w => ↑(extChartAt I z) (↑(extChartAt I z).symm w)) w = id w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_eq_zero_iff_deriv_eq_zero | [240, 1] | [256, 24] | by_cases d : DifferentiableAt ℂ f z | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
⊢ mfderiv I I f z = 0 ↔ deriv f z = 0 | case pos
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
⊢ mfderiv I I f z = 0 ↔ deriv f z = 0
case neg
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : ¬DifferentiableAt ℂ f z
⊢ mfderiv I I f z = 0 ↔ deriv f z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
⊢ mfderiv I I f z = 0 ↔ deriv f z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_eq_zero_iff_deriv_eq_zero | [240, 1] | [256, 24] | constructor | case pos
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
⊢ mfderiv I I f z = 0 ↔ deriv f z = 0 | case pos.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
⊢ mfderiv I I f z = 0 → deriv f z = 0
case pos.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
⊢ deriv f z = 0 → mfderiv I I f z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
⊢ mfderiv I I f z = 0 ↔ deriv f z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_eq_zero_iff_deriv_eq_zero | [240, 1] | [256, 24] | have h := d.mdifferentiableAt.hasMFDerivAt | case pos.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
⊢ mfderiv I I f z = 0 → deriv f z = 0 | case pos.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasMFDerivAt I I f z (mfderiv I I f z)
⊢ mfderiv I I f z = 0 → deriv f z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
⊢ mfderiv I I f z = 0 → deriv f z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_eq_zero_iff_deriv_eq_zero | [240, 1] | [256, 24] | intro e | case pos.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasMFDerivAt I I f z (mfderiv I I f z)
⊢ mfderiv I I f z = 0 → deriv f z = 0 | case pos.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasMFDerivAt I I f z (mfderiv I I f z)
e : mfderiv I I f z = 0
⊢ deriv f z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasMFDerivAt I I f z (mfderiv I I f z)
⊢ mfderiv I I f z = 0 → deriv f z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_eq_zero_iff_deriv_eq_zero | [240, 1] | [256, 24] | rw [e] at h | case pos.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasMFDerivAt I I f z (mfderiv I I f z)
e : mfderiv I I f z = 0
⊢ deriv f z = 0 | case pos.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasMFDerivAt I I f z 0
e : mfderiv I I f z = 0
⊢ deriv f z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasMFDerivAt I I f z (mfderiv I I f z)
e : mfderiv I I f z = 0
⊢ deriv f z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_eq_zero_iff_deriv_eq_zero | [240, 1] | [256, 24] | have p := h.hasFDerivAt.hasDerivAt | case pos.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasMFDerivAt I I f z 0
e : mfderiv I I f z = 0
⊢ deriv f z = 0 | case pos.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasMFDerivAt I I f z 0
e : mfderiv I I f z = 0
p : HasDerivAt f (0 1) z
⊢ deriv f z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasMFDerivAt I I f z 0
e : mfderiv I I f z = 0
⊢ deriv f z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_eq_zero_iff_deriv_eq_zero | [240, 1] | [256, 24] | exact p.deriv | case pos.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasMFDerivAt I I f z 0
e : mfderiv I I f z = 0
p : HasDerivAt f (0 1) z
⊢ deriv f z = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasMFDerivAt I I f z 0
e : mfderiv I I f z = 0
p : HasDerivAt f (0 1) z
⊢ deriv f z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_eq_zero_iff_deriv_eq_zero | [240, 1] | [256, 24] | have h := d.hasDerivAt | case pos.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
⊢ deriv f z = 0 → mfderiv I I f z = 0 | case pos.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f (deriv f z) z
⊢ deriv f z = 0 → mfderiv I I f z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
⊢ deriv f z = 0 → mfderiv I I f z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_eq_zero_iff_deriv_eq_zero | [240, 1] | [256, 24] | intro e | case pos.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f (deriv f z) z
⊢ deriv f z = 0 → mfderiv I I f z = 0 | case pos.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f (deriv f z) z
e : deriv f z = 0
⊢ mfderiv I I f z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f (deriv f z) z
⊢ deriv f z = 0 → mfderiv I I f z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_eq_zero_iff_deriv_eq_zero | [240, 1] | [256, 24] | rw [e] at h | case pos.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f (deriv f z) z
e : deriv f z = 0
⊢ mfderiv I I f z = 0 | case pos.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f 0 z
e : deriv f z = 0
⊢ mfderiv I I f z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f (deriv f z) z
e : deriv f z = 0
⊢ mfderiv I I f z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_eq_zero_iff_deriv_eq_zero | [240, 1] | [256, 24] | have s0 : (1 : ℂ →L[ℂ] ℂ).smulRight (0 : ℂ) = 0 := by
apply ContinuousLinearMap.ext
simp only [ContinuousLinearMap.smulRight_apply, Algebra.id.smul_eq_mul,
MulZeroClass.mul_zero, ContinuousLinearMap.zero_apply, forall_const] | case pos.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f 0 z
e : deriv f z = 0
⊢ mfderiv I I f z = 0 | case pos.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f 0 z
e : deriv f z = 0
s0 : ContinuousLinearMap.smulRight 1 0 = 0
⊢ mfderiv I I f z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f 0 z
e : deriv f z = 0
⊢ mfderiv I I f z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_eq_zero_iff_deriv_eq_zero | [240, 1] | [256, 24] | have p := h.hasFDerivAt.hasMFDerivAt | case pos.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f 0 z
e : deriv f z = 0
s0 : ContinuousLinearMap.smulRight 1 0 = 0
⊢ mfderiv I I f z = 0 | case pos.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f 0 z
e : deriv f z = 0
s0 : ContinuousLinearMap.smulRight 1 0 = 0
p : HasMFDerivAt I I f z (ContinuousLinearMap.smulRight 1 0)
⊢ mfderiv I I f z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f 0 z
e : deriv f z = 0
s0 : ContinuousLinearMap.smulRight 1 0 = 0
⊢ mfderiv I I f z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_eq_zero_iff_deriv_eq_zero | [240, 1] | [256, 24] | rw [s0] at p | case pos.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f 0 z
e : deriv f z = 0
s0 : ContinuousLinearMap.smulRight 1 0 = 0
p : HasMFDerivAt I I f z (ContinuousLinearMap.smulRight 1 0)
⊢ mfderiv I I f z = 0 | case pos.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f 0 z
e : deriv f z = 0
s0 : ContinuousLinearMap.smulRight 1 0 = 0
p : HasMFDerivAt I I f z 0
⊢ mfderiv I I f z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f 0 z
e : deriv f z = 0
s0 : ContinuousLinearMap.smulRight 1 0 = 0
p : HasMFDerivAt I I f z (ContinuousLinearMap.smulRight 1 0)
⊢ mfderiv I I f z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_eq_zero_iff_deriv_eq_zero | [240, 1] | [256, 24] | exact p.mfderiv | case pos.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f 0 z
e : deriv f z = 0
s0 : ContinuousLinearMap.smulRight 1 0 = 0
p : HasMFDerivAt I I f z 0
⊢ mfderiv I I f z = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f 0 z
e : deriv f z = 0
s0 : ContinuousLinearMap.smulRight 1 0 = 0
p : HasMFDerivAt I I f z 0
⊢ mfderiv I I f z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_eq_zero_iff_deriv_eq_zero | [240, 1] | [256, 24] | apply ContinuousLinearMap.ext | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f 0 z
e : deriv f z = 0
⊢ ContinuousLinearMap.smulRight 1 0 = 0 | case h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f 0 z
e : deriv f z = 0
⊢ ∀ (x : ℂ), (ContinuousLinearMap.smulRight 1 0) x = 0 x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f 0 z
e : deriv f z = 0
⊢ ContinuousLinearMap.smulRight 1 0 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_eq_zero_iff_deriv_eq_zero | [240, 1] | [256, 24] | simp only [ContinuousLinearMap.smulRight_apply, Algebra.id.smul_eq_mul,
MulZeroClass.mul_zero, ContinuousLinearMap.zero_apply, forall_const] | case h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f 0 z
e : deriv f z = 0
⊢ ∀ (x : ℂ), (ContinuousLinearMap.smulRight 1 0) x = 0 x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : DifferentiableAt ℂ f z
h : HasDerivAt f 0 z
e : deriv f z = 0
⊢ ∀ (x : ℂ), (ContinuousLinearMap.smulRight 1 0) x = 0 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_eq_zero_iff_deriv_eq_zero | [240, 1] | [256, 24] | have d' : ¬MDifferentiableAt I I f z := by
contrapose d; simp only [not_not] at d ⊢; exact d.differentiableAt | case neg
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : ¬DifferentiableAt ℂ f z
⊢ mfderiv I I f z = 0 ↔ deriv f z = 0 | case neg
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : ¬DifferentiableAt ℂ f z
d' : ¬MDifferentiableAt I I f z
⊢ mfderiv I I f z = 0 ↔ deriv f z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : ¬DifferentiableAt ℂ f z
⊢ mfderiv I I f z = 0 ↔ deriv f z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_eq_zero_iff_deriv_eq_zero | [240, 1] | [256, 24] | simp only [deriv_zero_of_not_differentiableAt d, mfderiv_zero_of_not_mdifferentiableAt d',
eq_self_iff_true] | case neg
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : ¬DifferentiableAt ℂ f z
d' : ¬MDifferentiableAt I I f z
⊢ mfderiv I I f z = 0 ↔ deriv f z = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : ¬DifferentiableAt ℂ f z
d' : ¬MDifferentiableAt I I f z
⊢ mfderiv I I f z = 0 ↔ deriv f z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_eq_zero_iff_deriv_eq_zero | [240, 1] | [256, 24] | contrapose d | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : ¬DifferentiableAt ℂ f z
⊢ ¬MDifferentiableAt I I f z | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : ¬¬MDifferentiableAt I I f z
⊢ ¬¬DifferentiableAt ℂ f z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : ¬DifferentiableAt ℂ f z
⊢ ¬MDifferentiableAt I I f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_eq_zero_iff_deriv_eq_zero | [240, 1] | [256, 24] | simp only [not_not] at d ⊢ | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : ¬¬MDifferentiableAt I I f z
⊢ ¬¬DifferentiableAt ℂ f z | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : MDifferentiableAt I I f z
⊢ DifferentiableAt ℂ f z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : ¬¬MDifferentiableAt I I f z
⊢ ¬¬DifferentiableAt ℂ f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_eq_zero_iff_deriv_eq_zero | [240, 1] | [256, 24] | exact d.differentiableAt | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : MDifferentiableAt I I f z
⊢ DifferentiableAt ℂ f z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
d : MDifferentiableAt I I f z
⊢ DifferentiableAt ℂ f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | mfderiv_ne_zero_iff_deriv_ne_zero | [259, 1] | [260, 98] | rw [not_iff_not, mfderiv_eq_zero_iff_deriv_eq_zero] | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
⊢ mfderiv I I f z ≠ 0 ↔ deriv f z ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → ℂ
z : ℂ
⊢ mfderiv I I f z ≠ 0 ↔ deriv f z ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | critical_iter | [271, 1] | [280, 43] | induction' n with n h | S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
n : ℕ
z : S
fa : Holomorphic I I f
c : Critical f^[n] z
⊢ Precritical f z | case zero
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
z : S
fa : Holomorphic I I f
c : Critical f^[0] z
⊢ Precritical f z
case succ
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
z : S
fa : Holomorphic I I f
n : ℕ
h : Critical f^[n] z → Precritical f z
c : Critical f^[n + 1] z
⊢ Precritical f z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
n : ℕ
z : S
fa : Holomorphic I I f
c : Critical f^[n] z
⊢ Precritical f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | critical_iter | [271, 1] | [280, 43] | rw [Function.iterate_zero, Critical, mfderiv_id, ← ContinuousLinearMap.opNorm_zero_iff,
ContinuousLinearMap.norm_id] at c | case zero
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
z : S
fa : Holomorphic I I f
c : Critical f^[0] z
⊢ Precritical f z | case zero
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
z : S
fa : Holomorphic I I f
c : 1 = 0
⊢ Precritical f z | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
z : S
fa : Holomorphic I I f
c : Critical f^[0] z
⊢ Precritical f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OneDimension.lean | critical_iter | [271, 1] | [280, 43] | norm_num at c | case zero
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
z : S
fa : Holomorphic I I f
c : 1 = 0
⊢ Precritical f z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
z : S
fa : Holomorphic I I f
c : 1 = 0
⊢ Precritical f z
TACTIC:
|
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