url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | monotone_maxLog | [49, 1] | [52, 34] | simp_rw [maxLog] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ Monotone fun x => maxLog b x | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ Monotone fun x => (max b.exp x).log | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ Monotone fun x => maxLog b x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | monotone_maxLog | [49, 1] | [52, 34] | intro x y xy | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ Monotone fun x => (max b.exp x).log | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
xy : x ≤ y
⊢ (fun x => (max b.exp x).log) x ≤ (fun x => (max b.exp x).log) y | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ Monotone fun x => (max b.exp x).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | monotone_maxLog | [49, 1] | [52, 34] | simp only [ge_iff_le] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
xy : x ≤ y
⊢ (fun x => (max b.exp x).log) x ≤ (fun x => (max b.exp x).log) y | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
xy : x ≤ y
⊢ (max b.exp x).log ≤ (max b.exp y).log | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
xy : x ≤ y
⊢ (fun x => (max b.exp x).log) x ≤ (fun x => (max b.exp x).log) y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | monotone_maxLog | [49, 1] | [52, 34] | rw [Real.log_le_log_iff max_exp_pos max_exp_pos] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
xy : x ≤ y
⊢ (max b.exp x).log ≤ (max b.exp y).log | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
xy : x ≤ y
⊢ max b.exp x ≤ max b.exp y | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
xy : x ≤ y
⊢ (max b.exp x).log ≤ (max b.exp y).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | monotone_maxLog | [49, 1] | [52, 34] | apply max_le_max (le_refl _) xy | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
xy : x ≤ y
⊢ max b.exp x ≤ max b.exp y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
xy : x ≤ y
⊢ max b.exp x ≤ max b.exp y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | continuous_maxLog | [55, 1] | [59, 26] | simp_rw [maxLog] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ Continuous fun x => maxLog b x | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ Continuous fun x => (max b.exp x).log | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ Continuous fun x => maxLog b x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | continuous_maxLog | [55, 1] | [59, 26] | rw [continuous_iff_continuousAt] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ Continuous fun x => (max b.exp x).log | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ ∀ (x : ℝ), ContinuousAt (fun x => (max b.exp x).log) x | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ Continuous fun x => (max b.exp x).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | continuous_maxLog | [55, 1] | [59, 26] | intro x | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ ∀ (x : ℝ), ContinuousAt (fun x => (max b.exp x).log) x | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x : ℝ
⊢ ContinuousAt (fun x => (max b.exp x).log) x | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ ∀ (x : ℝ), ContinuousAt (fun x => (max b.exp x).log) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | continuous_maxLog | [55, 1] | [59, 26] | refine (ContinuousAt.log ?_ max_exp_pos.ne').comp ?_ | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x : ℝ
⊢ ContinuousAt (fun x => (max b.exp x).log) x | case refine_1
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x : ℝ
⊢ ContinuousAt (max b.exp) x
case refine_2
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x : ℝ
⊢ ContinuousAt (fun x => x) x | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x : ℝ
⊢ ContinuousAt (fun x => (max b.exp x).log) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | continuous_maxLog | [55, 1] | [59, 26] | apply Continuous.continuousAt | case refine_1
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x : ℝ
⊢ ContinuousAt (max b.exp) x | case refine_1.h
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x : ℝ
⊢ Continuous (max b.exp) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x : ℝ
⊢ ContinuousAt (max b.exp) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | continuous_maxLog | [55, 1] | [59, 26] | apply Continuous.max | case refine_1.h
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x : ℝ
⊢ Continuous (max b.exp) | case refine_1.h.hf
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x : ℝ
⊢ Continuous fun b_1 => b.exp
case refine_1.h.hg
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x : ℝ
⊢ Continuous fun b => b | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.h
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x : ℝ
⊢ Continuous (max b.exp)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | continuous_maxLog | [55, 1] | [59, 26] | exact continuous_const | case refine_1.h.hf
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x : ℝ
⊢ Continuous fun b_1 => b.exp
case refine_1.h.hg
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x : ℝ
⊢ Continuous fun b => b | case refine_1.h.hg
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x : ℝ
⊢ Continuous fun b => b | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.h.hf
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x : ℝ
⊢ Continuous fun b_1 => b.exp
case refine_1.h.hg
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x : ℝ
⊢ Continuous fun b => b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | continuous_maxLog | [55, 1] | [59, 26] | exact continuous_id | case refine_1.h.hg
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x : ℝ
⊢ Continuous fun b => b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.h.hg
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x : ℝ
⊢ Continuous fun b => b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | continuous_maxLog | [55, 1] | [59, 26] | exact continuousAt_id | case refine_2
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x : ℝ
⊢ ContinuousAt (fun x => x) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x : ℝ
⊢ ContinuousAt (fun x => x) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | rw [lipschitzOnWith_iff_dist_le_mul] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ LipschitzOnWith (-b).exp.toNNReal Real.log (Ici b.exp) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ ∀ x ∈ Ici b.exp, ∀ y ∈ Ici b.exp, dist x.log y.log ≤ ↑(-b).exp.toNNReal * dist x y | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ LipschitzOnWith (-b).exp.toNNReal Real.log (Ici b.exp)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | have half : ∀ x y : ℝ, b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y| := by
intro x y yb xy
have yp : y > 0 := lt_of_lt_of_le (Real.exp_pos _) yb
have xp : x > 0 := lt_of_lt_of_le yp xy
have yi : y⁻¹ ≤ (-b).exp := by rw [Real.exp_neg]; bound
rw [abs_of_nonneg (sub_nonneg.mpr xy)]
rw [abs_of_nonneg (sub_nonneg.mpr ((Real.log_le_log_iff yp xp).mpr xy))]
rw [← Real.log_div xp.ne' yp.ne']
rw [Real.log_le_iff_le_exp (div_pos xp yp)]
trans (y⁻¹ * (x - y)).exp; swap; bound
have e : y⁻¹ * (x - y) = x / y - 1 := by field_simp [yp.ne']
rw [e]
have e1 := Real.add_one_le_exp (x / y - 1)
simp at e1; exact e1 | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ ∀ x ∈ Ici b.exp, ∀ y ∈ Ici b.exp, dist x.log y.log ≤ ↑(-b).exp.toNNReal * dist x y | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
⊢ ∀ x ∈ Ici b.exp, ∀ y ∈ Ici b.exp, dist x.log y.log ≤ ↑(-b).exp.toNNReal * dist x y | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ ∀ x ∈ Ici b.exp, ∀ y ∈ Ici b.exp, dist x.log y.log ≤ ↑(-b).exp.toNNReal * dist x y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | intro x xs y ys | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
⊢ ∀ x ∈ Ici b.exp, ∀ y ∈ Ici b.exp, dist x.log y.log ≤ ↑(-b).exp.toNNReal * dist x y | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x : ℝ
xs : x ∈ Ici b.exp
y : ℝ
ys : y ∈ Ici b.exp
⊢ dist x.log y.log ≤ ↑(-b).exp.toNNReal * dist x y | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
⊢ ∀ x ∈ Ici b.exp, ∀ y ∈ Ici b.exp, dist x.log y.log ≤ ↑(-b).exp.toNNReal * dist x y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | simp at xs ys ⊢ | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x : ℝ
xs : x ∈ Ici b.exp
y : ℝ
ys : y ∈ Ici b.exp
⊢ dist x.log y.log ≤ ↑(-b).exp.toNNReal * dist x y | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
⊢ dist x.log y.log ≤ max (-b).exp 0 * dist x y | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x : ℝ
xs : x ∈ Ici b.exp
y : ℝ
ys : y ∈ Ici b.exp
⊢ dist x.log y.log ≤ ↑(-b).exp.toNNReal * dist x y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | rw [max_eq_left (Real.exp_pos _).le] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
⊢ dist x.log y.log ≤ max (-b).exp 0 * dist x y | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
⊢ dist x.log y.log ≤ (-b).exp * dist x y | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
⊢ dist x.log y.log ≤ max (-b).exp 0 * dist x y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | simp_rw [Real.dist_eq] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
⊢ dist x.log y.log ≤ (-b).exp * dist x y | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
⊢ |x.log - y.log| ≤ (-b).exp * |x - y| | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
⊢ dist x.log y.log ≤ (-b).exp * dist x y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | by_cases xy : x ≥ y | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
⊢ |x.log - y.log| ≤ (-b).exp * |x - y| | case pos
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
xy : x ≥ y
⊢ |x.log - y.log| ≤ (-b).exp * |x - y|
case neg
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
xy : ¬x ≥ y
⊢ |x.log - y.log| ≤ (-b).exp * |x - y| | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
⊢ |x.log - y.log| ≤ (-b).exp * |x - y|
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | simp at xy | case neg
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
xy : ¬x ≥ y
⊢ |x.log - y.log| ≤ (-b).exp * |x - y| | case neg
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
xy : x < y
⊢ |x.log - y.log| ≤ (-b).exp * |x - y| | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
xy : ¬x ≥ y
⊢ |x.log - y.log| ≤ (-b).exp * |x - y|
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | rw [← neg_sub y x, abs_neg] | case neg
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
xy : x < y
⊢ |x.log - y.log| ≤ (-b).exp * |x - y| | case neg
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
xy : x < y
⊢ |x.log - y.log| ≤ (-b).exp * |y - x| | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
xy : x < y
⊢ |x.log - y.log| ≤ (-b).exp * |x - y|
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | rw [← neg_sub y.log x.log, abs_neg] | case neg
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
xy : x < y
⊢ |x.log - y.log| ≤ (-b).exp * |y - x| | case neg
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
xy : x < y
⊢ |y.log - x.log| ≤ (-b).exp * |y - x| | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
xy : x < y
⊢ |x.log - y.log| ≤ (-b).exp * |y - x|
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | exact half y x xs xy.le | case neg
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
xy : x < y
⊢ |y.log - x.log| ≤ (-b).exp * |y - x| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
xy : x < y
⊢ |y.log - x.log| ≤ (-b).exp * |y - x|
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | intro x y yb xy | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y| | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
⊢ |x.log - y.log| ≤ (-b).exp * |x - y| | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | have yp : y > 0 := lt_of_lt_of_le (Real.exp_pos _) yb | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
⊢ |x.log - y.log| ≤ (-b).exp * |x - y| | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
⊢ |x.log - y.log| ≤ (-b).exp * |x - y| | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
⊢ |x.log - y.log| ≤ (-b).exp * |x - y|
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | have xp : x > 0 := lt_of_lt_of_le yp xy | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
⊢ |x.log - y.log| ≤ (-b).exp * |x - y| | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
⊢ |x.log - y.log| ≤ (-b).exp * |x - y| | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
⊢ |x.log - y.log| ≤ (-b).exp * |x - y|
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | have yi : y⁻¹ ≤ (-b).exp := by rw [Real.exp_neg]; bound | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
⊢ |x.log - y.log| ≤ (-b).exp * |x - y| | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ |x.log - y.log| ≤ (-b).exp * |x - y| | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
⊢ |x.log - y.log| ≤ (-b).exp * |x - y|
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | rw [abs_of_nonneg (sub_nonneg.mpr xy)] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ |x.log - y.log| ≤ (-b).exp * |x - y| | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ |x.log - y.log| ≤ (-b).exp * (x - y) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ |x.log - y.log| ≤ (-b).exp * |x - y|
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | rw [abs_of_nonneg (sub_nonneg.mpr ((Real.log_le_log_iff yp xp).mpr xy))] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ |x.log - y.log| ≤ (-b).exp * (x - y) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ x.log - y.log ≤ (-b).exp * (x - y) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ |x.log - y.log| ≤ (-b).exp * (x - y)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | rw [← Real.log_div xp.ne' yp.ne'] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ x.log - y.log ≤ (-b).exp * (x - y) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ (x / y).log ≤ (-b).exp * (x - y) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ x.log - y.log ≤ (-b).exp * (x - y)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | rw [Real.log_le_iff_le_exp (div_pos xp yp)] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ (x / y).log ≤ (-b).exp * (x - y) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ x / y ≤ ((-b).exp * (x - y)).exp | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ (x / y).log ≤ (-b).exp * (x - y)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | trans (y⁻¹ * (x - y)).exp | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ x / y ≤ ((-b).exp * (x - y)).exp | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ x / y ≤ (y⁻¹ * (x - y)).exp
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ (y⁻¹ * (x - y)).exp ≤ ((-b).exp * (x - y)).exp | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ x / y ≤ ((-b).exp * (x - y)).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | swap | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ x / y ≤ (y⁻¹ * (x - y)).exp
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ (y⁻¹ * (x - y)).exp ≤ ((-b).exp * (x - y)).exp | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ (y⁻¹ * (x - y)).exp ≤ ((-b).exp * (x - y)).exp
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ x / y ≤ (y⁻¹ * (x - y)).exp | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ x / y ≤ (y⁻¹ * (x - y)).exp
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ (y⁻¹ * (x - y)).exp ≤ ((-b).exp * (x - y)).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | bound | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ (y⁻¹ * (x - y)).exp ≤ ((-b).exp * (x - y)).exp
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ x / y ≤ (y⁻¹ * (x - y)).exp | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ x / y ≤ (y⁻¹ * (x - y)).exp | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ (y⁻¹ * (x - y)).exp ≤ ((-b).exp * (x - y)).exp
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ x / y ≤ (y⁻¹ * (x - y)).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | have e : y⁻¹ * (x - y) = x / y - 1 := by field_simp [yp.ne'] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ x / y ≤ (y⁻¹ * (x - y)).exp | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
e : y⁻¹ * (x - y) = x / y - 1
⊢ x / y ≤ (y⁻¹ * (x - y)).exp | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ x / y ≤ (y⁻¹ * (x - y)).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | rw [e] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
e : y⁻¹ * (x - y) = x / y - 1
⊢ x / y ≤ (y⁻¹ * (x - y)).exp | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
e : y⁻¹ * (x - y) = x / y - 1
⊢ x / y ≤ (x / y - 1).exp | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
e : y⁻¹ * (x - y) = x / y - 1
⊢ x / y ≤ (y⁻¹ * (x - y)).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | have e1 := Real.add_one_le_exp (x / y - 1) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
e : y⁻¹ * (x - y) = x / y - 1
⊢ x / y ≤ (x / y - 1).exp | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
e : y⁻¹ * (x - y) = x / y - 1
e1 : x / y - 1 + 1 ≤ (x / y - 1).exp
⊢ x / y ≤ (x / y - 1).exp | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
e : y⁻¹ * (x - y) = x / y - 1
⊢ x / y ≤ (x / y - 1).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | simp at e1 | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
e : y⁻¹ * (x - y) = x / y - 1
e1 : x / y - 1 + 1 ≤ (x / y - 1).exp
⊢ x / y ≤ (x / y - 1).exp | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
e : y⁻¹ * (x - y) = x / y - 1
e1 : x / y ≤ (x / y - 1).exp
⊢ x / y ≤ (x / y - 1).exp | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
e : y⁻¹ * (x - y) = x / y - 1
e1 : x / y - 1 + 1 ≤ (x / y - 1).exp
⊢ x / y ≤ (x / y - 1).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | exact e1 | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
e : y⁻¹ * (x - y) = x / y - 1
e1 : x / y ≤ (x / y - 1).exp
⊢ x / y ≤ (x / y - 1).exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
e : y⁻¹ * (x - y) = x / y - 1
e1 : x / y ≤ (x / y - 1).exp
⊢ x / y ≤ (x / y - 1).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | rw [Real.exp_neg] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
⊢ y⁻¹ ≤ (-b).exp | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
⊢ y⁻¹ ≤ b.exp⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
⊢ y⁻¹ ≤ (-b).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | bound | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
⊢ y⁻¹ ≤ b.exp⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
⊢ y⁻¹ ≤ b.exp⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | field_simp [yp.ne'] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ y⁻¹ * (x - y) = x / y - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ y⁻¹ * (x - y) = x / y - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | exact half x y ys xy | case pos
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
xy : x ≥ y
⊢ |x.log - y.log| ≤ (-b).exp * |x - y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
xy : x ≥ y
⊢ |x.log - y.log| ≤ (-b).exp * |x - y|
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzWith.maxLog | [94, 1] | [100, 70] | rw [← lipschitzOn_univ] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ LipschitzWith (-b).exp.toNNReal (_root_.maxLog b) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ LipschitzOnWith (-b).exp.toNNReal (_root_.maxLog b) univ | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ LipschitzWith (-b).exp.toNNReal (_root_.maxLog b)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzWith.maxLog | [94, 1] | [100, 70] | have h := (LipschitzOnWith.log b).comp ((LipschitzWith.id.const_max b.exp).lipschitzOnWith univ)
(by simp only [id_eq, Set.mapsTo_univ_iff, Set.mem_Ici, le_max_iff, le_refl, true_or,
forall_const]) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ LipschitzOnWith (-b).exp.toNNReal (_root_.maxLog b) univ | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
⊢ LipschitzOnWith (-b).exp.toNNReal (_root_.maxLog b) univ | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ LipschitzOnWith (-b).exp.toNNReal (_root_.maxLog b) univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzWith.maxLog | [94, 1] | [100, 70] | have e : Real.log ∘ max (Real.exp b) = _root_.maxLog b := by funext x; simp [_root_.maxLog] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
⊢ LipschitzOnWith (-b).exp.toNNReal (_root_.maxLog b) univ | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
e : Real.log ∘ max b.exp = _root_.maxLog b
⊢ LipschitzOnWith (-b).exp.toNNReal (_root_.maxLog b) univ | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
⊢ LipschitzOnWith (-b).exp.toNNReal (_root_.maxLog b) univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzWith.maxLog | [94, 1] | [100, 70] | simpa only [e, mul_one, id_eq, ge_iff_le, lipschitzOn_univ] using h | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
e : Real.log ∘ max b.exp = _root_.maxLog b
⊢ LipschitzOnWith (-b).exp.toNNReal (_root_.maxLog b) univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
e : Real.log ∘ max b.exp = _root_.maxLog b
⊢ LipschitzOnWith (-b).exp.toNNReal (_root_.maxLog b) univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzWith.maxLog | [94, 1] | [100, 70] | simp only [id_eq, Set.mapsTo_univ_iff, Set.mem_Ici, le_max_iff, le_refl, true_or,
forall_const] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ Set.MapsTo (fun x => max b.exp (id x)) univ (Ici b.exp) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ Set.MapsTo (fun x => max b.exp (id x)) univ (Ici b.exp)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzWith.maxLog | [94, 1] | [100, 70] | funext x | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
⊢ Real.log ∘ max b.exp = _root_.maxLog b | case h
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
x : ℝ
⊢ (Real.log ∘ max b.exp) x = _root_.maxLog b x | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
⊢ Real.log ∘ max b.exp = _root_.maxLog b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzWith.maxLog | [94, 1] | [100, 70] | simp [_root_.maxLog] | case h
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
x : ℝ
⊢ (Real.log ∘ max b.exp) x = _root_.maxLog b x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
x : ℝ
⊢ (Real.log ∘ max b.exp) x = _root_.maxLog b x
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:
|
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