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stringclasses 147
values | file_path
stringlengths 7
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stringlengths 1
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stringlengths 6
10
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stringlengths 6
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stringlengths 1
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stringlengths 73
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|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Measurable.square | [129, 1] | [130, 54] | apply_rules [MeasurableSet.prod, measurableSet_Ioc] | r0 r1 : β
β’ MeasurableSet (_root_.square r0 r1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r0 r1 : β
β’ MeasurableSet (_root_.square r0 r1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | rw [β MeasurableEquiv.image_eq_preimage] | c : β
r0 r1 : β
r0p : 0 β€ r0
β’ βComplex.measurableEquivRealProd.symm β»ΒΉ' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1 | c : β
r0 r1 : β
r0p : 0 β€ r0
β’ βComplex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
β’ βComplex.measurableEquivRealProd.symm β»ΒΉ' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | have e : realCircleMap c =
fun x : β Γ β β¦ Complex.measurableEquivRealProd (circleMap c x.1 x.2) := by
funext
simp only [realCircleMap_eq_circleMap, Complex.measurableEquivRealProd,
Complex.equivRealProd_apply, Homeomorph.toMeasurableEquiv_coe,
ContinuousLinearEquiv.coe_toHomeomorph, Complex.equivRealProdCLM_apply] | c : β
r0 r1 : β
r0p : 0 β€ r0
β’ βComplex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1 | c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
β’ βComplex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
β’ βComplex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | have im := image_comp Complex.measurableEquivRealProd (fun x : β Γ β β¦ circleMap c x.1 x.2)
(square r0 r1) | c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
i : (fun x => circleMap c x.1 x.2) '' square r0 r1 = annulus_oc c r0 r1
β’ βComplex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1 | c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
i : (fun x => circleMap c x.1 x.2) '' square r0 r1 = annulus_oc c r0 r1
im :
(βComplex.measurableEquivRealProd β fun x => circleMap c x.1 x.2) '' square r0 r1 =
βComplex.measurableEquivRealProd '' ((fun x => circleMap c x.1 x.2) '' square r0 r1)
β’ βComplex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
i : (fun x => circleMap c x.1 x.2) '' square r0 r1 = annulus_oc c r0 r1
β’ βComplex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | simp only [Function.comp] at im | c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
i : (fun x => circleMap c x.1 x.2) '' square r0 r1 = annulus_oc c r0 r1
im :
(βComplex.measurableEquivRealProd β fun x => circleMap c x.1 x.2) '' square r0 r1 =
βComplex.measurableEquivRealProd '' ((fun x => circleMap c x.1 x.2) '' square r0 r1)
β’ βComplex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1 | c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
i : (fun x => circleMap c x.1 x.2) '' square r0 r1 = annulus_oc c r0 r1
im :
(fun a => Complex.measurableEquivRealProd (circleMap c a.1 a.2)) '' square r0 r1 =
βComplex.measurableEquivRealProd '' ((fun x => circleMap c x.1 x.2) '' square r0 r1)
β’ βComplex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
i : (fun x => circleMap c x.1 x.2) '' square r0 r1 = annulus_oc c r0 r1
im :
(βComplex.measurableEquivRealProd β fun x => circleMap c x.1 x.2) '' square r0 r1 =
βComplex.measurableEquivRealProd '' ((fun x => circleMap c x.1 x.2) '' square r0 r1)
β’ βComplex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | simp only [e, im, i] | c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
i : (fun x => circleMap c x.1 x.2) '' square r0 r1 = annulus_oc c r0 r1
im :
(fun a => Complex.measurableEquivRealProd (circleMap c a.1 a.2)) '' square r0 r1 =
βComplex.measurableEquivRealProd '' ((fun x => circleMap c x.1 x.2) '' square r0 r1)
β’ βComplex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
i : (fun x => circleMap c x.1 x.2) '' square r0 r1 = annulus_oc c r0 r1
im :
(fun a => Complex.measurableEquivRealProd (circleMap c a.1 a.2)) '' square r0 r1 =
βComplex.measurableEquivRealProd '' ((fun x => circleMap c x.1 x.2) '' square r0 r1)
β’ βComplex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | funext | c : β
r0 r1 : β
r0p : 0 β€ r0
β’ realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) | case h
c : β
r0 r1 : β
r0p : 0 β€ r0
xβ : β Γ β
β’ realCircleMap c xβ = Complex.measurableEquivRealProd (circleMap c xβ.1 xβ.2) | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
β’ realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | simp only [realCircleMap_eq_circleMap, Complex.measurableEquivRealProd,
Complex.equivRealProd_apply, Homeomorph.toMeasurableEquiv_coe,
ContinuousLinearEquiv.coe_toHomeomorph, Complex.equivRealProdCLM_apply] | case h
c : β
r0 r1 : β
r0p : 0 β€ r0
xβ : β Γ β
β’ realCircleMap c xβ = Complex.measurableEquivRealProd (circleMap c xβ.1 xβ.2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : β
r0 r1 : β
r0p : 0 β€ r0
xβ : β Γ β
β’ realCircleMap c xβ = Complex.measurableEquivRealProd (circleMap c xβ.1 xβ.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | ext z | c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
β’ (fun x => circleMap c x.1 x.2) '' square r0 r1 = annulus_oc c r0 r1 | case h
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
β’ z β (fun x => circleMap c x.1 x.2) '' square r0 r1 β z β annulus_oc c r0 r1 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
β’ (fun x => circleMap c x.1 x.2) '' square r0 r1 = annulus_oc c r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | rw [mem_image] | case h
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
β’ z β (fun x => circleMap c x.1 x.2) '' square r0 r1 β z β annulus_oc c r0 r1 | case h
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
β’ (β x β square r0 r1, circleMap c x.1 x.2 = z) β z β annulus_oc c r0 r1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
β’ z β (fun x => circleMap c x.1 x.2) '' square r0 r1 β z β annulus_oc c r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | constructor | case h
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
β’ (β x β square r0 r1, circleMap c x.1 x.2 = z) β z β annulus_oc c r0 r1 | case h.mp
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
β’ (β x β square r0 r1, circleMap c x.1 x.2 = z) β z β annulus_oc c r0 r1
case h.mpr
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
β’ z β annulus_oc c r0 r1 β β x β square r0 r1, circleMap c x.1 x.2 = z | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
β’ (β x β square r0 r1, circleMap c x.1 x.2 = z) β z β annulus_oc c r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | intro gp | case h.mp
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
β’ (β x β square r0 r1, circleMap c x.1 x.2 = z) β z β annulus_oc c r0 r1 | case h.mp
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
gp : β x β square r0 r1, circleMap c x.1 x.2 = z
β’ z β annulus_oc c r0 r1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
β’ (β x β square r0 r1, circleMap c x.1 x.2 = z) β z β annulus_oc c r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | rcases gp with β¨β¨s, tβ©, ss, tzβ© | case h.mp
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
gp : β x β square r0 r1, circleMap c x.1 x.2 = z
β’ z β annulus_oc c r0 r1 | case h.mp.intro.mk.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
s t : β
ss : (s, t) β square r0 r1
tz : circleMap c (s, t).1 (s, t).2 = z
β’ z β annulus_oc c r0 r1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
gp : β x β square r0 r1, circleMap c x.1 x.2 = z
β’ z β annulus_oc c r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | simp only at tz | case h.mp.intro.mk.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
s t : β
ss : (s, t) β square r0 r1
tz : circleMap c (s, t).1 (s, t).2 = z
β’ z β annulus_oc c r0 r1 | case h.mp.intro.mk.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
s t : β
ss : (s, t) β square r0 r1
tz : circleMap c s t = z
β’ z β annulus_oc c r0 r1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.mk.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
s t : β
ss : (s, t) β square r0 r1
tz : circleMap c (s, t).1 (s, t).2 = z
β’ z β annulus_oc c r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | simp only [square, prod_mk_mem_set_prod_eq, mem_Ioc] at ss | case h.mp.intro.mk.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
s t : β
ss : (s, t) β square r0 r1
tz : circleMap c s t = z
β’ z β annulus_oc c r0 r1 | case h.mp.intro.mk.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
s t : β
tz : circleMap c s t = z
ss : (r0 < s β§ s β€ r1) β§ 0 < t β§ t β€ 2 * Ο
β’ z β annulus_oc c r0 r1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.mk.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
s t : β
ss : (s, t) β square r0 r1
tz : circleMap c s t = z
β’ z β annulus_oc c r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | rw [β tz] | case h.mp.intro.mk.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
s t : β
tz : circleMap c s t = z
ss : (r0 < s β§ s β€ r1) β§ 0 < t β§ t β€ 2 * Ο
β’ z β annulus_oc c r0 r1 | case h.mp.intro.mk.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
s t : β
tz : circleMap c s t = z
ss : (r0 < s β§ s β€ r1) β§ 0 < t β§ t β€ 2 * Ο
β’ circleMap c s t β annulus_oc c r0 r1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.mk.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
s t : β
tz : circleMap c s t = z
ss : (r0 < s β§ s β€ r1) β§ 0 < t β§ t β€ 2 * Ο
β’ z β annulus_oc c r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | have s0 : 0 < s := by linarith | case h.mp.intro.mk.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
s t : β
tz : circleMap c s t = z
ss : (r0 < s β§ s β€ r1) β§ 0 < t β§ t β€ 2 * Ο
β’ circleMap c s t β annulus_oc c r0 r1 | case h.mp.intro.mk.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
s t : β
tz : circleMap c s t = z
ss : (r0 < s β§ s β€ r1) β§ 0 < t β§ t β€ 2 * Ο
s0 : 0 < s
β’ circleMap c s t β annulus_oc c r0 r1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.mk.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
s t : β
tz : circleMap c s t = z
ss : (r0 < s β§ s β€ r1) β§ 0 < t β§ t β€ 2 * Ο
β’ circleMap c s t β annulus_oc c r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | simp only [circleMap, add_comm c, annulus_oc, mem_diff, Metric.mem_closedBall,
dist_add_self_left, norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal,
Complex.abs_exp_ofReal_mul_I, mul_one, not_le, abs_of_pos s0, ss.1, true_and] | case h.mp.intro.mk.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
s t : β
tz : circleMap c s t = z
ss : (r0 < s β§ s β€ r1) β§ 0 < t β§ t β€ 2 * Ο
s0 : 0 < s
β’ circleMap c s t β annulus_oc c r0 r1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.mk.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
s t : β
tz : circleMap c s t = z
ss : (r0 < s β§ s β€ r1) β§ 0 < t β§ t β€ 2 * Ο
s0 : 0 < s
β’ circleMap c s t β annulus_oc c r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | linarith | c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
s t : β
tz : circleMap c s t = z
ss : (r0 < s β§ s β€ r1) β§ 0 < t β§ t β€ 2 * Ο
β’ 0 < s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
s t : β
tz : circleMap c s t = z
ss : (r0 < s β§ s β€ r1) β§ 0 < t β§ t β€ 2 * Ο
β’ 0 < s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | intro zr | case h.mpr
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
β’ z β annulus_oc c r0 r1 β β x β square r0 r1, circleMap c x.1 x.2 = z | case h.mpr
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
zr : z β annulus_oc c r0 r1
β’ β x β square r0 r1, circleMap c x.1 x.2 = z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
β’ z β annulus_oc c r0 r1 β β x β square r0 r1, circleMap c x.1 x.2 = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | simp only [mem_diff, Metric.mem_closedBall, mem_singleton_iff, annulus_oc,
not_le] at zr | case h.mpr
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
zr : z β annulus_oc c r0 r1
β’ β x β square r0 r1, circleMap c x.1 x.2 = z | case h.mpr
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
zr : dist z c β€ r1 β§ r0 < dist z c
β’ β x β square r0 r1, circleMap c x.1 x.2 = z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
zr : z β annulus_oc c r0 r1
β’ β x β square r0 r1, circleMap c x.1 x.2 = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | rw [dist_comm] at zr | case h.mpr
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
zr : dist z c β€ r1 β§ r0 < dist z c
β’ β x β square r0 r1, circleMap c x.1 x.2 = z | case h.mpr
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
zr : dist c z β€ r1 β§ r0 < dist c z
β’ β x β square r0 r1, circleMap c x.1 x.2 = z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
zr : dist z c β€ r1 β§ r0 < dist z c
β’ β x β square r0 r1, circleMap c x.1 x.2 = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | have zz : z β sphere c (dist c z) := by
simp only [Complex.dist_eq, mem_sphere_iff_norm, Complex.norm_eq_abs, Complex.abs.map_sub] | case h.mpr
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
zr : dist c z β€ r1 β§ r0 < dist c z
β’ β x β square r0 r1, circleMap c x.1 x.2 = z | case h.mpr
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
zr : dist c z β€ r1 β§ r0 < dist c z
zz : z β sphere c (dist c z)
β’ β x β square r0 r1, circleMap c x.1 x.2 = z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
zr : dist c z β€ r1 β§ r0 < dist c z
β’ β x β square r0 r1, circleMap c x.1 x.2 = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | rcases circleMap_Ioc zz with β¨t, ts, tzβ© | case h.mpr
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
zr : dist c z β€ r1 β§ r0 < dist c z
zz : z β sphere c (dist c z)
β’ β x β square r0 r1, circleMap c x.1 x.2 = z | case h.mpr.intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
zr : dist c z β€ r1 β§ r0 < dist c z
zz : z β sphere c (dist c z)
t : β
ts : t β Ioc 0 (2 * Ο)
tz : z = circleMap c (dist c z) t
β’ β x β square r0 r1, circleMap c x.1 x.2 = z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
zr : dist c z β€ r1 β§ r0 < dist c z
zz : z β sphere c (dist c z)
β’ β x β square r0 r1, circleMap c x.1 x.2 = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | use (dist c z, t) | case h.mpr.intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
zr : dist c z β€ r1 β§ r0 < dist c z
zz : z β sphere c (dist c z)
t : β
ts : t β Ioc 0 (2 * Ο)
tz : z = circleMap c (dist c z) t
β’ β x β square r0 r1, circleMap c x.1 x.2 = z | case h
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
zr : dist c z β€ r1 β§ r0 < dist c z
zz : z β sphere c (dist c z)
t : β
ts : t β Ioc 0 (2 * Ο)
tz : z = circleMap c (dist c z) t
β’ (dist c z, t) β square r0 r1 β§ circleMap c (dist c z, t).1 (dist c z, t).2 = z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
zr : dist c z β€ r1 β§ r0 < dist c z
zz : z β sphere c (dist c z)
t : β
ts : t β Ioc 0 (2 * Ο)
tz : z = circleMap c (dist c z) t
β’ β x β square r0 r1, circleMap c x.1 x.2 = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | simpa only [square, gt_iff_lt, not_lt, ge_iff_le, zero_lt_two, mul_pos_iff_of_pos_left,
mem_prod, mem_Ioc, dist_pos, ne_eq, not_false_eq_true, zr, and_self, true_and,
tz.symm, and_true] using ts | case h
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
zr : dist c z β€ r1 β§ r0 < dist c z
zz : z β sphere c (dist c z)
t : β
ts : t β Ioc 0 (2 * Ο)
tz : z = circleMap c (dist c z) t
β’ (dist c z, t) β square r0 r1 β§ circleMap c (dist c z, t).1 (dist c z, t).2 = z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
zr : dist c z β€ r1 β§ r0 < dist c z
zz : z β sphere c (dist c z)
t : β
ts : t β Ioc 0 (2 * Ο)
tz : z = circleMap c (dist c z) t
β’ (dist c z, t) β square r0 r1 β§ circleMap c (dist c z, t).1 (dist c z, t).2 = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square_eq | [139, 1] | [175, 23] | simp only [Complex.dist_eq, mem_sphere_iff_norm, Complex.norm_eq_abs, Complex.abs.map_sub] | c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
zr : dist c z β€ r1 β§ r0 < dist c z
β’ z β sphere c (dist c z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
z : β
zr : dist c z β€ r1 β§ r0 < dist c z
β’ z β sphere c (dist c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | exp_of_im | [178, 1] | [179, 71] | simp [Complex.ext_iff, Complex.cos_ofReal_re, Complex.sin_ofReal_re] | t : β
β’ (βt * I).exp = βt.cos + βt.sin * I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
β’ (βt * I).exp = βt.cos + βt.sin * I
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Complex.cos_eq_cos | [181, 1] | [181, 78] | simp | t : β
β’ (βt).cos = βt.cos | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
β’ (βt).cos = βt.cos
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Complex.sin_eq_sin | [183, 1] | [183, 78] | simp | t : β
β’ (βt).sin = βt.sin | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
β’ (βt).sin = βt.sin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | generalize hn : βt / (2 * Ο) - 1 / 2β = n | t : β
β’ β n, (βt * I).exp.arg = t - 2 * Ο * βn | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
β’ β n, (βt * I).exp.arg = t - 2 * Ο * βn | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
β’ β n, (βt * I).exp.arg = t - 2 * Ο * βn
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | exists n | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
β’ β n, (βt * I).exp.arg = t - 2 * Ο * βn | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
β’ (βt * I).exp.arg = t - 2 * Ο * βn | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
β’ β n, (βt * I).exp.arg = t - 2 * Ο * βn
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | have en : exp (2 * Ο * n * I) = 1 := by
rw [mul_comm _ (n:β), mul_assoc, Complex.exp_int_mul]
simp only [Complex.exp_two_pi_mul_I, one_zpow] | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
β’ (βt * I).exp.arg = t - 2 * Ο * βn | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
β’ (βt * I).exp.arg = t - 2 * Ο * βn | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
β’ (βt * I).exp.arg = t - 2 * Ο * βn
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | have e : exp (t * I) = exp (β(t - 2 * Ο * n) * I) := by
simp [mul_sub_right_distrib, Complex.exp_sub, en] | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
β’ (βt * I).exp.arg = t - 2 * Ο * βn | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ (βt * I).exp.arg = t - 2 * Ο * βn | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
β’ (βt * I).exp.arg = t - 2 * Ο * βn
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | rw [e, exp_of_im, β Complex.cos_eq_cos, β Complex.sin_eq_sin, Complex.arg_cos_add_sin_mul_I ts] | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
ts : t - 2 * Ο * βn β Ioc (-Ο) Ο
β’ (βt * I).exp.arg = t - 2 * Ο * βn | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
ts : t - 2 * Ο * βn β Ioc (-Ο) Ο
β’ (βt * I).exp.arg = t - 2 * Ο * βn
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | rw [mul_comm _ (n:β), mul_assoc, Complex.exp_int_mul] | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
β’ (2 * βΟ * βn * I).exp = 1 | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
β’ (2 * βΟ * I).exp ^ n = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
β’ (2 * βΟ * βn * I).exp = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | simp only [Complex.exp_two_pi_mul_I, one_zpow] | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
β’ (2 * βΟ * I).exp ^ n = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
β’ (2 * βΟ * I).exp ^ n = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | simp [mul_sub_right_distrib, Complex.exp_sub, en] | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
β’ (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
β’ (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | simp only [mem_Ioc, neg_lt_sub_iff_lt_add, tsub_le_iff_right] | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ t - 2 * Ο * βn β Ioc (-Ο) Ο | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ 2 * Ο * βn < Ο + t β§ t β€ Ο + 2 * Ο * βn | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ t - 2 * Ο * βn β Ioc (-Ο) Ο
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | constructor | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ 2 * Ο * βn < Ο + t β§ t β€ Ο + 2 * Ο * βn | case left
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ 2 * Ο * βn < Ο + t
case right
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ t β€ Ο + 2 * Ο * βn | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ 2 * Ο * βn < Ο + t β§ t β€ Ο + 2 * Ο * βn
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | have h : βn < t * (2 * Ο)β»ΒΉ - 1 / 2 + 1 := by rw [β hn]; exact Int.ceil_lt_add_one _ | case left
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ 2 * Ο * βn < Ο + t | case left
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
h : βn < t * (2 * Ο)β»ΒΉ - 1 / 2 + 1
β’ 2 * Ο * βn < Ο + t | Please generate a tactic in lean4 to solve the state.
STATE:
case left
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ 2 * Ο * βn < Ο + t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | calc 2 * Ο * βn
_ < 2 * Ο * (t * (2 * Ο)β»ΒΉ - 1 / 2 + 1) := by bound
_ = Ο + 2 * Ο * (2 * Ο)β»ΒΉ * t := by ring
_ = Ο + t := by field_simp [Real.two_pi_pos.ne'] | case left
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
h : βn < t * (2 * Ο)β»ΒΉ - 1 / 2 + 1
β’ 2 * Ο * βn < Ο + t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
h : βn < t * (2 * Ο)β»ΒΉ - 1 / 2 + 1
β’ 2 * Ο * βn < Ο + t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | rw [β hn] | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ βn < t * (2 * Ο)β»ΒΉ - 1 / 2 + 1 | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ ββt / (2 * Ο) - 1 / 2β < t * (2 * Ο)β»ΒΉ - 1 / 2 + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ βn < t * (2 * Ο)β»ΒΉ - 1 / 2 + 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | exact Int.ceil_lt_add_one _ | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ ββt / (2 * Ο) - 1 / 2β < t * (2 * Ο)β»ΒΉ - 1 / 2 + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ ββt / (2 * Ο) - 1 / 2β < t * (2 * Ο)β»ΒΉ - 1 / 2 + 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | bound | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
h : βn < t * (2 * Ο)β»ΒΉ - 1 / 2 + 1
β’ 2 * Ο * βn < 2 * Ο * (t * (2 * Ο)β»ΒΉ - 1 / 2 + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
h : βn < t * (2 * Ο)β»ΒΉ - 1 / 2 + 1
β’ 2 * Ο * βn < 2 * Ο * (t * (2 * Ο)β»ΒΉ - 1 / 2 + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | ring | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
h : βn < t * (2 * Ο)β»ΒΉ - 1 / 2 + 1
β’ 2 * Ο * (t * (2 * Ο)β»ΒΉ - 1 / 2 + 1) = Ο + 2 * Ο * (2 * Ο)β»ΒΉ * t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
h : βn < t * (2 * Ο)β»ΒΉ - 1 / 2 + 1
β’ 2 * Ο * (t * (2 * Ο)β»ΒΉ - 1 / 2 + 1) = Ο + 2 * Ο * (2 * Ο)β»ΒΉ * t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | field_simp [Real.two_pi_pos.ne'] | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
h : βn < t * (2 * Ο)β»ΒΉ - 1 / 2 + 1
β’ Ο + 2 * Ο * (2 * Ο)β»ΒΉ * t = Ο + t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
h : βn < t * (2 * Ο)β»ΒΉ - 1 / 2 + 1
β’ Ο + 2 * Ο * (2 * Ο)β»ΒΉ * t = Ο + t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | have h : βn β₯ t * (2 * Ο)β»ΒΉ - 1 / 2 := by rw [β hn]; exact Int.le_ceil _ | case right
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ t β€ Ο + 2 * Ο * βn | case right
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
h : βn β₯ t * (2 * Ο)β»ΒΉ - 1 / 2
β’ t β€ Ο + 2 * Ο * βn | Please generate a tactic in lean4 to solve the state.
STATE:
case right
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ t β€ Ο + 2 * Ο * βn
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | calc Ο + 2 * Ο * βn
_ β₯ Ο + 2 * Ο * (t * (2 * Ο)β»ΒΉ - 1 / 2) := by bound
_ = 2 * Ο * (2 * Ο)β»ΒΉ * t := by ring
_ = t := by field_simp [Real.two_pi_pos.ne'] | case right
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
h : βn β₯ t * (2 * Ο)β»ΒΉ - 1 / 2
β’ t β€ Ο + 2 * Ο * βn | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
h : βn β₯ t * (2 * Ο)β»ΒΉ - 1 / 2
β’ t β€ Ο + 2 * Ο * βn
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | rw [β hn] | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ βn β₯ t * (2 * Ο)β»ΒΉ - 1 / 2 | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ ββt / (2 * Ο) - 1 / 2β β₯ t * (2 * Ο)β»ΒΉ - 1 / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ βn β₯ t * (2 * Ο)β»ΒΉ - 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | exact Int.le_ceil _ | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ ββt / (2 * Ο) - 1 / 2β β₯ t * (2 * Ο)β»ΒΉ - 1 / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
β’ ββt / (2 * Ο) - 1 / 2β β₯ t * (2 * Ο)β»ΒΉ - 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | bound | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
h : βn β₯ t * (2 * Ο)β»ΒΉ - 1 / 2
β’ Ο + 2 * Ο * βn β₯ Ο + 2 * Ο * (t * (2 * Ο)β»ΒΉ - 1 / 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
h : βn β₯ t * (2 * Ο)β»ΒΉ - 1 / 2
β’ Ο + 2 * Ο * βn β₯ Ο + 2 * Ο * (t * (2 * Ο)β»ΒΉ - 1 / 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | ring | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
h : βn β₯ t * (2 * Ο)β»ΒΉ - 1 / 2
β’ Ο + 2 * Ο * (t * (2 * Ο)β»ΒΉ - 1 / 2) = 2 * Ο * (2 * Ο)β»ΒΉ * t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
h : βn β₯ t * (2 * Ο)β»ΒΉ - 1 / 2
β’ Ο + 2 * Ο * (t * (2 * Ο)β»ΒΉ - 1 / 2) = 2 * Ο * (2 * Ο)β»ΒΉ * t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | field_simp [Real.two_pi_pos.ne'] | t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
h : βn β₯ t * (2 * Ο)β»ΒΉ - 1 / 2
β’ 2 * Ο * (2 * Ο)β»ΒΉ * t = t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : β
n : β€
hn : βt / (2 * Ο) - 1 / 2β = n
en : (2 * βΟ * βn * I).exp = 1
e : (βt * I).exp = (β(t - 2 * Ο * βn) * I).exp
h : βn β₯ t * (2 * Ο)β»ΒΉ - 1 / 2
β’ 2 * Ο * (2 * Ο)β»ΒΉ * t = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | intro x xs y ys e | c : β
r0 r1 : β
r0p : 0 β€ r0
β’ InjOn (realCircleMap c) (square r0 r1) | c : β
r0 r1 : β
r0p : 0 β€ r0
x : β Γ β
xs : x β square r0 r1
y : β Γ β
ys : y β square r0 r1
e : realCircleMap c x = realCircleMap c y
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
β’ InjOn (realCircleMap c) (square r0 r1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | simp [square] at xs ys | c : β
r0 r1 : β
r0p : 0 β€ r0
x : β Γ β
xs : x β square r0 r1
y : β Γ β
ys : y β square r0 r1
e : realCircleMap c x = realCircleMap c y
β’ x = y | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
e : realCircleMap c x = realCircleMap c y
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x : β Γ β
xs : x β square r0 r1
y : β Γ β
ys : y β square r0 r1
e : realCircleMap c x = realCircleMap c y
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | simp_rw [realCircleMap_eq_circleMap, Equiv.apply_eq_iff_eq] at e | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
e : realCircleMap c x = realCircleMap c y
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
β’ x = y | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : circleMap c x.1 x.2 = circleMap c y.1 y.2
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
e : realCircleMap c x = realCircleMap c y
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | simp_rw [circleMap] at e | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : circleMap c x.1 x.2 = circleMap c y.1 y.2
β’ x = y | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : c + βx.1 * (βx.2 * I).exp = c + βy.1 * (βy.2 * I).exp
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : circleMap c x.1 x.2 = circleMap c y.1 y.2
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | simp at e | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : c + βx.1 * (βx.2 * I).exp = c + βy.1 * (βy.2 * I).exp
β’ x = y | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : c + βx.1 * (βx.2 * I).exp = c + βy.1 * (βy.2 * I).exp
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have re : abs (βx.1 * exp (x.2 * I)) = abs (βy.1 * exp (y.2 * I)) := by rw [e] | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
β’ x = y | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
re : Complex.abs (βx.1 * (βx.2 * I).exp) = Complex.abs (βy.1 * (βy.2 * I).exp)
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have x0 : 0 < x.1 := by linarith | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
re : Complex.abs (βx.1 * (βx.2 * I).exp) = Complex.abs (βy.1 * (βy.2 * I).exp)
β’ x = y | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
re : Complex.abs (βx.1 * (βx.2 * I).exp) = Complex.abs (βy.1 * (βy.2 * I).exp)
x0 : 0 < x.1
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
re : Complex.abs (βx.1 * (βx.2 * I).exp) = Complex.abs (βy.1 * (βy.2 * I).exp)
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have y0 : 0 < y.1 := by linarith | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
re : Complex.abs (βx.1 * (βx.2 * I).exp) = Complex.abs (βy.1 * (βy.2 * I).exp)
x0 : 0 < x.1
β’ x = y | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
re : Complex.abs (βx.1 * (βx.2 * I).exp) = Complex.abs (βy.1 * (βy.2 * I).exp)
x0 : 0 < x.1
y0 : 0 < y.1
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
re : Complex.abs (βx.1 * (βx.2 * I).exp) = Complex.abs (βy.1 * (βy.2 * I).exp)
x0 : 0 < x.1
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | simp only [map_mul, Complex.abs_ofReal, abs_of_pos x0, Complex.abs_exp_ofReal_mul_I, mul_one,
abs_of_pos y0] at re | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
re : Complex.abs (βx.1 * (βx.2 * I).exp) = Complex.abs (βy.1 * (βy.2 * I).exp)
x0 : 0 < x.1
y0 : 0 < y.1
β’ x = y | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
re : Complex.abs (βx.1 * (βx.2 * I).exp) = Complex.abs (βy.1 * (βy.2 * I).exp)
x0 : 0 < x.1
y0 : 0 < y.1
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have ae : arg (βx.1 * exp (x.2 * I)) = arg (βy.1 * exp (y.2 * I)) := by rw [e] | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
β’ x = y | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (βx.1 * (βx.2 * I).exp).arg = (βy.1 * (βy.2 * I).exp).arg
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | simp [Complex.arg_real_mul _ x0, Complex.arg_real_mul _ y0] at ae | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (βx.1 * (βx.2 * I).exp).arg = (βy.1 * (βy.2 * I).exp).arg
β’ x = y | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (βx.2 * I).exp.arg = (βy.2 * I).exp.arg
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (βx.1 * (βx.2 * I).exp).arg = (βy.1 * (βy.2 * I).exp).arg
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rcases arg_exp_of_im x.2 with β¨nx, hxβ© | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (βx.2 * I).exp.arg = (βy.2 * I).exp.arg
β’ x = y | case intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (βx.2 * I).exp.arg = (βy.2 * I).exp.arg
nx : β€
hx : (βx.2 * I).exp.arg = x.2 - 2 * Ο * βnx
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (βx.2 * I).exp.arg = (βy.2 * I).exp.arg
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rcases arg_exp_of_im y.2 with β¨ny, hβ© | case intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (βx.2 * I).exp.arg = (βy.2 * I).exp.arg
nx : β€
hx : (βx.2 * I).exp.arg = x.2 - 2 * Ο * βnx
β’ x = y | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (βx.2 * I).exp.arg = (βy.2 * I).exp.arg
nx : β€
hx : (βx.2 * I).exp.arg = x.2 - 2 * Ο * βnx
ny : β€
h : (βy.2 * I).exp.arg = y.2 - 2 * Ο * βny
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (βx.2 * I).exp.arg = (βy.2 * I).exp.arg
nx : β€
hx : (βx.2 * I).exp.arg = x.2 - 2 * Ο * βnx
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rw [β ae, hx] at h | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (βx.2 * I).exp.arg = (βy.2 * I).exp.arg
nx : β€
hx : (βx.2 * I).exp.arg = x.2 - 2 * Ο * βnx
ny : β€
h : (βy.2 * I).exp.arg = y.2 - 2 * Ο * βny
β’ x = y | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (βx.2 * I).exp.arg = (βy.2 * I).exp.arg
nx : β€
hx : (βx.2 * I).exp.arg = x.2 - 2 * Ο * βnx
ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (βx.2 * I).exp.arg = (βy.2 * I).exp.arg
nx : β€
hx : (βx.2 * I).exp.arg = x.2 - 2 * Ο * βnx
ny : β€
h : (βy.2 * I).exp.arg = y.2 - 2 * Ο * βny
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | clear e ae hx | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (βx.2 * I).exp.arg = (βy.2 * I).exp.arg
nx : β€
hx : (βx.2 * I).exp.arg = x.2 - 2 * Ο * βnx
ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
β’ x = y | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (βx.2 * I).exp.arg = (βy.2 * I).exp.arg
nx : β€
hx : (βx.2 * I).exp.arg = x.2 - 2 * Ο * βnx
ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have n0 : 2 * Ο * (nx - ny) < 2 * Ο * 1 := by linarith | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
β’ x = y | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have n1 : 2 * Ο * -1 < 2 * Ο * (nx - ny) := by linarith | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
β’ x = y | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
n1 : 2 * Ο * -1 < 2 * Ο * (βnx - βny)
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have hn : (nx : β) - ny = β(nx - ny) := by simp only [Int.cast_sub] | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
n1 : 2 * Ο * -1 < 2 * Ο * (βnx - βny)
β’ x = y | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
n1 : 2 * Ο * -1 < 2 * Ο * (βnx - βny)
hn : βnx - βny = β(nx - ny)
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
n1 : 2 * Ο * -1 < 2 * Ο * (βnx - βny)
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have hn1 : (-1 : β) = β(-1 : β€) := by norm_num | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
n1 : 2 * Ο * -1 < 2 * Ο * (βnx - βny)
hn : βnx - βny = β(nx - ny)
β’ x = y | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
n1 : 2 * Ο * -1 < 2 * Ο * (βnx - βny)
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
n1 : 2 * Ο * -1 < 2 * Ο * (βnx - βny)
hn : βnx - βny = β(nx - ny)
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have h1 : (1 : β) = β(1 : β€) := by norm_num | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
n1 : 2 * Ο * -1 < 2 * Ο * (βnx - βny)
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
β’ x = y | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
n1 : 2 * Ο * -1 < 2 * Ο * (βnx - βny)
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
n1 : 2 * Ο * -1 < 2 * Ο * (βnx - βny)
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rw [mul_lt_mul_left Real.two_pi_pos, hn] at n0 n1 | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
n1 : 2 * Ο * -1 < 2 * Ο * (βnx - βny)
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
β’ x = y | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : β(nx - ny) < 1
n1 : -1 < β(nx - ny)
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
n1 : 2 * Ο * -1 < 2 * Ο * (βnx - βny)
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rw [hn1] at n1 | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : β(nx - ny) < 1
n1 : -1 < β(nx - ny)
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
β’ x = y | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : β(nx - ny) < 1
n1 : β(-1) < β(nx - ny)
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : β(nx - ny) < 1
n1 : -1 < β(nx - ny)
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rw [h1] at n0 | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : β(nx - ny) < 1
n1 : β(-1) < β(nx - ny)
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
β’ x = y | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : β(nx - ny) < β1
n1 : β(-1) < β(nx - ny)
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : β(nx - ny) < 1
n1 : β(-1) < β(nx - ny)
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rw [Int.cast_lt] at n0 n1 | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : β(nx - ny) < β1
n1 : β(-1) < β(nx - ny)
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
β’ x = y | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : β(nx - ny) < β1
n1 : β(-1) < β(nx - ny)
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have n : nx = ny := by linarith | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
β’ x = y | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
n : nx = ny
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rw [n] at h | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
n : nx = ny
β’ x = y | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βny = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
n : nx = ny
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
n : nx = ny
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have i : x.2 = y.2 := by linarith | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βny = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
n : nx = ny
β’ x = y | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βny = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
n : nx = ny
i : x.2 = y.2
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βny = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
n : nx = ny
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have g : (x.1, x.2) = (y.1, y.2) := by rw [re, i] | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βny = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
n : nx = ny
i : x.2 = y.2
β’ x = y | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βny = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
n : nx = ny
i : x.2 = y.2
g : (x.1, x.2) = (y.1, y.2)
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βny = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
n : nx = ny
i : x.2 = y.2
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | simp only [Prod.mk.eta] at g | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βny = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
n : nx = ny
i : x.2 = y.2
g : (x.1, x.2) = (y.1, y.2)
β’ x = y | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βny = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
n : nx = ny
i : x.2 = y.2
g : x = y
β’ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βny = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
n : nx = ny
i : x.2 = y.2
g : (x.1, x.2) = (y.1, y.2)
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | exact g | case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βny = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
n : nx = ny
i : x.2 = y.2
g : x = y
β’ x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βny = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
n : nx = ny
i : x.2 = y.2
g : x = y
β’ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rw [e] | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
β’ Complex.abs (βx.1 * (βx.2 * I).exp) = Complex.abs (βy.1 * (βy.2 * I).exp) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
β’ Complex.abs (βx.1 * (βx.2 * I).exp) = Complex.abs (βy.1 * (βy.2 * I).exp)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | linarith | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
re : Complex.abs (βx.1 * (βx.2 * I).exp) = Complex.abs (βy.1 * (βy.2 * I).exp)
β’ 0 < x.1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
re : Complex.abs (βx.1 * (βx.2 * I).exp) = Complex.abs (βy.1 * (βy.2 * I).exp)
β’ 0 < x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | linarith | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
re : Complex.abs (βx.1 * (βx.2 * I).exp) = Complex.abs (βy.1 * (βy.2 * I).exp)
x0 : 0 < x.1
β’ 0 < y.1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
re : Complex.abs (βx.1 * (βx.2 * I).exp) = Complex.abs (βy.1 * (βy.2 * I).exp)
x0 : 0 < x.1
β’ 0 < y.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rw [e] | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
β’ (βx.1 * (βx.2 * I).exp).arg = (βy.1 * (βy.2 * I).exp).arg | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
e : βx.1 * (βx.2 * I).exp = βy.1 * (βy.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
β’ (βx.1 * (βx.2 * I).exp).arg = (βy.1 * (βy.2 * I).exp).arg
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | linarith | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
β’ 2 * Ο * (βnx - βny) < 2 * Ο * 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
β’ 2 * Ο * (βnx - βny) < 2 * Ο * 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | linarith | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
β’ 2 * Ο * -1 < 2 * Ο * (βnx - βny) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
β’ 2 * Ο * -1 < 2 * Ο * (βnx - βny)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | simp only [Int.cast_sub] | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
n1 : 2 * Ο * -1 < 2 * Ο * (βnx - βny)
β’ βnx - βny = β(nx - ny) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
n1 : 2 * Ο * -1 < 2 * Ο * (βnx - βny)
β’ βnx - βny = β(nx - ny)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | norm_num | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
n1 : 2 * Ο * -1 < 2 * Ο * (βnx - βny)
hn : βnx - βny = β(nx - ny)
β’ -1 = β(-1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
n1 : 2 * Ο * -1 < 2 * Ο * (βnx - βny)
hn : βnx - βny = β(nx - ny)
β’ -1 = β(-1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | norm_num | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
n1 : 2 * Ο * -1 < 2 * Ο * (βnx - βny)
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
β’ 1 = β1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : 2 * Ο * (βnx - βny) < 2 * Ο * 1
n1 : 2 * Ο * -1 < 2 * Ο * (βnx - βny)
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
β’ 1 = β1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | linarith | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
β’ nx = ny | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βnx = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
β’ nx = ny
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | linarith | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βny = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
n : nx = ny
β’ x.2 = y.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βny = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
n : nx = ny
β’ x.2 = y.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rw [re, i] | c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βny = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
n : nx = ny
i : x.2 = y.2
β’ (x.1, x.2) = (y.1, y.2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r0 r1 : β
r0p : 0 β€ r0
x y : β Γ β
xs : (r0 < x.1 β§ x.1 β€ r1) β§ 0 < x.2 β§ x.2 β€ 2 * Ο
ys : (r0 < y.1 β§ y.1 β€ r1) β§ 0 < y.2 β§ y.2 β€ 2 * Ο
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : β€
h : x.2 - 2 * Ο * βny = y.2 - 2 * Ο * βny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : βnx - βny = β(nx - ny)
hn1 : -1 = β(-1)
h1 : 1 = β1
n : nx = ny
i : x.2 = y.2
β’ (x.1, x.2) = (y.1, y.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | measurable_symm_equiv_inverse | [239, 1] | [245, 55] | simp only [Complex.equivRealProd_apply] | z : β
β’ Complex.measurableEquivRealProd.symm (Complex.equivRealProd z) = z | z : β
β’ Complex.measurableEquivRealProd.symm (z.re, z.im) = z | Please generate a tactic in lean4 to solve the state.
STATE:
z : β
β’ Complex.measurableEquivRealProd.symm (Complex.equivRealProd z) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | measurable_symm_equiv_inverse | [239, 1] | [245, 55] | rw [Complex.measurableEquivRealProd, Homeomorph.toMeasurableEquiv_symm_coe] | z : β
β’ Complex.measurableEquivRealProd.symm (z.re, z.im) = z | z : β
β’ Complex.equivRealProdCLM.toHomeomorph.symm (z.re, z.im) = z | Please generate a tactic in lean4 to solve the state.
STATE:
z : β
β’ Complex.measurableEquivRealProd.symm (z.re, z.im) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | measurable_symm_equiv_inverse | [239, 1] | [245, 55] | simp only [ContinuousLinearEquiv.symm_toHomeomorph, ContinuousLinearEquiv.coe_toHomeomorph] | z : β
β’ Complex.equivRealProdCLM.toHomeomorph.symm (z.re, z.im) = z | z : β
β’ Complex.equivRealProdCLM.symm (z.re, z.im) = z | Please generate a tactic in lean4 to solve the state.
STATE:
z : β
β’ Complex.equivRealProdCLM.toHomeomorph.symm (z.re, z.im) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | measurable_symm_equiv_inverse | [239, 1] | [245, 55] | apply Complex.ext | z : β
β’ Complex.equivRealProdCLM.symm (z.re, z.im) = z | case a
z : β
β’ (Complex.equivRealProdCLM.symm (z.re, z.im)).re = z.re
case a
z : β
β’ (Complex.equivRealProdCLM.symm (z.re, z.im)).im = z.im | Please generate a tactic in lean4 to solve the state.
STATE:
z : β
β’ Complex.equivRealProdCLM.symm (z.re, z.im) = z
TACTIC:
|
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