url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_basin | [287, 1] | [296, 88] | simp only [Function.iterate_succ_apply'] | case h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
n : β
h : (c, (f d c)^[n] 0) β s.near
β’ (c, (f d c)^[n + 1] 0) β β―.near | case h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
n : β
h : (c, (f d c)^[n] 0) β s.near
β’ (c, f d c ((f d c)^[n] 0)) β β―.near | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
n : β
h : (c, (f d c)^[n] 0) β s.near
β’ (c, (f d c)^[n + 1] 0) β β―.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_basin | [287, 1] | [296, 88] | exact s.stays_near h | case h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
n : β
h : (c, (f d c)^[n] 0) β s.near
β’ (c, f d c ((f d c)^[n] 0)) β β―.near | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
n : β
h : (c, (f d c)^[n] 0) β s.near
β’ (c, f d c ((f d c)^[n] 0)) β β―.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | set s := superF d | c : β
d : β
instβ : Fact (2 β€ d)
β’ β―.p c = β―.potential c 0 | c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ s.p c = s.potential c 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ β―.p c = β―.potential c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | simp only [Super.p, e, csInf_pair] | c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : s.ps c = {1, s.potential c 0}
β’ s.p c = s.potential c 0 | c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : s.ps c = {1, s.potential c 0}
β’ 1 β s.potential c 0 = s.potential c 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : s.ps c = {1, s.potential c 0}
β’ s.p c = s.potential c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | exact inf_of_le_right s.potential_le_one | c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : s.ps c = {1, s.potential c 0}
β’ 1 β s.potential c 0 = s.potential c 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : s.ps c = {1, s.potential c 0}
β’ 1 β s.potential c 0 = s.potential c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | apply Set.ext | c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ s.ps c = {1, s.potential c 0} | case h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ β (x : β), x β s.ps c β x β {1, s.potential c 0} | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ s.ps c = {1, s.potential c 0}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | intro p | case h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ β (x : β), x β s.ps c β x β {1, s.potential c 0} | case h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
β’ p β s.ps c β p β {1, s.potential c 0} | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ β (x : β), x β s.ps c β x β {1, s.potential c 0}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | simp only [Super.ps, mem_singleton_iff, mem_setOf, critical_f, Ne, mem_insert_iff,
mem_singleton_iff] | case h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
β’ p β s.ps c β p β {1, s.potential c 0} | case h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
β’ (p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)) β p = 1 β¨ p = s.potential c 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
β’ p β s.ps c β p β {1, s.potential c 0}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | constructor | case h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
β’ (p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)) β p = 1 β¨ p = s.potential c 0 | case h.mp
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
β’ (p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)) β p = 1 β¨ p = s.potential c 0
case h.mpr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
β’ p = 1 β¨ p = s.potential c 0 β p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
β’ (p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)) β p = 1 β¨ p = s.potential c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | intro h | case h.mp
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
β’ (p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)) β p = 1 β¨ p = s.potential c 0 | case h.mp
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
β’ p = 1 β¨ p = s.potential c 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
β’ (p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)) β p = 1 β¨ p = s.potential c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | cases' h with h h | case h.mp
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
β’ p = 1 β¨ p = s.potential c 0 | case h.mp.inl
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = 1
β’ p = 1 β¨ p = s.potential c 0
case h.mp.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
β’ p = 1 β¨ p = s.potential c 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
β’ p = 1 β¨ p = s.potential c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | left | case h.mp.inl
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = 1
β’ p = 1 β¨ p = s.potential c 0
case h.mp.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
β’ p = 1 β¨ p = s.potential c 0 | case h.mp.inl.h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = 1
β’ p = 1
case h.mp.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
β’ p = 1 β¨ p = s.potential c 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.inl
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = 1
β’ p = 1 β¨ p = s.potential c 0
case h.mp.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
β’ p = 1 β¨ p = s.potential c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | exact h | case h.mp.inl.h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = 1
β’ p = 1
case h.mp.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
β’ p = 1 β¨ p = s.potential c 0 | case h.mp.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
β’ p = 1 β¨ p = s.potential c 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.inl.h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = 1
β’ p = 1
case h.mp.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
β’ p = 1 β¨ p = s.potential c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | right | case h.mp.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
β’ p = 1 β¨ p = s.potential c 0 | case h.mp.inr.h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
β’ p = s.potential c 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
β’ p = 1 β¨ p = s.potential c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | rcases h with β¨p0, z, e, hβ© | case h.mp.inr.h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
β’ p = s.potential c 0 | case h.mp.inr.h.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : s.potential c z = p
h : z = 0 β¨ z = β
β’ p = s.potential c 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.inr.h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
β’ p = s.potential c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | cases' h with h h | case h.mp.inr.h.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : s.potential c z = p
h : z = 0 β¨ z = β
β’ p = s.potential c 0 | case h.mp.inr.h.intro.intro.intro.inl
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : s.potential c z = p
h : z = 0
β’ p = s.potential c 0
case h.mp.inr.h.intro.intro.intro.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : s.potential c z = p
h : z = β
β’ p = s.potential c 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.inr.h.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : s.potential c z = p
h : z = 0 β¨ z = β
β’ p = s.potential c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | rw [h] at e | case h.mp.inr.h.intro.intro.intro.inl
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : s.potential c z = p
h : z = 0
β’ p = s.potential c 0
case h.mp.inr.h.intro.intro.intro.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : s.potential c z = p
h : z = β
β’ p = s.potential c 0 | case h.mp.inr.h.intro.intro.intro.inl
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : s.potential c 0 = p
h : z = 0
β’ p = s.potential c 0
case h.mp.inr.h.intro.intro.intro.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : s.potential c z = p
h : z = β
β’ p = s.potential c 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.inr.h.intro.intro.intro.inl
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : s.potential c z = p
h : z = 0
β’ p = s.potential c 0
case h.mp.inr.h.intro.intro.intro.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : s.potential c z = p
h : z = β
β’ p = s.potential c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | exact e.symm | case h.mp.inr.h.intro.intro.intro.inl
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : s.potential c 0 = p
h : z = 0
β’ p = s.potential c 0
case h.mp.inr.h.intro.intro.intro.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : s.potential c z = p
h : z = β
β’ p = s.potential c 0 | case h.mp.inr.h.intro.intro.intro.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : s.potential c z = p
h : z = β
β’ p = s.potential c 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.inr.h.intro.intro.intro.inl
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : s.potential c 0 = p
h : z = 0
β’ p = s.potential c 0
case h.mp.inr.h.intro.intro.intro.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : s.potential c z = p
h : z = β
β’ p = s.potential c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | rw [h, s.potential_a] at e | case h.mp.inr.h.intro.intro.intro.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : s.potential c z = p
h : z = β
β’ p = s.potential c 0 | case h.mp.inr.h.intro.intro.intro.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : 0 = p
h : z = β
β’ p = s.potential c 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.inr.h.intro.intro.intro.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : s.potential c z = p
h : z = β
β’ p = s.potential c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | exfalso | case h.mp.inr.h.intro.intro.intro.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : 0 = p
h : z = β
β’ p = s.potential c 0 | case h.mp.inr.h.intro.intro.intro.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : 0 = p
h : z = β
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.inr.h.intro.intro.intro.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : 0 = p
h : z = β
β’ p = s.potential c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | exact p0 e.symm | case h.mp.inr.h.intro.intro.intro.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : 0 = p
h : z = β
β’ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.inr.h.intro.intro.intro.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
p0 : Β¬p = 0
z : π
e : 0 = p
h : z = β
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | intro h | case h.mpr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
β’ p = 1 β¨ p = s.potential c 0 β p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β) | case h.mpr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = 1 β¨ p = s.potential c 0
β’ p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
β’ p = 1 β¨ p = s.potential c 0 β p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | cases' h with h h | case h.mpr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = 1 β¨ p = s.potential c 0
β’ p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β) | case h.mpr.inl
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = 1
β’ p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
case h.mpr.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = s.potential c 0
β’ p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = 1 β¨ p = s.potential c 0
β’ p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | left | case h.mpr.inl
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = 1
β’ p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
case h.mpr.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = s.potential c 0
β’ p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β) | case h.mpr.inl.h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = 1
β’ p = 1
case h.mpr.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = s.potential c 0
β’ p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.inl
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = 1
β’ p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
case h.mpr.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = s.potential c 0
β’ p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | exact h | case h.mpr.inl.h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = 1
β’ p = 1
case h.mpr.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = s.potential c 0
β’ p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β) | case h.mpr.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = s.potential c 0
β’ p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.inl.h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = 1
β’ p = 1
case h.mpr.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = s.potential c 0
β’ p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | right | case h.mpr.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = s.potential c 0
β’ p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β) | case h.mpr.inr.h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = s.potential c 0
β’ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.inr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = s.potential c 0
β’ p = 1 β¨ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | constructor | case h.mpr.inr.h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = s.potential c 0
β’ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β) | case h.mpr.inr.h.left
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = s.potential c 0
β’ Β¬p = 0
case h.mpr.inr.h.right
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = s.potential c 0
β’ β z, s.potential c z = p β§ (z = 0 β¨ z = β) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.inr.h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = s.potential c 0
β’ Β¬p = 0 β§ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | simp only [h, β ne_eq, s.potential_ne_zero] | case h.mpr.inr.h.left
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = s.potential c 0
β’ Β¬p = 0 | case h.mpr.inr.h.left
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = s.potential c 0
β’ 0 β β | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.inr.h.left
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = s.potential c 0
β’ Β¬p = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | exact inf_ne_zero.symm | case h.mpr.inr.h.left
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = s.potential c 0
β’ 0 β β | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.inr.h.left
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = s.potential c 0
β’ 0 β β
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | use 0, h.symm, Or.inl rfl | case h.mpr.inr.h.right
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = s.potential c 0
β’ β z, s.potential c z = p β§ (z = 0 β¨ z = β) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.inr.h.right
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
p : β
h : p = s.potential c 0
β’ β z, s.potential c z = p β§ (z = 0 β¨ z = β)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrotPost | [315, 1] | [320, 40] | set s := superF d | c : β
d : β
instβ : Fact (2 β€ d)
m : c β multibrot d
β’ Postcritical β― c βc | c : β
d : β
instβ : Fact (2 β€ d)
m : c β multibrot d
s : Super (f d) d β := superF d
β’ Postcritical s c βc | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
m : c β multibrot d
β’ Postcritical β― c βc
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrotPost | [315, 1] | [320, 40] | simp only [Postcritical, multibrot_p, β f_0 d, s.potential_eqn] | c : β
d : β
instβ : Fact (2 β€ d)
m : c β multibrot d
s : Super (f d) d β := superF d
β’ Postcritical s c βc | c : β
d : β
instβ : Fact (2 β€ d)
m : c β multibrot d
s : Super (f d) d β := superF d
β’ s.potential c 0 ^ d < β―.potential c 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
m : c β multibrot d
s : Super (f d) d β := superF d
β’ Postcritical s c βc
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrotPost | [315, 1] | [320, 40] | simp only [multibrot_basin, not_not] at m | c : β
d : β
instβ : Fact (2 β€ d)
m : c β multibrot d
s : Super (f d) d β := superF d
β’ s.potential c 0 ^ d < β―.potential c 0 | c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
m : (c, 0) β β―.basin
β’ s.potential c 0 ^ d < β―.potential c 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
m : c β multibrot d
s : Super (f d) d β := superF d
β’ s.potential c 0 ^ d < β―.potential c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrotPost | [315, 1] | [320, 40] | exact pow_lt_self_of_lt_one ((s.potential_pos c).mpr inf_ne_zero.symm)
(s.potential_lt_one m) (d_gt_one d) | c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
m : (c, 0) β β―.basin
β’ s.potential c 0 ^ d < β―.potential c 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
m : (c, 0) β β―.basin
β’ s.potential c 0 ^ d < β―.potential c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | induction' n with n h | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
β’ (b - 1) ^ n * Complex.abs z β€ Complex.abs ((f' d c)^[n] z) | case zero
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
β’ (b - 1) ^ 0 * Complex.abs z β€ Complex.abs ((f' d c)^[0] z)
case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs ((f' d c)^[n] z)
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs ((f' d c)^[n + 1] z) | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
β’ (b - 1) ^ n * Complex.abs z β€ Complex.abs ((f' d c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | simp only [Nat.zero_eq, pow_zero, one_mul, Function.iterate_zero_apply, le_refl] | case zero
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
β’ (b - 1) ^ 0 * Complex.abs z β€ Complex.abs ((f' d c)^[0] z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
β’ (b - 1) ^ 0 * Complex.abs z β€ Complex.abs ((f' d c)^[0] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | simp only [Function.iterate_succ_apply'] | case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs ((f' d c)^[n] z)
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs ((f' d c)^[n + 1] z) | case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs ((f' d c)^[n] z)
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c ((f' d c)^[n] z)) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs ((f' d c)^[n] z)
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs ((f' d c)^[n + 1] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | generalize hw : (f' d c)^[n] z = w | case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs ((f' d c)^[n] z)
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c ((f' d c)^[n] z)) | case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs ((f' d c)^[n] z)
w : β
hw : (f' d c)^[n] z = w
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c w) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs ((f' d c)^[n] z)
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c ((f' d c)^[n] z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | rw [hw] at h | case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs ((f' d c)^[n] z)
w : β
hw : (f' d c)^[n] z = w
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c w) | case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
hw : (f' d c)^[n] z = w
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c w) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs ((f' d c)^[n] z)
w : β
hw : (f' d c)^[n] z = w
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c w)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | clear hw | case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
hw : (f' d c)^[n] z = w
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c w) | case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c w) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
hw : (f' d c)^[n] z = w
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c w)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | have z1 : 1 β€ abs z := le_trans (by norm_num) (le_trans b2 bz) | case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c w) | case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c w) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c w)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | have b1 : 1 β€ b - 1 := by linarith | case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c w) | case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c w) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c w)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | have b0 : 0 β€ b - 1 := by linarith | case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c w) | case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c w) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c w)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | have nd : n + 1 β€ n * d + 1 := by bound | case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c w) | case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c w) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c w)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | calc abs (w ^ d + c)
_ β₯ abs (w ^ d) - abs c := by bound
_ = abs w ^ d - abs c := by rw [Complex.abs.map_pow]
_ β₯ ((b-1) ^ n * abs z) ^ d - abs c := by bound
_ = (b-1) ^ (n*d) * abs z ^ d - abs c := by rw [mul_pow, pow_mul]
_ β₯ (b-1) ^ (n*d) * abs z ^ 2 - abs c := by bound
_ = (b-1) ^ (n*d) * (abs z * abs z) - abs c := by rw [pow_two]
_ β₯ (b-1) ^ (n*d) * (b * abs z) - abs c := by bound
_ = (b-1) ^ (n*d) * (b-1) * abs z + ((b-1) ^ (n*d) * abs z - abs c) := by ring
_ = (b-1) ^ (n*d + 1) * abs z + ((b-1) ^ (n * d) * abs z - abs c) := by rw [pow_succ]
_ β₯ (b-1) ^ (n + 1) * abs z + (1 * abs z - abs c) := by bound
_ = (b-1) ^ (n + 1) * abs z + (abs z - abs c) := by rw [one_mul]
_ β₯ (b-1) ^ (n + 1) * abs z := by bound | case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c w) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ (b - 1) ^ (n + 1) * Complex.abs z β€ Complex.abs (f' d c w)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | norm_num | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
β’ 1 β€ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
β’ 1 β€ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | linarith | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
β’ 1 β€ b - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
β’ 1 β€ b - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | linarith | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
β’ 0 β€ b - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
β’ 0 β€ b - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | bound | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
β’ n + 1 β€ n * d + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
β’ n + 1 β€ n * d + 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | bound | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ Complex.abs (w ^ d + c) β₯ Complex.abs (w ^ d) - Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ Complex.abs (w ^ d + c) β₯ Complex.abs (w ^ d) - Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | rw [Complex.abs.map_pow] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ Complex.abs (w ^ d) - Complex.abs c = Complex.abs w ^ d - Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ Complex.abs (w ^ d) - Complex.abs c = Complex.abs w ^ d - Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | bound | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ Complex.abs w ^ d - Complex.abs c β₯ ((b - 1) ^ n * Complex.abs z) ^ d - Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ Complex.abs w ^ d - Complex.abs c β₯ ((b - 1) ^ n * Complex.abs z) ^ d - Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | rw [mul_pow, pow_mul] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ ((b - 1) ^ n * Complex.abs z) ^ d - Complex.abs c = (b - 1) ^ (n * d) * Complex.abs z ^ d - Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ ((b - 1) ^ n * Complex.abs z) ^ d - Complex.abs c = (b - 1) ^ (n * d) * Complex.abs z ^ d - Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | bound | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ (b - 1) ^ (n * d) * Complex.abs z ^ d - Complex.abs c β₯ (b - 1) ^ (n * d) * Complex.abs z ^ 2 - Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ (b - 1) ^ (n * d) * Complex.abs z ^ d - Complex.abs c β₯ (b - 1) ^ (n * d) * Complex.abs z ^ 2 - Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | rw [pow_two] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ (b - 1) ^ (n * d) * Complex.abs z ^ 2 - Complex.abs c =
(b - 1) ^ (n * d) * (Complex.abs z * Complex.abs z) - Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ (b - 1) ^ (n * d) * Complex.abs z ^ 2 - Complex.abs c =
(b - 1) ^ (n * d) * (Complex.abs z * Complex.abs z) - Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | bound | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ (b - 1) ^ (n * d) * (Complex.abs z * Complex.abs z) - Complex.abs c β₯
(b - 1) ^ (n * d) * (b * Complex.abs z) - Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ (b - 1) ^ (n * d) * (Complex.abs z * Complex.abs z) - Complex.abs c β₯
(b - 1) ^ (n * d) * (b * Complex.abs z) - Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | ring | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ (b - 1) ^ (n * d) * (b * Complex.abs z) - Complex.abs c =
(b - 1) ^ (n * d) * (b - 1) * Complex.abs z + ((b - 1) ^ (n * d) * Complex.abs z - Complex.abs c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ (b - 1) ^ (n * d) * (b * Complex.abs z) - Complex.abs c =
(b - 1) ^ (n * d) * (b - 1) * Complex.abs z + ((b - 1) ^ (n * d) * Complex.abs z - Complex.abs c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | rw [pow_succ] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ (b - 1) ^ (n * d) * (b - 1) * Complex.abs z + ((b - 1) ^ (n * d) * Complex.abs z - Complex.abs c) =
(b - 1) ^ (n * d + 1) * Complex.abs z + ((b - 1) ^ (n * d) * Complex.abs z - Complex.abs c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ (b - 1) ^ (n * d) * (b - 1) * Complex.abs z + ((b - 1) ^ (n * d) * Complex.abs z - Complex.abs c) =
(b - 1) ^ (n * d + 1) * Complex.abs z + ((b - 1) ^ (n * d) * Complex.abs z - Complex.abs c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | bound | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ (b - 1) ^ (n * d + 1) * Complex.abs z + ((b - 1) ^ (n * d) * Complex.abs z - Complex.abs c) β₯
(b - 1) ^ (n + 1) * Complex.abs z + (1 * Complex.abs z - Complex.abs c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ (b - 1) ^ (n * d + 1) * Complex.abs z + ((b - 1) ^ (n * d) * Complex.abs z - Complex.abs c) β₯
(b - 1) ^ (n + 1) * Complex.abs z + (1 * Complex.abs z - Complex.abs c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | rw [one_mul] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ (b - 1) ^ (n + 1) * Complex.abs z + (1 * Complex.abs z - Complex.abs c) =
(b - 1) ^ (n + 1) * Complex.abs z + (Complex.abs z - Complex.abs c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ (b - 1) ^ (n + 1) * Complex.abs z + (1 * Complex.abs z - Complex.abs c) =
(b - 1) ^ (n + 1) * Complex.abs z + (Complex.abs z - Complex.abs c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | bound | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ (b - 1) ^ (n + 1) * Complex.abs z + (Complex.abs z - Complex.abs c) β₯ (b - 1) ^ (n + 1) * Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
b : β
c z : β
b2 : 2 β€ b
bz : b β€ Complex.abs z
cz : Complex.abs c β€ Complex.abs z
n : β
w : β
h : (b - 1) ^ n * Complex.abs z β€ Complex.abs w
z1 : 1 β€ Complex.abs z
b1 : 1 β€ b - 1
b0 : 0 β€ b - 1
nd : n + 1 β€ n * d + 1
β’ (b - 1) ^ (n + 1) * Complex.abs z + (Complex.abs z - Complex.abs c) β₯ (b - 1) ^ (n + 1) * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_small | [367, 1] | [371, 57] | calc abs (z^d + c)
_ β€ abs (z^d) + abs c := by bound
_ β€ abs z ^ d + abs c := by rw [Complex.abs.map_pow] | cβ : β
dβ : β
instβ : Fact (2 β€ dβ)
d : β
c z : β
β’ Complex.abs (f' d c z) β€ Complex.abs z ^ d + Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβ : Fact (2 β€ dβ)
d : β
c z : β
β’ Complex.abs (f' d c z) β€ Complex.abs z ^ d + Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_small | [367, 1] | [371, 57] | bound | cβ : β
dβ : β
instβ : Fact (2 β€ dβ)
d : β
c z : β
β’ Complex.abs (z ^ d + c) β€ Complex.abs (z ^ d) + Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβ : Fact (2 β€ dβ)
d : β
c z : β
β’ Complex.abs (z ^ d + c) β€ Complex.abs (z ^ d) + Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_small | [367, 1] | [371, 57] | rw [Complex.abs.map_pow] | cβ : β
dβ : β
instβ : Fact (2 β€ dβ)
d : β
c z : β
β’ Complex.abs (z ^ d) + Complex.abs c β€ Complex.abs z ^ d + Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβ : Fact (2 β€ dβ)
d : β
c z : β
β’ Complex.abs (z ^ d) + Complex.abs c β€ Complex.abs z ^ d + Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | f_f'_iter | [378, 1] | [381, 32] | induction' n with n h | c : β
d : β
instβ : Fact (2 β€ d)
n : β
z : β
β’ (f d c)^[n] βz = β((f' d c)^[n] z) | case zero
c : β
d : β
instβ : Fact (2 β€ d)
z : β
β’ (f d c)^[0] βz = β((f' d c)^[0] z)
case succ
c : β
d : β
instβ : Fact (2 β€ d)
z : β
n : β
h : (f d c)^[n] βz = β((f' d c)^[n] z)
β’ (f d c)^[n + 1] βz = β((f' d c)^[n + 1] z) | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
z : β
β’ (f d c)^[n] βz = β((f' d c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | f_f'_iter | [378, 1] | [381, 32] | simp only [Function.iterate_zero, id] | case zero
c : β
d : β
instβ : Fact (2 β€ d)
z : β
β’ (f d c)^[0] βz = β((f' d c)^[0] z)
case succ
c : β
d : β
instβ : Fact (2 β€ d)
z : β
n : β
h : (f d c)^[n] βz = β((f' d c)^[n] z)
β’ (f d c)^[n + 1] βz = β((f' d c)^[n + 1] z) | case succ
c : β
d : β
instβ : Fact (2 β€ d)
z : β
n : β
h : (f d c)^[n] βz = β((f' d c)^[n] z)
β’ (f d c)^[n + 1] βz = β((f' d c)^[n + 1] z) | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
c : β
d : β
instβ : Fact (2 β€ d)
z : β
β’ (f d c)^[0] βz = β((f' d c)^[0] z)
case succ
c : β
d : β
instβ : Fact (2 β€ d)
z : β
n : β
h : (f d c)^[n] βz = β((f' d c)^[n] z)
β’ (f d c)^[n + 1] βz = β((f' d c)^[n + 1] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | f_f'_iter | [378, 1] | [381, 32] | simp only [h, Function.iterate_succ_apply'] | case succ
c : β
d : β
instβ : Fact (2 β€ d)
z : β
n : β
h : (f d c)^[n] βz = β((f' d c)^[n] z)
β’ (f d c)^[n + 1] βz = β((f' d c)^[n + 1] z) | case succ
c : β
d : β
instβ : Fact (2 β€ d)
z : β
n : β
h : (f d c)^[n] βz = β((f' d c)^[n] z)
β’ f d c β((f' d c)^[n] z) = β(f' d c ((f' d c)^[n] z)) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : β
d : β
instβ : Fact (2 β€ d)
z : β
n : β
h : (f d c)^[n] βz = β((f' d c)^[n] z)
β’ (f d c)^[n + 1] βz = β((f' d c)^[n + 1] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | f_f'_iter | [378, 1] | [381, 32] | simp only [f, lift', rec_coe] | case succ
c : β
d : β
instβ : Fact (2 β€ d)
z : β
n : β
h : (f d c)^[n] βz = β((f' d c)^[n] z)
β’ f d c β((f' d c)^[n] z) = β(f' d c ((f' d c)^[n] z)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : β
d : β
instβ : Fact (2 β€ d)
z : β
n : β
h : (f d c)^[n] βz = β((f' d c)^[n] z)
β’ f d c β((f' d c)^[n] z) = β(f' d c ((f' d c)^[n] z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_coe | [383, 1] | [384, 90] | simp only [multibrot, mem_setOf, f_f'_iter, not_iff_not, tendsto_inf_iff_tendsto_atInf] | c : β
d : β
instβ : Fact (2 β€ d)
β’ c β multibrot d β Β¬Tendsto (fun n => (f' d c)^[n] c) atTop atInf | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ c β multibrot d β Β¬Tendsto (fun n => (f' d c)^[n] c) atTop atInf
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | julia_two_lt | [387, 1] | [393, 52] | simp only [multibrot_coe, not_le, not_not, (superF d).basin_iff_attracts, Attracts, f_f'_iter,
tendsto_inf_iff_tendsto_atInf, tendsto_atInf_iff_norm_tendsto_atTop,
Complex.norm_eq_abs] at z2 β’ | c : β
d : β
instβ : Fact (2 β€ d)
z : β
z2 : 2 < Complex.abs z
cz : Complex.abs c β€ Complex.abs z
β’ (c, βz) β β―.basin | c : β
d : β
instβ : Fact (2 β€ d)
z : β
z2 : 2 < Complex.abs z
cz : Complex.abs c β€ Complex.abs z
β’ Tendsto (fun x => Complex.abs ((f' d c)^[x] z)) atTop atTop | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
z : β
z2 : 2 < Complex.abs z
cz : Complex.abs c β€ Complex.abs z
β’ (c, βz) β β―.basin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | julia_two_lt | [387, 1] | [393, 52] | apply Filter.tendsto_atTop_mono (iter_large d (abs z) z2.le (le_refl _) cz) | c : β
d : β
instβ : Fact (2 β€ d)
z : β
z2 : 2 < Complex.abs z
cz : Complex.abs c β€ Complex.abs z
β’ Tendsto (fun x => Complex.abs ((f' d c)^[x] z)) atTop atTop | c : β
d : β
instβ : Fact (2 β€ d)
z : β
z2 : 2 < Complex.abs z
cz : Complex.abs c β€ Complex.abs z
β’ Tendsto (fun n => (Complex.abs z - 1) ^ n * Complex.abs z) atTop atTop | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
z : β
z2 : 2 < Complex.abs z
cz : Complex.abs c β€ Complex.abs z
β’ Tendsto (fun x => Complex.abs ((f' d c)^[x] z)) atTop atTop
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | julia_two_lt | [387, 1] | [393, 52] | refine Filter.Tendsto.atTop_mul (by linarith) ?_ tendsto_const_nhds | c : β
d : β
instβ : Fact (2 β€ d)
z : β
z2 : 2 < Complex.abs z
cz : Complex.abs c β€ Complex.abs z
β’ Tendsto (fun n => (Complex.abs z - 1) ^ n * Complex.abs z) atTop atTop | c : β
d : β
instβ : Fact (2 β€ d)
z : β
z2 : 2 < Complex.abs z
cz : Complex.abs c β€ Complex.abs z
β’ Tendsto (fun n => (Complex.abs z - 1) ^ n) atTop atTop | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
z : β
z2 : 2 < Complex.abs z
cz : Complex.abs c β€ Complex.abs z
β’ Tendsto (fun n => (Complex.abs z - 1) ^ n * Complex.abs z) atTop atTop
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | julia_two_lt | [387, 1] | [393, 52] | apply tendsto_pow_atTop_atTop_of_one_lt | c : β
d : β
instβ : Fact (2 β€ d)
z : β
z2 : 2 < Complex.abs z
cz : Complex.abs c β€ Complex.abs z
β’ Tendsto (fun n => (Complex.abs z - 1) ^ n) atTop atTop | case h
c : β
d : β
instβ : Fact (2 β€ d)
z : β
z2 : 2 < Complex.abs z
cz : Complex.abs c β€ Complex.abs z
β’ 1 < Complex.abs z - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
z : β
z2 : 2 < Complex.abs z
cz : Complex.abs c β€ Complex.abs z
β’ Tendsto (fun n => (Complex.abs z - 1) ^ n) atTop atTop
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | julia_two_lt | [387, 1] | [393, 52] | linarith | case h
c : β
d : β
instβ : Fact (2 β€ d)
z : β
z2 : 2 < Complex.abs z
cz : Complex.abs c β€ Complex.abs z
β’ 1 < Complex.abs z - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : β
d : β
instβ : Fact (2 β€ d)
z : β
z2 : 2 < Complex.abs z
cz : Complex.abs c β€ Complex.abs z
β’ 1 < Complex.abs z - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | julia_two_lt | [387, 1] | [393, 52] | linarith | c : β
d : β
instβ : Fact (2 β€ d)
z : β
z2 : 2 < Complex.abs z
cz : Complex.abs c β€ Complex.abs z
β’ 0 < Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
z : β
z2 : 2 < Complex.abs z
cz : Complex.abs c β€ Complex.abs z
β’ 0 < Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | julia_le_two | [396, 1] | [399, 26] | contrapose m | c : β
d : β
instβ : Fact (2 β€ d)
z : β
m : (c, βz) β β―.basin
cz : Complex.abs c β€ Complex.abs z
β’ Complex.abs z β€ 2 | c : β
d : β
instβ : Fact (2 β€ d)
z : β
cz : Complex.abs c β€ Complex.abs z
m : Β¬Complex.abs z β€ 2
β’ Β¬(c, βz) β β―.basin | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
z : β
m : (c, βz) β β―.basin
cz : Complex.abs c β€ Complex.abs z
β’ Complex.abs z β€ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | julia_le_two | [396, 1] | [399, 26] | simp only [not_le, not_not] at m β’ | c : β
d : β
instβ : Fact (2 β€ d)
z : β
cz : Complex.abs c β€ Complex.abs z
m : Β¬Complex.abs z β€ 2
β’ Β¬(c, βz) β β―.basin | c : β
d : β
instβ : Fact (2 β€ d)
z : β
cz : Complex.abs c β€ Complex.abs z
m : 2 < Complex.abs z
β’ (c, βz) β β―.basin | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
z : β
cz : Complex.abs c β€ Complex.abs z
m : Β¬Complex.abs z β€ 2
β’ Β¬(c, βz) β β―.basin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | julia_le_two | [396, 1] | [399, 26] | exact julia_two_lt m cz | c : β
d : β
instβ : Fact (2 β€ d)
z : β
cz : Complex.abs c β€ Complex.abs z
m : 2 < Complex.abs z
β’ (c, βz) β β―.basin | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
z : β
cz : Complex.abs c β€ Complex.abs z
m : 2 < Complex.abs z
β’ (c, βz) β β―.basin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_le_two | [411, 1] | [413, 35] | rw [multibrot_basin' (d := d)] at m | c : β
d : β
instβ : Fact (2 β€ d)
m : c β multibrot d
β’ Complex.abs c β€ 2 | c : β
d : β
instβ : Fact (2 β€ d)
m : (c, βc) β β―.basin
β’ Complex.abs c β€ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
m : c β multibrot d
β’ Complex.abs c β€ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_le_two | [411, 1] | [413, 35] | exact julia_le_two m (le_refl _) | c : β
d : β
instβ : Fact (2 β€ d)
m : (c, βc) β β―.basin
β’ Complex.abs c β€ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
m : (c, βc) β β―.basin
β’ Complex.abs c β€ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_subset_closedBall | [416, 1] | [417, 93] | intro c m | c : β
d : β
instβ : Fact (2 β€ d)
β’ multibrot d β closedBall 0 2 | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
m : c β multibrot d
β’ c β closedBall 0 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ multibrot d β closedBall 0 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_subset_closedBall | [416, 1] | [417, 93] | simp only [mem_closedBall, Complex.dist_eq, sub_zero] | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
m : c β multibrot d
β’ c β closedBall 0 2 | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
m : c β multibrot d
β’ Complex.abs c β€ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
m : c β multibrot d
β’ c β closedBall 0 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_subset_closedBall | [416, 1] | [417, 93] | exact multibrot_le_two m | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
m : c β multibrot d
β’ Complex.abs c β€ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
m : c β multibrot d
β’ Complex.abs c β€ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_two_lt | [420, 1] | [421, 77] | contrapose a | c : β
d : β
instβ : Fact (2 β€ d)
a : 2 < Complex.abs c
β’ c β multibrot d | c : β
d : β
instβ : Fact (2 β€ d)
a : Β¬c β multibrot d
β’ Β¬2 < Complex.abs c | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
a : 2 < Complex.abs c
β’ c β multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_two_lt | [420, 1] | [421, 77] | simp only [not_lt, not_not] at a β’ | c : β
d : β
instβ : Fact (2 β€ d)
a : Β¬c β multibrot d
β’ Β¬2 < Complex.abs c | c : β
d : β
instβ : Fact (2 β€ d)
a : c β multibrot d
β’ Complex.abs c β€ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
a : Β¬c β multibrot d
β’ Β¬2 < Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_two_lt | [420, 1] | [421, 77] | exact multibrot_le_two a | c : β
d : β
instβ : Fact (2 β€ d)
a : c β multibrot d
β’ Complex.abs c β€ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
a : c β multibrot d
β’ Complex.abs c β€ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | generalize hg : (fun n β¦ (f' d c)^[n] c) = g | c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
h : (f d c)^[a] βc = (f d c)^[b] βc
β’ c β multibrot d | c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
h : (f d c)^[a] βc = (f d c)^[b] βc
g : β β β
hg : (fun n => (f' d c)^[n] c) = g
β’ c β multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
h : (f d c)^[a] βc = (f d c)^[b] βc
β’ c β multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | replace hg : β n, (f' d c)^[n] c = g n := fun n β¦ by rw [β hg] | c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
h : (f d c)^[a] βc = (f d c)^[b] βc
g : β β β
hg : (fun n => (f' d c)^[n] c) = g
β’ c β multibrot d | c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
h : (f d c)^[a] βc = (f d c)^[b] βc
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
β’ c β multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
h : (f d c)^[a] βc = (f d c)^[b] βc
g : β β β
hg : (fun n => (f' d c)^[n] c) = g
β’ c β multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | simp only [f_f'_iter, β coe_zero, coe_eq_coe, hg] at h | c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
h : (f d c)^[a] βc = (f d c)^[b] βc
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
β’ c β multibrot d | c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
β’ c β multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
h : (f d c)^[a] βc = (f d c)^[b] βc
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
β’ c β multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | simp only [multibrot_coe, atInf_basis.tendsto_right_iff, true_imp_iff, not_forall,
Filter.not_eventually, mem_setOf, not_lt, Complex.norm_eq_abs, hg] | c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
lo : β (n : β), β k β€ b, g n = g k
β’ c β multibrot d | c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
lo : β (n : β), β k β€ b, g n = g k
β’ β x, βαΆ (x_1 : β) in atTop, Complex.abs (g x_1) β€ x | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
lo : β (n : β), β k β€ b, g n = g k
β’ c β multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | use partialSups (fun k β¦ Complex.abs (g k)) b | c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
lo : β (n : β), β k β€ b, g n = g k
β’ β x, βαΆ (x_1 : β) in atTop, Complex.abs (g x_1) β€ x | case h
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
lo : β (n : β), β k β€ b, g n = g k
β’ βαΆ (x : β) in atTop, Complex.abs (g x) β€ (partialSups fun k => Complex.abs (g k)) b | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
lo : β (n : β), β k β€ b, g n = g k
β’ β x, βαΆ (x_1 : β) in atTop, Complex.abs (g x_1) β€ x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | apply Filter.frequently_of_forall | case h
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
lo : β (n : β), β k β€ b, g n = g k
β’ βαΆ (x : β) in atTop, Complex.abs (g x) β€ (partialSups fun k => Complex.abs (g k)) b | case h.h
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
lo : β (n : β), β k β€ b, g n = g k
β’ β (x : β), Complex.abs (g x) β€ (partialSups fun k => Complex.abs (g k)) b | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
lo : β (n : β), β k β€ b, g n = g k
β’ βαΆ (x : β) in atTop, Complex.abs (g x) β€ (partialSups fun k => Complex.abs (g k)) b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | intro k | case h.h
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
lo : β (n : β), β k β€ b, g n = g k
β’ β (x : β), Complex.abs (g x) β€ (partialSups fun k => Complex.abs (g k)) b | case h.h
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
lo : β (n : β), β k β€ b, g n = g k
k : β
β’ Complex.abs (g k) β€ (partialSups fun k => Complex.abs (g k)) b | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
lo : β (n : β), β k β€ b, g n = g k
β’ β (x : β), Complex.abs (g x) β€ (partialSups fun k => Complex.abs (g k)) b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | rcases lo k with β¨l, lb, klβ© | case h.h
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
lo : β (n : β), β k β€ b, g n = g k
k : β
β’ Complex.abs (g k) β€ (partialSups fun k => Complex.abs (g k)) b | case h.h.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
lo : β (n : β), β k β€ b, g n = g k
k l : β
lb : l β€ b
kl : g k = g l
β’ Complex.abs (g k) β€ (partialSups fun k => Complex.abs (g k)) b | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
lo : β (n : β), β k β€ b, g n = g k
k : β
β’ Complex.abs (g k) β€ (partialSups fun k => Complex.abs (g k)) b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | rw [kl] | case h.h.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
lo : β (n : β), β k β€ b, g n = g k
k l : β
lb : l β€ b
kl : g k = g l
β’ Complex.abs (g k) β€ (partialSups fun k => Complex.abs (g k)) b | case h.h.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
lo : β (n : β), β k β€ b, g n = g k
k l : β
lb : l β€ b
kl : g k = g l
β’ Complex.abs (g l) β€ (partialSups fun k => Complex.abs (g k)) b | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
lo : β (n : β), β k β€ b, g n = g k
k l : β
lb : l β€ b
kl : g k = g l
β’ Complex.abs (g k) β€ (partialSups fun k => Complex.abs (g k)) b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | exact le_partialSups_of_le (fun k β¦ abs (g k)) lb | case h.h.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
lo : β (n : β), β k β€ b, g n = g k
k l : β
lb : l β€ b
kl : g k = g l
β’ Complex.abs (g l) β€ (partialSups fun k => Complex.abs (g k)) b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
lo : β (n : β), β k β€ b, g n = g k
k l : β
lb : l β€ b
kl : g k = g l
β’ Complex.abs (g l) β€ (partialSups fun k => Complex.abs (g k)) b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | rw [β hg] | c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
h : (f d c)^[a] βc = (f d c)^[b] βc
g : β β β
hg : (fun n => (f' d c)^[n] c) = g
n : β
β’ (f' d c)^[n] c = g n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
h : (f d c)^[a] βc = (f d c)^[b] βc
g : β β β
hg : (fun n => (f' d c)^[n] c) = g
n : β
β’ (f' d c)^[n] c = g n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | intro n | c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
β’ β (n : β), β k β€ b, g n = g k | c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n : β
β’ β k β€ b, g n = g k | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
β’ β (n : β), β k β€ b, g n = g k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | induction' n with n h | c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n : β
β’ β k β€ b, g n = g k | case zero
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
β’ β k β€ b, g 0 = g k
case succ
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
hβ : g a = g b
n : β
h : β k β€ b, g n = g k
β’ β k β€ b, g (n + 1) = g k | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n : β
β’ β k β€ b, g n = g k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | use 0, Nat.zero_le _ | case zero
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
β’ β k β€ b, g 0 = g k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
β’ β k β€ b, g 0 = g k
TACTIC:
|
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