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/FubiniBall.lean | measurable_symm_equiv_inverse | [239, 1] | [245, 55] | simp only [Complex.equivRealProdCLM_symm_apply_re] | case a
z : ℂ
⊢ (Complex.equivRealProdCLM.symm (z.re, z.im)).re = z.re | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
z : ℂ
⊢ (Complex.equivRealProdCLM.symm (z.re, z.im)).re = z.re
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | measurable_symm_equiv_inverse | [239, 1] | [245, 55] | simp only [Complex.equivRealProdCLM_symm_apply_im] | case a
z : ℂ
⊢ (Complex.equivRealProdCLM.symm (z.re, z.im)).im = z.im | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
z : ℂ
⊢ (Complex.equivRealProdCLM.symm (z.re, z.im)).im = z.im
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | continuous_circleMap_full | [248, 1] | [249, 13] | continuity | c : ℂ
⊢ Continuous fun x => circleMap c x.1 x.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
⊢ Continuous fun x => circleMap c x.1 x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | invert_toReal | [252, 1] | [255, 56] | intro h | x : ENNReal
y : ℝ
yp : y > 0
⊢ x.toReal = y → x = ENNReal.ofReal y | x : ENNReal
y : ℝ
yp : y > 0
h : x.toReal = y
⊢ x = ENNReal.ofReal y | Please generate a tactic in lean4 to solve the state.
STATE:
x : ENNReal
y : ℝ
yp : y > 0
⊢ x.toReal = y → x = ENNReal.ofReal y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | invert_toReal | [252, 1] | [255, 56] | rw [← h] | x : ENNReal
y : ℝ
yp : y > 0
h : x.toReal = y
⊢ x = ENNReal.ofReal y | x : ENNReal
y : ℝ
yp : y > 0
h : x.toReal = y
⊢ x = ENNReal.ofReal x.toReal | Please generate a tactic in lean4 to solve the state.
STATE:
x : ENNReal
y : ℝ
yp : y > 0
h : x.toReal = y
⊢ x = ENNReal.ofReal y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | invert_toReal | [252, 1] | [255, 56] | refine (ENNReal.ofReal_toReal ?_).symm | x : ENNReal
y : ℝ
yp : y > 0
h : x.toReal = y
⊢ x = ENNReal.ofReal x.toReal | x : ENNReal
y : ℝ
yp : y > 0
h : x.toReal = y
⊢ x ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
x : ENNReal
y : ℝ
yp : y > 0
h : x.toReal = y
⊢ x = ENNReal.ofReal x.toReal
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | invert_toReal | [252, 1] | [255, 56] | contrapose yp | x : ENNReal
y : ℝ
yp : y > 0
h : x.toReal = y
⊢ x ≠ ⊤ | x : ENNReal
y : ℝ
h : x.toReal = y
yp : ¬x ≠ ⊤
⊢ ¬y > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
x : ENNReal
y : ℝ
yp : y > 0
h : x.toReal = y
⊢ x ≠ ⊤
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | invert_toReal | [252, 1] | [255, 56] | simp only [ne_eq, not_not] at yp | x : ENNReal
y : ℝ
h : x.toReal = y
yp : ¬x ≠ ⊤
⊢ ¬y > 0 | x : ENNReal
y : ℝ
h : x.toReal = y
yp : x = ⊤
⊢ ¬y > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
x : ENNReal
y : ℝ
h : x.toReal = y
yp : ¬x ≠ ⊤
⊢ ¬y > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | invert_toReal | [252, 1] | [255, 56] | simp only [yp, ENNReal.top_toReal] at h | x : ENNReal
y : ℝ
h : x.toReal = y
yp : x = ⊤
⊢ ¬y > 0 | x : ENNReal
y : ℝ
yp : x = ⊤
h : 0 = y
⊢ ¬y > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
x : ENNReal
y : ℝ
h : x.toReal = y
yp : x = ⊤
⊢ ¬y > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | invert_toReal | [252, 1] | [255, 56] | simp only [← h, lt_self_iff_false, not_false_eq_true] | x : ENNReal
y : ℝ
yp : x = ⊤
h : 0 = y
⊢ ¬y > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ENNReal
y : ℝ
yp : x = ⊤
h : 0 = y
⊢ ¬y > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | have im := MeasurePreserving.symm _ Complex.volume_preserving_equiv_real_prod | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (z : ℂ) in annulus_oc c r0 r1, f z = ∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
im : MeasurePreserving (⇑Complex.measurableEquivRealProd.symm) volume volume
⊢ ∫ (z : ℂ) in annulus_oc c r0 r1, f z = ∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (z : ℂ) in annulus_oc c r0 r1, f z = ∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | rw [←MeasurePreserving.setIntegral_preimage_emb im
Complex.measurableEquivRealProd.symm.measurableEmbedding f _] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
im : MeasurePreserving (⇑Complex.measurableEquivRealProd.symm) volume volume
⊢ ∫ (z : ℂ) in annulus_oc c r0 r1, f z = ∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
im : MeasurePreserving (⇑Complex.measurableEquivRealProd.symm) volume volume
⊢ ∫ (x : ℝ × ℝ) in ⇑Complex.measurableEquivRealProd.symm ⁻¹' annulus_oc c r0 r1,
f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
im : MeasurePreserving (⇑Complex.measurableEquivRealProd.symm) volume volume
⊢ ∫ (z : ℂ) in annulus_oc c r0 r1, f z = ∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | clear im | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
im : MeasurePreserving (⇑Complex.measurableEquivRealProd.symm) volume volume
⊢ ∫ (x : ℝ × ℝ) in ⇑Complex.measurableEquivRealProd.symm ⁻¹' annulus_oc c r0 r1,
f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in ⇑Complex.measurableEquivRealProd.symm ⁻¹' annulus_oc c r0 r1,
f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
im : MeasurePreserving (⇑Complex.measurableEquivRealProd.symm) volume volume
⊢ ∫ (x : ℝ × ℝ) in ⇑Complex.measurableEquivRealProd.symm ⁻¹' annulus_oc c r0 r1,
f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | rw [square_eq r0p] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in ⇑Complex.measurableEquivRealProd.symm ⁻¹' annulus_oc c r0 r1,
f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in realCircleMap c '' square r0 r1, f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in ⇑Complex.measurableEquivRealProd.symm ⁻¹' annulus_oc c r0 r1,
f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | have dc : ∀ x, x ∈ square r0 r1 →
HasFDerivWithinAt (realCircleMap c) (rcmDeriv x) (square r0 r1) x :=
fun _ _ ↦ realCircleMap.fderiv.hasFDerivWithinAt | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in realCircleMap c '' square r0 r1, f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
dc : ∀ x ∈ square r0 r1, HasFDerivWithinAt (realCircleMap c) (rcmDeriv x) (square r0 r1) x
⊢ ∫ (x : ℝ × ℝ) in realCircleMap c '' square r0 r1, f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in realCircleMap c '' square r0 r1, f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | rw [integral_image_eq_integral_abs_det_fderiv_smul volume Measurable.square dc (rcm_inj r0p)] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
dc : ∀ x ∈ square r0 r1, HasFDerivWithinAt (realCircleMap c) (rcmDeriv x) (square r0 r1) x
⊢ ∫ (x : ℝ × ℝ) in realCircleMap c '' square r0 r1, f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
dc : ∀ x ∈ square r0 r1, HasFDerivWithinAt (realCircleMap c) (rcmDeriv x) (square r0 r1) x
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |(rcmDeriv x).det| • f (Complex.measurableEquivRealProd.symm (realCircleMap c x)) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
dc : ∀ x ∈ square r0 r1, HasFDerivWithinAt (realCircleMap c) (rcmDeriv x) (square r0 r1) x
⊢ ∫ (x : ℝ × ℝ) in realCircleMap c '' square r0 r1, f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | clear dc | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
dc : ∀ x ∈ square r0 r1, HasFDerivWithinAt (realCircleMap c) (rcmDeriv x) (square r0 r1) x
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |(rcmDeriv x).det| • f (Complex.measurableEquivRealProd.symm (realCircleMap c x)) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |(rcmDeriv x).det| • f (Complex.measurableEquivRealProd.symm (realCircleMap c x)) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
dc : ∀ x ∈ square r0 r1, HasFDerivWithinAt (realCircleMap c) (rcmDeriv x) (square r0 r1) x
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |(rcmDeriv x).det| • f (Complex.measurableEquivRealProd.symm (realCircleMap c x)) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | simp_rw [rcmDeriv.det] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |(rcmDeriv x).det| • f (Complex.measurableEquivRealProd.symm (realCircleMap c x)) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |x.1| • f (Complex.measurableEquivRealProd.symm (realCircleMap c x)) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |(rcmDeriv x).det| • f (Complex.measurableEquivRealProd.symm (realCircleMap c x)) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | simp_rw [realCircleMap_eq_circleMap] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |x.1| • f (Complex.measurableEquivRealProd.symm (realCircleMap c x)) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1,
|x.1| • f (Complex.measurableEquivRealProd.symm (Complex.equivRealProd (circleMap c x.1 x.2))) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |x.1| • f (Complex.measurableEquivRealProd.symm (realCircleMap c x)) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | simp_rw [measurable_symm_equiv_inverse] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1,
|x.1| • f (Complex.measurableEquivRealProd.symm (Complex.equivRealProd (circleMap c x.1 x.2))) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |x.1| • f (circleMap c x.1 x.2) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1,
|x.1| • f (Complex.measurableEquivRealProd.symm (Complex.equivRealProd (circleMap c x.1 x.2))) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | have e : ∀ x : ℝ × ℝ, x ∈ square r0 r1 → |x.1| • f (circleMap c x.1 x.2) =
x.1 • f (circleMap c x.1 x.2) := by
intro x xs; rw [abs_of_pos (square.rp r0p xs)] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |x.1| • f (circleMap c x.1 x.2) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
e : ∀ x ∈ square r0 r1, |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2)
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |x.1| • f (circleMap c x.1 x.2) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |x.1| • f (circleMap c x.1 x.2) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | rw [MeasureTheory.setIntegral_congr Measurable.square e] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
e : ∀ x ∈ square r0 r1, |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2)
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |x.1| • f (circleMap c x.1 x.2) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
e : ∀ x ∈ square r0 r1, |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2)
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, x.1 • f (circleMap c x.1 x.2) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
e : ∀ x ∈ square r0 r1, |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2)
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |x.1| • f (circleMap c x.1 x.2) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | clear e | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
e : ∀ x ∈ square r0 r1, |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2)
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, x.1 • f (circleMap c x.1 x.2) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, x.1 • f (circleMap c x.1 x.2) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
e : ∀ x ∈ square r0 r1, |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2)
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, x.1 • f (circleMap c x.1 x.2) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | rw [square, Measure.volume_eq_prod, MeasureTheory.setIntegral_prod] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, x.1 • f (circleMap c x.1 x.2) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ) in Ioc r0 r1, ∫ (y : ℝ) in Ioc 0 (2 * π), (x, y).1 • f (circleMap c (x, y).1 (x, y).2) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ IntegrableOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Ioc r0 r1 ×ˢ Ioc 0 (2 * π)) (volume.prod volume) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, x.1 • f (circleMap c x.1 x.2) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | simp [integral_smul] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ) in Ioc r0 r1, ∫ (y : ℝ) in Ioc 0 (2 * π), (x, y).1 • f (circleMap c (x, y).1 (x, y).2) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ IntegrableOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Ioc r0 r1 ×ˢ Ioc 0 (2 * π)) (volume.prod volume) | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ IntegrableOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Ioc r0 r1 ×ˢ Ioc 0 (2 * π)) (volume.prod volume) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ) in Ioc r0 r1, ∫ (y : ℝ) in Ioc 0 (2 * π), (x, y).1 • f (circleMap c (x, y).1 (x, y).2) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ IntegrableOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Ioc r0 r1 ×ˢ Ioc 0 (2 * π)) (volume.prod volume)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | exact fi.mono_set (prod_mono Ioc_subset_Icc_self Ioc_subset_Icc_self) | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
fi : IntegrableOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) volume
⊢ IntegrableOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Ioc r0 r1 ×ˢ Ioc 0 (2 * π)) (volume.prod volume) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
fi : IntegrableOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) volume
⊢ IntegrableOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Ioc r0 r1 ×ˢ Ioc 0 (2 * π)) (volume.prod volume)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | intro x xs | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∀ x ∈ square r0 r1, |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : x ∈ square r0 r1
⊢ |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∀ x ∈ square r0 r1, |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | rw [abs_of_pos (square.rp r0p xs)] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : x ∈ square r0 r1
⊢ |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : x ∈ square r0 r1
⊢ |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | apply ContinuousOn.integrableOn_compact | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ IntegrableOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) volume | case hK
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ IsCompact (Icc r0 r1 ×ˢ Icc 0 (2 * π))
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ContinuousOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ IntegrableOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) volume
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | exact IsCompact.prod isCompact_Icc isCompact_Icc | case hK
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ IsCompact (Icc r0 r1 ×ˢ Icc 0 (2 * π)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hK
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ IsCompact (Icc r0 r1 ×ˢ Icc 0 (2 * π))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | apply ContinuousOn.smul continuous_fst.continuousOn | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ContinuousOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ContinuousOn (fun x => f (circleMap c x.1 x.2)) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ContinuousOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Icc r0 r1 ×ˢ Icc 0 (2 * π))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | apply fc.comp continuous_circleMap_full.continuousOn | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ContinuousOn (fun x => f (circleMap c x.1 x.2)) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ MapsTo (fun x => circleMap c x.1 x.2) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) (annulus_cc c r0 r1) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ContinuousOn (fun x => f (circleMap c x.1 x.2)) (Icc r0 r1 ×ˢ Icc 0 (2 * π))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | intro x xs | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ MapsTo (fun x => circleMap c x.1 x.2) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) (annulus_cc c r0 r1) | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : x ∈ Icc r0 r1 ×ˢ Icc 0 (2 * π)
⊢ (fun x => circleMap c x.1 x.2) x ∈ annulus_cc c r0 r1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ MapsTo (fun x => circleMap c x.1 x.2) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) (annulus_cc c r0 r1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | simp only [Icc_prod_Icc, mem_Icc, Prod.le_def] at xs | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : x ∈ Icc r0 r1 ×ˢ Icc 0 (2 * π)
⊢ (fun x => circleMap c x.1 x.2) x ∈ annulus_cc c r0 r1 | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : (r0 ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ r1 ∧ x.2 ≤ 2 * π
⊢ (fun x => circleMap c x.1 x.2) x ∈ annulus_cc c r0 r1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : x ∈ Icc r0 r1 ×ˢ Icc 0 (2 * π)
⊢ (fun x => circleMap c x.1 x.2) x ∈ annulus_cc c r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | have x0 : 0 ≤ x.1 := by linarith | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : (r0 ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ r1 ∧ x.2 ≤ 2 * π
⊢ (fun x => circleMap c x.1 x.2) x ∈ annulus_cc c r0 r1 | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : (r0 ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ r1 ∧ x.2 ≤ 2 * π
x0 : 0 ≤ x.1
⊢ (fun x => circleMap c x.1 x.2) x ∈ annulus_cc c r0 r1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : (r0 ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ r1 ∧ x.2 ≤ 2 * π
⊢ (fun x => circleMap c x.1 x.2) x ∈ annulus_cc c r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | simp only [circleMap, annulus_cc, mem_diff, Metric.mem_closedBall, dist_self_add_left,
norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_of_nonneg x0,
Complex.abs_exp_ofReal_mul_I, mul_one, xs.2.1, Metric.mem_ball, not_lt, xs.1.1, and_self] | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : (r0 ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ r1 ∧ x.2 ≤ 2 * π
x0 : 0 ≤ x.1
⊢ (fun x => circleMap c x.1 x.2) x ∈ annulus_cc c r0 r1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : (r0 ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ r1 ∧ x.2 ≤ 2 * π
x0 : 0 ≤ x.1
⊢ (fun x => circleMap c x.1 x.2) x ∈ annulus_cc c r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | linarith | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : (r0 ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ r1 ∧ x.2 ≤ 2 * π
⊢ 0 ≤ x.1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : (r0 ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ r1 ∧ x.2 ≤ 2 * π
⊢ 0 ≤ x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_ball | [296, 1] | [305, 8] | have center : closedBall c r =ᵐ[volume] (closedBall c r \ {c} : Set ℂ) := ae_minus_point | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ ∫ (z : ℂ) in closedBall c r, f z = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
center : volume.ae.EventuallyEq (closedBall c r) (closedBall c r \ {c})
⊢ ∫ (z : ℂ) in closedBall c r, f z = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ ∫ (z : ℂ) in closedBall c r, f z = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_ball | [296, 1] | [305, 8] | rw [MeasureTheory.setIntegral_congr_set_ae center] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
center : volume.ae.EventuallyEq (closedBall c r) (closedBall c r \ {c})
⊢ ∫ (z : ℂ) in closedBall c r, f z = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
center : volume.ae.EventuallyEq (closedBall c r) (closedBall c r \ {c})
⊢ ∫ (x : ℂ) in closedBall c r \ {c}, f x = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
center : volume.ae.EventuallyEq (closedBall c r) (closedBall c r \ {c})
⊢ ∫ (z : ℂ) in closedBall c r, f z = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_ball | [296, 1] | [305, 8] | clear center | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
center : volume.ae.EventuallyEq (closedBall c r) (closedBall c r \ {c})
⊢ ∫ (x : ℂ) in closedBall c r \ {c}, f x = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ ∫ (x : ℂ) in closedBall c r \ {c}, f x = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
center : volume.ae.EventuallyEq (closedBall c r) (closedBall c r \ {c})
⊢ ∫ (x : ℂ) in closedBall c r \ {c}, f x = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_ball | [296, 1] | [305, 8] | rw [←Metric.closedBall_zero, ←annulus_oc] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ ∫ (x : ℂ) in closedBall c r \ {c}, f x = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ ∫ (x : ℂ) in annulus_oc c 0 r, f x = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ ∫ (x : ℂ) in closedBall c r \ {c}, f x = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_ball | [296, 1] | [305, 8] | apply fubini_annulus | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ ∫ (x : ℂ) in annulus_oc c 0 r, f x = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | case fc
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ ContinuousOn (fun z => f z) (annulus_cc c 0 r)
case r0p
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ 0 ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ ∫ (x : ℂ) in annulus_oc c 0 r, f x = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_ball | [296, 1] | [305, 8] | simpa only [annulus_cc, Metric.ball_zero, diff_empty] | case fc
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ ContinuousOn (fun z => f z) (annulus_cc c 0 r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case fc
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ ContinuousOn (fun z => f z) (annulus_cc c 0 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_ball | [296, 1] | [305, 8] | rfl | case r0p
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ 0 ≤ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case r0p
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ 0 ≤ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Complex.volume_closedBall' | [308, 1] | [317, 26] | have c : ContinuousOn (fun _ : ℂ ↦ (1 : ℝ)) (closedBall c r) := continuousOn_const | c : ℂ
r : ℝ
rp : r ≥ 0
⊢ (↑volume (closedBall c r)).toReal = π * r ^ 2 | c✝ : ℂ
r : ℝ
rp : r ≥ 0
c : ContinuousOn (fun x => 1) (closedBall c✝ r)
⊢ (↑volume (closedBall c✝ r)).toReal = π * r ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r : ℝ
rp : r ≥ 0
⊢ (↑volume (closedBall c r)).toReal = π * r ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Complex.volume_closedBall' | [308, 1] | [317, 26] | have f := fubini_ball c | c✝ : ℂ
r : ℝ
rp : r ≥ 0
c : ContinuousOn (fun x => 1) (closedBall c✝ r)
⊢ (↑volume (closedBall c✝ r)).toReal = π * r ^ 2 | c✝ : ℂ
r : ℝ
rp : r ≥ 0
c : ContinuousOn (fun x => 1) (closedBall c✝ r)
f : ∫ (z : ℂ) in closedBall c✝ r, 1 = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), 1
⊢ (↑volume (closedBall c✝ r)).toReal = π * r ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
r : ℝ
rp : r ≥ 0
c : ContinuousOn (fun x => 1) (closedBall c✝ r)
⊢ (↑volume (closedBall c✝ r)).toReal = π * r ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Complex.volume_closedBall' | [308, 1] | [317, 26] | clear c | c✝ : ℂ
r : ℝ
rp : r ≥ 0
c : ContinuousOn (fun x => 1) (closedBall c✝ r)
f : ∫ (z : ℂ) in closedBall c✝ r, 1 = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), 1
⊢ (↑volume (closedBall c✝ r)).toReal = π * r ^ 2 | c : ℂ
r : ℝ
rp : r ≥ 0
f : ∫ (z : ℂ) in closedBall c r, 1 = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), 1
⊢ (↑volume (closedBall c r)).toReal = π * r ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
r : ℝ
rp : r ≥ 0
c : ContinuousOn (fun x => 1) (closedBall c✝ r)
f : ∫ (z : ℂ) in closedBall c✝ r, 1 = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), 1
⊢ (↑volume (closedBall c✝ r)).toReal = π * r ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Complex.volume_closedBall' | [308, 1] | [317, 26] | simp only [ENNReal.toReal_ofReal Real.two_pi_pos.le, ←
intervalIntegral.integral_of_le rp, integral_const, Measure.restrict_apply, MeasurableSet.univ,
univ_inter, Algebra.id.smul_eq_mul, mul_one, Real.volume_Ioc, tsub_zero,
intervalIntegral.integral_mul_const, integral_id, zero_pow, Ne, bit0_eq_zero,
Nat.one_ne_zero, not_false_iff] at f | c : ℂ
r : ℝ
rp : r ≥ 0
f : ∫ (z : ℂ) in closedBall c r, 1 = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), 1
⊢ (↑volume (closedBall c r)).toReal = π * r ^ 2 | c : ℂ
r : ℝ
rp : r ≥ 0
f : (↑volume (closedBall c r)).toReal = r ^ 2 / 2 * (2 * π)
⊢ (↑volume (closedBall c r)).toReal = π * r ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r : ℝ
rp : r ≥ 0
f : ∫ (z : ℂ) in closedBall c r, 1 = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), 1
⊢ (↑volume (closedBall c r)).toReal = π * r ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Complex.volume_closedBall' | [308, 1] | [317, 26] | ring_nf at f ⊢ | c : ℂ
r : ℝ
rp : r ≥ 0
f : (↑volume (closedBall c r)).toReal = r ^ 2 / 2 * (2 * π)
⊢ (↑volume (closedBall c r)).toReal = π * r ^ 2 | c : ℂ
r : ℝ
rp : r ≥ 0
f : (↑volume (closedBall c r)).toReal = r ^ 2 * π
⊢ (↑volume (closedBall c r)).toReal = r ^ 2 * π | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r : ℝ
rp : r ≥ 0
f : (↑volume (closedBall c r)).toReal = r ^ 2 / 2 * (2 * π)
⊢ (↑volume (closedBall c r)).toReal = π * r ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Complex.volume_closedBall' | [308, 1] | [317, 26] | exact f | c : ℂ
r : ℝ
rp : r ≥ 0
f : (↑volume (closedBall c r)).toReal = r ^ 2 * π
⊢ (↑volume (closedBall c r)).toReal = r ^ 2 * π | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r : ℝ
rp : r ≥ 0
f : (↑volume (closedBall c r)).toReal = r ^ 2 * π
⊢ (↑volume (closedBall c r)).toReal = r ^ 2 * π
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | NiceVolume.closedBall | [320, 1] | [332, 14] | simp only [Complex.volume_closedBall] | c : ℂ
r : ℝ
rp : r > 0
⊢ ↑volume (Metric.closedBall c r) < ⊤ | c : ℂ
r : ℝ
rp : r > 0
⊢ ENNReal.ofReal r ^ 2 * ↑NNReal.pi < ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r : ℝ
rp : r > 0
⊢ ↑volume (Metric.closedBall c r) < ⊤
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | NiceVolume.closedBall | [320, 1] | [332, 14] | apply ENNReal.mul_lt_top | c : ℂ
r : ℝ
rp : r > 0
⊢ ENNReal.ofReal r ^ 2 * ↑NNReal.pi < ⊤ | case a
c : ℂ
r : ℝ
rp : r > 0
⊢ ENNReal.ofReal r ^ 2 ≠ ⊤
case a
c : ℂ
r : ℝ
rp : r > 0
⊢ ↑NNReal.pi ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r : ℝ
rp : r > 0
⊢ ENNReal.ofReal r ^ 2 * ↑NNReal.pi < ⊤
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | NiceVolume.closedBall | [320, 1] | [332, 14] | simp only [ne_eq, ENNReal.pow_eq_top_iff, ENNReal.ofReal_ne_top, OfNat.ofNat_ne_zero,
not_false_eq_true, and_true] | case a
c : ℂ
r : ℝ
rp : r > 0
⊢ ENNReal.ofReal r ^ 2 ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
c : ℂ
r : ℝ
rp : r > 0
⊢ ENNReal.ofReal r ^ 2 ≠ ⊤
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | NiceVolume.closedBall | [320, 1] | [332, 14] | simp only [ne_eq, ENNReal.coe_ne_top, not_false_eq_true] | case a
c : ℂ
r : ℝ
rp : r > 0
⊢ ↑NNReal.pi ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
c : ℂ
r : ℝ
rp : r > 0
⊢ ↑NNReal.pi ≠ ⊤
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | NiceVolume.closedBall | [320, 1] | [332, 14] | simp only [Complex.volume_closedBall, gt_iff_lt, CanonicallyOrderedCommSemiring.mul_pos,
ENNReal.coe_pos, NNReal.pi_pos, and_true] | c : ℂ
r : ℝ
rp : r > 0
⊢ ↑volume (Metric.closedBall c r) > 0 | c : ℂ
r : ℝ
rp : r > 0
⊢ 0 < ENNReal.ofReal r ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r : ℝ
rp : r > 0
⊢ ↑volume (Metric.closedBall c r) > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | NiceVolume.closedBall | [320, 1] | [332, 14] | apply ENNReal.pow_pos | c : ℂ
r : ℝ
rp : r > 0
⊢ 0 < ENNReal.ofReal r ^ 2 | case a
c : ℂ
r : ℝ
rp : r > 0
⊢ 0 < ENNReal.ofReal r | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r : ℝ
rp : r > 0
⊢ 0 < ENNReal.ofReal r ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | NiceVolume.closedBall | [320, 1] | [332, 14] | bound | case a
c : ℂ
r : ℝ
rp : r > 0
⊢ 0 < ENNReal.ofReal r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
c : ℂ
r : ℝ
rp : r > 0
⊢ 0 < ENNReal.ofReal r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | LocalVolume.closedBall | [335, 1] | [343, 73] | apply LocalVolume.closure_interior | c : ℂ
r : ℝ
rp : r > 0
⊢ LocalVolumeSet (Metric.closedBall c r) | case bp
c : ℂ
r : ℝ
rp : r > 0
⊢ ∀ (x : ℂ), ∀ r > 0, ↑volume (ball x r) > 0
case ci
c : ℂ
r : ℝ
rp : r > 0
⊢ Metric.closedBall c r ⊆ closure (interior (Metric.closedBall c r)) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r : ℝ
rp : r > 0
⊢ LocalVolumeSet (Metric.closedBall c r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | LocalVolume.closedBall | [335, 1] | [343, 73] | intro x r rp | case bp
c : ℂ
r : ℝ
rp : r > 0
⊢ ∀ (x : ℂ), ∀ r > 0, ↑volume (ball x r) > 0 | case bp
c : ℂ
r✝ : ℝ
rp✝ : r✝ > 0
x : ℂ
r : ℝ
rp : r > 0
⊢ ↑volume (ball x r) > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case bp
c : ℂ
r : ℝ
rp : r > 0
⊢ ∀ (x : ℂ), ∀ r > 0, ↑volume (ball x r) > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | LocalVolume.closedBall | [335, 1] | [343, 73] | simp only [Complex.volume_ball, gt_iff_lt, CanonicallyOrderedCommSemiring.mul_pos,
ENNReal.coe_pos, NNReal.pi_pos, and_true] | case bp
c : ℂ
r✝ : ℝ
rp✝ : r✝ > 0
x : ℂ
r : ℝ
rp : r > 0
⊢ ↑volume (ball x r) > 0 | case bp
c : ℂ
r✝ : ℝ
rp✝ : r✝ > 0
x : ℂ
r : ℝ
rp : r > 0
⊢ 0 < ENNReal.ofReal r ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case bp
c : ℂ
r✝ : ℝ
rp✝ : r✝ > 0
x : ℂ
r : ℝ
rp : r > 0
⊢ ↑volume (ball x r) > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | LocalVolume.closedBall | [335, 1] | [343, 73] | apply ENNReal.pow_pos | case bp
c : ℂ
r✝ : ℝ
rp✝ : r✝ > 0
x : ℂ
r : ℝ
rp : r > 0
⊢ 0 < ENNReal.ofReal r ^ 2 | case bp.a
c : ℂ
r✝ : ℝ
rp✝ : r✝ > 0
x : ℂ
r : ℝ
rp : r > 0
⊢ 0 < ENNReal.ofReal r | Please generate a tactic in lean4 to solve the state.
STATE:
case bp
c : ℂ
r✝ : ℝ
rp✝ : r✝ > 0
x : ℂ
r : ℝ
rp : r > 0
⊢ 0 < ENNReal.ofReal r ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | LocalVolume.closedBall | [335, 1] | [343, 73] | bound | case bp.a
c : ℂ
r✝ : ℝ
rp✝ : r✝ > 0
x : ℂ
r : ℝ
rp : r > 0
⊢ 0 < ENNReal.ofReal r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case bp.a
c : ℂ
r✝ : ℝ
rp✝ : r✝ > 0
x : ℂ
r : ℝ
rp : r > 0
⊢ 0 < ENNReal.ofReal r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | LocalVolume.closedBall | [335, 1] | [343, 73] | have rz := rp.ne' | case ci
c : ℂ
r : ℝ
rp : r > 0
⊢ Metric.closedBall c r ⊆ closure (interior (Metric.closedBall c r)) | case ci
c : ℂ
r : ℝ
rp : r > 0
rz : r ≠ 0
⊢ Metric.closedBall c r ⊆ closure (interior (Metric.closedBall c r)) | Please generate a tactic in lean4 to solve the state.
STATE:
case ci
c : ℂ
r : ℝ
rp : r > 0
⊢ Metric.closedBall c r ⊆ closure (interior (Metric.closedBall c r))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | LocalVolume.closedBall | [335, 1] | [343, 73] | simp only [interior_closedBall c rz, closure_ball c rz, subset_refl] | case ci
c : ℂ
r : ℝ
rp : r > 0
rz : r ≠ 0
⊢ Metric.closedBall c r ⊆ closure (interior (Metric.closedBall c r)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ci
c : ℂ
r : ℝ
rp : r > 0
rz : r ≠ 0
⊢ Metric.closedBall c r ⊆ closure (interior (Metric.closedBall c r))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Mandelbrot.lean | approx_bad_potential_image | [60, 1] | [75, 9] | rw [potential_image, bad_potential_image] | c : ℚ × ℚ
n : ℕ
r : Floating
⊢ potential_image ↑c ∈ approx (bad_potential_image n r c) | c : ℚ × ℚ
n : ℕ
r : Floating
⊢ (let p := potential 2 ↑↑c;
let t := p ^ 16;
lerp t ↑clear ↑outside) ∈
approx
(let c := Box.ofRat c;
let p := c.potential c n r;
let t := p.1.sqr.sqr.sqr.sqr;
let i := lerp t clear' outside';
i.quantize) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℚ × ℚ
n : ℕ
r : Floating
⊢ potential_image ↑c ∈ approx (bad_potential_image n r c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Mandelbrot.lean | approx_bad_potential_image | [60, 1] | [75, 9] | have e : ∀ p : ℝ, p^16 = (((p^2)^2)^2)^2 := by intro p; rw [←pow_mul, ←pow_mul, ←pow_mul] | c : ℚ × ℚ
n : ℕ
r : Floating
⊢ (let p := potential 2 ↑↑c;
let t := p ^ 16;
lerp t ↑clear ↑outside) ∈
approx
(let c := Box.ofRat c;
let p := c.potential c n r;
let t := p.1.sqr.sqr.sqr.sqr;
let i := lerp t clear' outside';
i.quantize) | c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ (let p := potential 2 ↑↑c;
let t := p ^ 16;
lerp t ↑clear ↑outside) ∈
approx
(let c := Box.ofRat c;
let p := c.potential c n r;
let t := p.1.sqr.sqr.sqr.sqr;
let i := lerp t clear' outside';
i.quantize) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℚ × ℚ
n : ℕ
r : Floating
⊢ (let p := potential 2 ↑↑c;
let t := p ^ 16;
lerp t ↑clear ↑outside) ∈
approx
(let c := Box.ofRat c;
let p := c.potential c n r;
let t := p.1.sqr.sqr.sqr.sqr;
let i := lerp t clear' outside';
i.quantize)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Mandelbrot.lean | approx_bad_potential_image | [60, 1] | [75, 9] | simp only [far', outside', clear', green', e] | c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ (let p := potential 2 ↑↑c;
let t := p ^ 16;
lerp t ↑clear ↑outside) ∈
approx
(let c := Box.ofRat c;
let p := c.potential c n r;
let t := p.1.sqr.sqr.sqr.sqr;
let i := lerp t clear' outside';
i.quantize) | c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ lerp ((((potential 2 ↑↑c ^ 2) ^ 2) ^ 2) ^ 2) ↑clear ↑outside ∈
approx (lerp ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr.sqr clear.ofRat outside.ofRat).quantize | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ (let p := potential 2 ↑↑c;
let t := p ^ 16;
lerp t ↑clear ↑outside) ∈
approx
(let c := Box.ofRat c;
let p := c.potential c n r;
let t := p.1.sqr.sqr.sqr.sqr;
let i := lerp t clear' outside';
i.quantize)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Mandelbrot.lean | approx_bad_potential_image | [60, 1] | [75, 9] | apply Color.mem_approx_quantize | c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ lerp ((((potential 2 ↑↑c ^ 2) ^ 2) ^ 2) ^ 2) ↑clear ↑outside ∈
approx (lerp ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr.sqr clear.ofRat outside.ofRat).quantize | case cm
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ lerp ((((potential 2 ↑↑c ^ 2) ^ 2) ^ 2) ^ 2) ↑clear ↑outside ∈
approx (lerp ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr.sqr clear.ofRat outside.ofRat) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ lerp ((((potential 2 ↑↑c ^ 2) ^ 2) ^ 2) ^ 2) ↑clear ↑outside ∈
approx (lerp ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr.sqr clear.ofRat outside.ofRat).quantize
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Mandelbrot.lean | approx_bad_potential_image | [60, 1] | [75, 9] | apply mem_approx_lerp | case cm
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ lerp ((((potential 2 ↑↑c ^ 2) ^ 2) ^ 2) ^ 2) ↑clear ↑outside ∈
approx (lerp ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr.sqr clear.ofRat outside.ofRat) | case cm.tm
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ (((potential 2 ↑↑c ^ 2) ^ 2) ^ 2) ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr.sqr
case cm.xm
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ ↑clear ∈ approx clear.ofRat
case cm.ym
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ ↑outside ∈ approx outside.ofRat | Please generate a tactic in lean4 to solve the state.
STATE:
case cm
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ lerp ((((potential 2 ↑↑c ^ 2) ^ 2) ^ 2) ^ 2) ↑clear ↑outside ∈
approx (lerp ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr.sqr clear.ofRat outside.ofRat)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Mandelbrot.lean | approx_bad_potential_image | [60, 1] | [75, 9] | intro p | c : ℚ × ℚ
n : ℕ
r : Floating
⊢ ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 | c : ℚ × ℚ
n : ℕ
r : Floating
p : ℝ
⊢ p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℚ × ℚ
n : ℕ
r : Floating
⊢ ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Mandelbrot.lean | approx_bad_potential_image | [60, 1] | [75, 9] | rw [←pow_mul, ←pow_mul, ←pow_mul] | c : ℚ × ℚ
n : ℕ
r : Floating
p : ℝ
⊢ p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℚ × ℚ
n : ℕ
r : Floating
p : ℝ
⊢ p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Mandelbrot.lean | approx_bad_potential_image | [60, 1] | [75, 9] | apply Interval.mem_approx_sqr | case cm.tm
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ (((potential 2 ↑↑c ^ 2) ^ 2) ^ 2) ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr.sqr | case cm.tm.ax
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ ((potential 2 ↑↑c ^ 2) ^ 2) ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr | Please generate a tactic in lean4 to solve the state.
STATE:
case cm.tm
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ (((potential 2 ↑↑c ^ 2) ^ 2) ^ 2) ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr.sqr
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Mandelbrot.lean | approx_bad_potential_image | [60, 1] | [75, 9] | apply Interval.mem_approx_sqr | case cm.tm.ax
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ ((potential 2 ↑↑c ^ 2) ^ 2) ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr | case cm.tm.ax.ax
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ (potential 2 ↑↑c ^ 2) ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr | Please generate a tactic in lean4 to solve the state.
STATE:
case cm.tm.ax
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ ((potential 2 ↑↑c ^ 2) ^ 2) ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Mandelbrot.lean | approx_bad_potential_image | [60, 1] | [75, 9] | apply Interval.mem_approx_sqr | case cm.tm.ax.ax
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ (potential 2 ↑↑c ^ 2) ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr | case cm.tm.ax.ax.ax
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ potential 2 ↑↑c ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr | Please generate a tactic in lean4 to solve the state.
STATE:
case cm.tm.ax.ax
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ (potential 2 ↑↑c ^ 2) ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Mandelbrot.lean | approx_bad_potential_image | [60, 1] | [75, 9] | apply Interval.mem_approx_sqr | case cm.tm.ax.ax.ax
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ potential 2 ↑↑c ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr | case cm.tm.ax.ax.ax.ax
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ potential 2 ↑↑c ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1 | Please generate a tactic in lean4 to solve the state.
STATE:
case cm.tm.ax.ax.ax
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ potential 2 ↑↑c ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Mandelbrot.lean | approx_bad_potential_image | [60, 1] | [75, 9] | mono | case cm.tm.ax.ax.ax.ax
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ potential 2 ↑↑c ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cm.tm.ax.ax.ax.ax
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ potential 2 ↑↑c ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Mandelbrot.lean | approx_bad_potential_image | [60, 1] | [75, 9] | mono | case cm.xm
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ ↑clear ∈ approx clear.ofRat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cm.xm
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ ↑clear ∈ approx clear.ofRat
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Mandelbrot.lean | approx_bad_potential_image | [60, 1] | [75, 9] | mono | case cm.ym
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ ↑outside ∈ approx outside.ofRat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cm.ym
c : ℚ × ℚ
n : ℕ
r : Floating
e : ∀ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
⊢ ↑outside ∈ approx outside.ofRat
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.coe_eq_zero | [40, 9] | [41, 37] | simp only [← coe_zero, coe_eq_coe] | z : ℂ
⊢ ↑z = 0 ↔ z = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ ↑z = 0 ↔ z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.map_rec | [54, 1] | [58, 39] | induction z using OnePoint.rec | A : Sort u_1
B : Sort u_2
g : A → B
f : ℂ → A
i : A
z : 𝕊
⊢ g (OnePoint.rec i f z) = OnePoint.rec (g i) (g ∘ f) z | case h₁
A : Sort u_1
B : Sort u_2
g : A → B
f : ℂ → A
i : A
⊢ g (OnePoint.rec i f ∞) = OnePoint.rec (g i) (g ∘ f) ∞
case h₂
A : Sort u_1
B : Sort u_2
g : A → B
f : ℂ → A
i : A
x✝ : ℂ
⊢ g (OnePoint.rec i f ↑x✝) = OnePoint.rec (g i) (g ∘ f) ↑x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
A : Sort u_1
B : Sort u_2
g : A → B
f : ℂ → A
i : A
z : 𝕊
⊢ g (OnePoint.rec i f z) = OnePoint.rec (g i) (g ∘ f) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.map_rec | [54, 1] | [58, 39] | simp only [rec_inf] | case h₁
A : Sort u_1
B : Sort u_2
g : A → B
f : ℂ → A
i : A
⊢ g (OnePoint.rec i f ∞) = OnePoint.rec (g i) (g ∘ f) ∞ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
A : Sort u_1
B : Sort u_2
g : A → B
f : ℂ → A
i : A
⊢ g (OnePoint.rec i f ∞) = OnePoint.rec (g i) (g ∘ f) ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.map_rec | [54, 1] | [58, 39] | simp only [rec_coe, Function.comp] | case h₂
A : Sort u_1
B : Sort u_2
g : A → B
f : ℂ → A
i : A
x✝ : ℂ
⊢ g (OnePoint.rec i f ↑x✝) = OnePoint.rec (g i) (g ∘ f) ↑x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
A : Sort u_1
B : Sort u_2
g : A → B
f : ℂ → A
i : A
x✝ : ℂ
⊢ g (OnePoint.rec i f ↑x✝) = OnePoint.rec (g i) (g ∘ f) ↑x✝
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inf_ne_coe | [61, 1] | [62, 55] | simp only [Ne, OnePoint.infty_ne_coe, not_false_iff] | z : ℂ
⊢ ∞ ≠ ↑z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ ∞ ≠ ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inf_ne_zero | [63, 1] | [64, 68] | have e : (0 : 𝕊) = ((0 : ℂ) : 𝕊) := rfl | ⊢ ∞ ≠ 0 | e : 0 = ↑0
⊢ ∞ ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∞ ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inf_ne_zero | [63, 1] | [64, 68] | rw [e] | e : 0 = ↑0
⊢ ∞ ≠ 0 | e : 0 = ↑0
⊢ ∞ ≠ ↑0 | Please generate a tactic in lean4 to solve the state.
STATE:
e : 0 = ↑0
⊢ ∞ ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inf_ne_zero | [63, 1] | [64, 68] | exact inf_ne_coe | e : 0 = ↑0
⊢ ∞ ≠ ↑0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
e : 0 = ↑0
⊢ ∞ ≠ ↑0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.coe_toComplex | [71, 1] | [74, 30] | induction z using OnePoint.rec | z : 𝕊
h : z ≠ ∞
⊢ ↑z.toComplex = z | case h₁
h : ∞ ≠ ∞
⊢ ↑∞.toComplex = ∞
case h₂
x✝ : ℂ
h : ↑x✝ ≠ ∞
⊢ ↑(↑x✝).toComplex = ↑x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
z : 𝕊
h : z ≠ ∞
⊢ ↑z.toComplex = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.coe_toComplex | [71, 1] | [74, 30] | simp only [ne_eq, not_true_eq_false] at h | case h₁
h : ∞ ≠ ∞
⊢ ↑∞.toComplex = ∞ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
h : ∞ ≠ ∞
⊢ ↑∞.toComplex = ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.coe_toComplex | [71, 1] | [74, 30] | simp only [toComplex_coe] | case h₂
x✝ : ℂ
h : ↑x✝ ≠ ∞
⊢ ↑(↑x✝).toComplex = ↑x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
x✝ : ℂ
h : ↑x✝ ≠ ∞
⊢ ↑(↑x✝).toComplex = ↑x✝
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.toComplex_zero | [75, 9] | [75, 92] | rw [← coe_zero, toComplex_coe] | ⊢ OnePoint.toComplex 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ OnePoint.toComplex 0 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_toComplex | [76, 1] | [77, 93] | simp only [OnePoint.continuousAt_coe, Function.comp, toComplex_coe] | z : ℂ
⊢ ContinuousAt OnePoint.toComplex ↑z | z : ℂ
⊢ ContinuousAt (fun x => x) z | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ ContinuousAt OnePoint.toComplex ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousAt_toComplex | [76, 1] | [77, 93] | exact continuousAt_id | z : ℂ
⊢ ContinuousAt (fun x => x) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ ContinuousAt (fun x => x) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousOn_toComplex | [78, 1] | [81, 52] | intro z m | ⊢ ContinuousOn OnePoint.toComplex {∞}ᶜ | z : 𝕊
m : z ∈ {∞}ᶜ
⊢ ContinuousWithinAt OnePoint.toComplex {∞}ᶜ z | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ContinuousOn OnePoint.toComplex {∞}ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousOn_toComplex | [78, 1] | [81, 52] | induction z using OnePoint.rec | z : 𝕊
m : z ∈ {∞}ᶜ
⊢ ContinuousWithinAt OnePoint.toComplex {∞}ᶜ z | case h₁
m : ∞ ∈ {∞}ᶜ
⊢ ContinuousWithinAt OnePoint.toComplex {∞}ᶜ ∞
case h₂
x✝ : ℂ
m : ↑x✝ ∈ {∞}ᶜ
⊢ ContinuousWithinAt OnePoint.toComplex {∞}ᶜ ↑x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
z : 𝕊
m : z ∈ {∞}ᶜ
⊢ ContinuousWithinAt OnePoint.toComplex {∞}ᶜ z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousOn_toComplex | [78, 1] | [81, 52] | simp only [mem_compl_iff, mem_singleton_iff, not_true] at m | case h₁
m : ∞ ∈ {∞}ᶜ
⊢ ContinuousWithinAt OnePoint.toComplex {∞}ᶜ ∞ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
m : ∞ ∈ {∞}ᶜ
⊢ ContinuousWithinAt OnePoint.toComplex {∞}ᶜ ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.continuousOn_toComplex | [78, 1] | [81, 52] | exact continuousAt_toComplex.continuousWithinAt | case h₂
x✝ : ℂ
m : ↑x✝ ∈ {∞}ᶜ
⊢ ContinuousWithinAt OnePoint.toComplex {∞}ᶜ ↑x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
x✝ : ℂ
m : ↑x✝ ∈ {∞}ᶜ
⊢ ContinuousWithinAt OnePoint.toComplex {∞}ᶜ ↑x✝
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_zero' | [97, 9] | [97, 100] | simp only [inv_def, inv, eq_self_iff_true, if_true] | ⊢ 0⁻¹ = ∞ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 0⁻¹ = ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_inf | [98, 9] | [98, 85] | simp [inv_def, inv, inf_ne_zero] | ⊢ ∞⁻¹ = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∞⁻¹ = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_coe | [100, 1] | [101, 90] | simp only [inv_def, inv, z0, WithTop.coe_eq_zero, toComplex_coe, if_false, coe_eq_zero] | z : ℂ
z0 : z ≠ 0
⊢ (↑z)⁻¹ = ↑z⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z0 : z ≠ 0
⊢ (↑z)⁻¹ = ↑z⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/RiemannSphere.lean | RiemannSphere.inv_eq_inf | [102, 9] | [105, 89] | induction z using OnePoint.rec | z : 𝕊
⊢ z⁻¹ = ∞ ↔ z = 0 | case h₁
⊢ ∞⁻¹ = ∞ ↔ ∞ = 0
case h₂
x✝ : ℂ
⊢ (↑x✝)⁻¹ = ∞ ↔ ↑x✝ = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
z : 𝕊
⊢ z⁻¹ = ∞ ↔ z = 0
TACTIC:
|
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