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/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | norm_num | c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
β’ 4 β€ 6 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
β’ 4 β€ 6
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | intro n | case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
β’ (0.8095 * (z.normSq.log * -0.9635).exp).hi β nan β
potential_error 2 c' z' β€ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val | case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : (0.8095 * (z.normSq.log * -0.9635).exp).hi β nan
β’ potential_error 2 c' z' β€ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val | Please generate a tactic in lean4 to solve the state.
STATE:
case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
β’ (0.8095 * (z.normSq.log * -0.9635).exp).hi β nan β
potential_error 2 c' z' β€ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | rw [Ne, Interval.hi_eq_nan] at n | case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : (0.8095 * (z.normSq.log * -0.9635).exp).hi β nan
β’ potential_error 2 c' z' β€ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val | case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
β’ potential_error 2 c' z' β€ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val | Please generate a tactic in lean4 to solve the state.
STATE:
case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : (0.8095 * (z.normSq.log * -0.9635).exp).hi β nan
β’ potential_error 2 c' z' β€ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | refine le_trans (potential_error_le_of_z6 _ z6 cz) ?_ | case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
β’ potential_error 2 c' z' β€ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val | case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
β’ 0.8095 / Complex.abs z' ^ 1.927 β€ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val | Please generate a tactic in lean4 to solve the state.
STATE:
case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
β’ potential_error 2 c' z' β€ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | apply Interval.le_hi n | case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
β’ 0.8095 / Complex.abs z' ^ 1.927 β€ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val | case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
β’ 0.8095 / Complex.abs z' ^ 1.927 β approx (0.8095 * (z.normSq.log * -0.9635).exp) | Please generate a tactic in lean4 to solve the state.
STATE:
case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
β’ 0.8095 / Complex.abs z' ^ 1.927 β€ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | rw [div_eq_mul_inv, βReal.rpow_neg (Complex.abs.nonneg _), Real.rpow_def_of_pos (by linarith)] | case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
β’ 0.8095 / Complex.abs z' ^ 1.927 β approx (0.8095 * (z.normSq.log * -0.9635).exp) | case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
β’ 0.8095 * ((Complex.abs z').log * -1.927).exp β approx (0.8095 * (z.normSq.log * -0.9635).exp) | Please generate a tactic in lean4 to solve the state.
STATE:
case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
β’ 0.8095 / Complex.abs z' ^ 1.927 β approx (0.8095 * (z.normSq.log * -0.9635).exp)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | have e : Real.log (Complex.abs z') * -1.927 = Real.log (Complex.abs z' ^ 2) * -0.9635 := by
rw [Real.log_pow, Nat.cast_two, mul_comm (2:β), mul_assoc]; norm_num | case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
β’ 0.8095 * ((Complex.abs z').log * -1.927).exp β approx (0.8095 * (z.normSq.log * -0.9635).exp) | case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
e : (Complex.abs z').log * -1.927 = (Complex.abs z' ^ 2).log * -0.9635
β’ 0.8095 * ((Complex.abs z').log * -1.927).exp β approx (0.8095 * (z.normSq.log * -0.9635).exp) | Please generate a tactic in lean4 to solve the state.
STATE:
case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
β’ 0.8095 * ((Complex.abs z').log * -1.927).exp β approx (0.8095 * (z.normSq.log * -0.9635).exp)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | rw [e] | case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
e : (Complex.abs z').log * -1.927 = (Complex.abs z' ^ 2).log * -0.9635
β’ 0.8095 * ((Complex.abs z').log * -1.927).exp β approx (0.8095 * (z.normSq.log * -0.9635).exp) | case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
e : (Complex.abs z').log * -1.927 = (Complex.abs z' ^ 2).log * -0.9635
β’ 0.8095 * ((Complex.abs z' ^ 2).log * -0.9635).exp β approx (0.8095 * (z.normSq.log * -0.9635).exp) | Please generate a tactic in lean4 to solve the state.
STATE:
case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
e : (Complex.abs z').log * -1.927 = (Complex.abs z' ^ 2).log * -0.9635
β’ 0.8095 * ((Complex.abs z').log * -1.927).exp β approx (0.8095 * (z.normSq.log * -0.9635).exp)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | mono | case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
e : (Complex.abs z').log * -1.927 = (Complex.abs z' ^ 2).log * -0.9635
β’ 0.8095 * ((Complex.abs z' ^ 2).log * -0.9635).exp β approx (0.8095 * (z.normSq.log * -0.9635).exp) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case be
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
e : (Complex.abs z').log * -1.927 = (Complex.abs z' ^ 2).log * -0.9635
β’ 0.8095 * ((Complex.abs z' ^ 2).log * -0.9635).exp β approx (0.8095 * (z.normSq.log * -0.9635).exp)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | linarith | c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
β’ 0 < Complex.abs z' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
β’ 0 < Complex.abs z'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | rw [Real.log_pow, Nat.cast_two, mul_comm (2:β), mul_assoc] | c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
β’ (Complex.abs z').log * -1.927 = (Complex.abs z' ^ 2).log * -0.9635 | c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
β’ (Complex.abs z').log * -1.927 = (Complex.abs z').log * (2 * -0.9635) | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
β’ (Complex.abs z').log * -1.927 = (Complex.abs z' ^ 2).log * -0.9635
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | norm_num | c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
β’ (Complex.abs z').log * -1.927 = (Complex.abs z').log * (2 * -0.9635) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
n : Β¬0.8095 * (z.normSq.log * -0.9635).exp = nan
β’ (Complex.abs z').log * -1.927 = (Complex.abs z').log * (2 * -0.9635)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | have e : 1 / Complex.abs z' = Real.exp (-(Real.log (Complex.abs z' ^ 2) / 2)) := by
simp only [one_div, Real.log_pow, Nat.cast_ofNat, neg_mul, Real.rpow_neg zero_le_two,
Real.rpow_one, βmul_assoc, mul_comm _ (2:β)β»ΒΉ]
rw [mul_div_cancel_leftβ _ two_ne_zero, Real.exp_neg, Real.exp_log (by linarith)] | case zx
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
β’ 1 / Complex.abs z' β approx (-z.normSq.log.div2).exp | case zx
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
e : 1 / Complex.abs z' = (-((Complex.abs z' ^ 2).log / 2)).exp
β’ 1 / Complex.abs z' β approx (-z.normSq.log.div2).exp | Please generate a tactic in lean4 to solve the state.
STATE:
case zx
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
β’ 1 / Complex.abs z' β approx (-z.normSq.log.div2).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | rw [e] | case zx
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
e : 1 / Complex.abs z' = (-((Complex.abs z' ^ 2).log / 2)).exp
β’ 1 / Complex.abs z' β approx (-z.normSq.log.div2).exp | case zx
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
e : 1 / Complex.abs z' = (-((Complex.abs z' ^ 2).log / 2)).exp
β’ (-((Complex.abs z' ^ 2).log / 2)).exp β approx (-z.normSq.log.div2).exp | Please generate a tactic in lean4 to solve the state.
STATE:
case zx
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
e : 1 / Complex.abs z' = (-((Complex.abs z' ^ 2).log / 2)).exp
β’ 1 / Complex.abs z' β approx (-z.normSq.log.div2).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | mono | case zx
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
e : 1 / Complex.abs z' = (-((Complex.abs z' ^ 2).log / 2)).exp
β’ (-((Complex.abs z' ^ 2).log / 2)).exp β approx (-z.normSq.log.div2).exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zx
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
e : 1 / Complex.abs z' = (-((Complex.abs z' ^ 2).log / 2)).exp
β’ (-((Complex.abs z' ^ 2).log / 2)).exp β approx (-z.normSq.log.div2).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | simp only [one_div, Real.log_pow, Nat.cast_ofNat, neg_mul, Real.rpow_neg zero_le_two,
Real.rpow_one, βmul_assoc, mul_comm _ (2:β)β»ΒΉ] | c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
β’ 1 / Complex.abs z' = (-((Complex.abs z' ^ 2).log / 2)).exp | c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
β’ (Complex.abs z')β»ΒΉ = (-(2 * (Complex.abs z').log / 2)).exp | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
β’ 1 / Complex.abs z' = (-((Complex.abs z' ^ 2).log / 2)).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | rw [mul_div_cancel_leftβ _ two_ne_zero, Real.exp_neg, Real.exp_log (by linarith)] | c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
β’ (Complex.abs z')β»ΒΉ = (-(2 * (Complex.abs z').log / 2)).exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
β’ (Complex.abs z')β»ΒΉ = (-(2 * (Complex.abs z').log / 2)).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | linarith | c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
β’ 0 < Complex.abs z' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
β’ 0 < Complex.abs z'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | set s := superF 2 | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
β’ β―.potential c' βz' β approx (c.potential z n r).1 | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
β’ s.potential c' βz' β approx (c.potential z n r).1 | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
β’ β―.potential c' βz' β approx (c.potential z n r).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [Box.potential] | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
β’ s.potential c' βz' β approx (c.potential z n r).1 | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
β’ s.potential c' βz' β
approx
(let cs := c.normSq.hi;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
β’ s.potential c' βz' β approx (c.potential z n r).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | generalize hcs : (normSq c).hi = cs | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
β’ s.potential c' βz' β
approx
(let cs := c.normSq.hi;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
β’ s.potential c' βz' β
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
β’ s.potential c' βz' β
approx
(let cs := c.normSq.hi;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | generalize hi : iterate c z (cs.max 9) n = i | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
β’ s.potential c' βz' β
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
β’ s.potential c' βz' β
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
β’ s.potential c' βz' β
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | by_cases csn : cs = nan | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
β’ s.potential c' βz' β
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case pos
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : cs = nan
β’ s.potential c' βz' β
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
β’ s.potential c' βz' β
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
β’ s.potential c' βz' β
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [hi, Interval.hi_eq_nan, Floating.val_lt_val] | case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
β’ s.potential c' βz' β
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
β’ s.potential c' βz' β
approx
(match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
β’ s.potential c' βz' β
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | generalize hie : i.exit = ie | case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
β’ s.potential c' βz' β
approx
(match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
ie : Exit
hie : i.exit = ie
β’ s.potential c' βz' β
approx
(match ie with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
β’ s.potential c' βz' β
approx
(match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | induction ie | case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
ie : Exit
hie : i.exit = ie
β’ s.potential c' βz' β
approx
(match ie with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case neg.count
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
β’ s.potential c' βz' β
approx
(match Exit.count with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
case neg.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
β’ s.potential c' βz' β
approx
(match Exit.large with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
case neg.nan
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.nan
β’ s.potential c' βz' β
approx
(match Exit.nan with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
ie : Exit
hie : i.exit = ie
β’ s.potential c' βz' β
approx
(match ie with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [csn, Floating.nan_max, iterate_nan, Interval.approx_nan, mem_univ] | case pos
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : cs = nan
β’ s.potential c' βz' β
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : cs = nan
β’ s.potential c' βz' β
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | generalize hzs : (normSq i.z) = zs | case neg.count
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
β’ s.potential c' βz' β
approx
(match Exit.count with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case neg.count
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
β’ s.potential c' βz' β
approx
(match Exit.count with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.count
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
β’ s.potential c' βz' β
approx
(match Exit.count with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | by_cases bad : zs = nan β¨ (16 : Floating).val < zs.hi.val β¨ (16 : Floating).val < cs.val | case neg.count
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
β’ s.potential c' βz' β
approx
(match Exit.count with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case pos
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val
β’ s.potential c' βz' β
approx
(match Exit.count with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : Β¬(zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val)
β’ s.potential c' βz' β
approx
(match Exit.count with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.count
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
β’ s.potential c' βz' β
approx
(match Exit.count with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [Floating.val_lt_val, bad, βreduceIte, Interval.approx_nan, mem_univ] | case pos
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val
β’ s.potential c' βz' β
approx
(match Exit.count with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val
β’ s.potential c' βz' β
approx
(match Exit.count with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [bad, βreduceIte] | case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : Β¬(zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val)
β’ s.potential c' βz' β
approx
(match Exit.count with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : Β¬(zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val)
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : Β¬(zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val)
β’ s.potential c' βz' β
approx
(match Exit.count with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [not_or, not_lt, βhzs] at bad | case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : Β¬(zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val)
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n) | case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : Β¬i.z.normSq = nan β§ i.z.normSq.hi.val β€ 16.val β§ cs.val β€ 16.val
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : Β¬(zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val)
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rcases bad with β¨zsn, z4, c4β© | case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : Β¬i.z.normSq = nan β§ i.z.normSq.hi.val β€ 16.val β§ cs.val β€ 16.val
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n) | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16.val
c4 : cs.val β€ 16.val
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : Β¬i.z.normSq = nan β§ i.z.normSq.hi.val β€ 16.val β§ cs.val β€ 16.val
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [Floating.val_ofNat] at c4 z4 | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16.val
c4 : cs.val β€ 16.val
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n) | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ β16
c4 : cs.val β€ β16
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16.val
c4 : cs.val β€ 16.val
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [β hcs, Nat.cast_ofNat, Interval.hi_eq_nan] at c4 z4 csn zsn | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ β16
c4 : cs.val β€ β16
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n) | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ β16
c4 : cs.val β€ β16
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | apply Interval.mem_approx_iter_sqrt' s.potential_nonneg | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n) | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
β’ s.potential c' βz' ^ 2 ^ i.n β approx potential_small | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [βs.potential_eqn_iter, f_f'_iter] | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
β’ s.potential c' βz' ^ 2 ^ i.n β approx potential_small | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
β’ s.potential c' β((f' 2 c')^[i.n] z') β approx potential_small | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
β’ s.potential c' βz' ^ 2 ^ i.n β approx potential_small
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | generalize hw' : (f' 2 c')^[i.n] z' = w' | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
β’ s.potential c' β((f' 2 c')^[i.n] z') β approx potential_small | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
β’ s.potential c' βw' β approx potential_small | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
β’ s.potential c' β((f' 2 c')^[i.n] z') β approx potential_small
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | have le4 : Real.sqrt 16 β€ 4 := by rw [Real.sqrt_le_iff]; norm_num | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
β’ s.potential c' βw' β approx potential_small | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ s.potential c' βw' β approx potential_small | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
β’ s.potential c' βw' β approx potential_small
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | apply approx_potential_small | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ s.potential c' βw' β approx potential_small | case neg.intro.intro.c4
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ Complex.abs c' β€ 4
case neg.intro.intro.z4
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ Complex.abs w' β€ 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ s.potential c' βw' β approx potential_small
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [Real.sqrt_le_iff] | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
β’ β16 β€ 4 | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
β’ 0 β€ 4 β§ 16 β€ 4 ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
β’ β16 β€ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | norm_num | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
β’ 0 β€ 4 β§ 16 β€ 4 ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
β’ 0 β€ 4 β§ 16 β€ 4 ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | exact le_trans (Box.abs_le_sqrt_normSq cm csn) (le_trans (Real.sqrt_le_sqrt c4) le4) | case neg.intro.intro.c4
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ Complex.abs c' β€ 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.c4
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ Complex.abs c' β€ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | refine le_trans (Box.abs_le_sqrt_normSq ?_ zsn) (le_trans (Real.sqrt_le_sqrt z4) le4) | case neg.intro.intro.z4
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ Complex.abs w' β€ 4 | case neg.intro.intro.z4
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ w' β approx i.z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.z4
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ Complex.abs w' β€ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [βhw', βhi] | case neg.intro.intro.z4
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ w' β approx i.z | case neg.intro.intro.z4
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ (f' 2 c')^[(iterate c z (cs.max 9) n).n] z' β approx (iterate c z (cs.max 9) n).z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.z4
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ w' β approx i.z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | exact mem_approx_iterate cm zm _ | case neg.intro.intro.z4
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ (f' 2 c')^[(iterate c z (cs.max 9) n).n] z' β approx (iterate c z (cs.max 9) n).z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.z4
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ (f' 2 c')^[(iterate c z (cs.max 9) n).n] z' β approx (iterate c z (cs.max 9) n).z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | generalize hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j | case neg.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
β’ s.potential c' βz' β
approx
(match Exit.large with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case neg.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
β’ s.potential c' βz' β
approx
(match Exit.large with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
β’ s.potential c' βz' β
approx
(match Exit.large with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [hj] | case neg.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
β’ s.potential c' βz' β
approx
(match Exit.large with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case neg.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
β’ s.potential c' βz' β
approx
(match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
β’ s.potential c' βz' β
approx
(match Exit.large with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | generalize hje : j.exit = je | case neg.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
β’ s.potential c' βz' β
approx
(match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1 | case neg.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
je : Exit
hje : j.exit = je
β’ s.potential c' βz' β
approx
(match je with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
β’ s.potential c' βz' β
approx
(match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | induction je | case neg.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
je : Exit
hje : j.exit = je
β’ s.potential c' βz' β
approx
(match je with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1 | case neg.large.count
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.count
β’ s.potential c' βz' β
approx
(match Exit.count with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.large
β’ s.potential c' βz' β
approx
(match Exit.large with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1
case neg.large.nan
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.nan
β’ s.potential c' βz' β
approx
(match Exit.nan with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
je : Exit
hje : j.exit = je
β’ s.potential c' βz' β
approx
(match je with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [Interval.approx_nan, mem_univ] | case neg.large.count
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.count
β’ s.potential c' βz' β
approx
(match Exit.count with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.count
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.count
β’ s.potential c' βz' β
approx
(match Exit.count with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.large
β’ s.potential c' βz' β
approx
(match Exit.large with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1 | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.large
β’ s.potential c' βz' β approx (j.z.potential_large.iter_sqrt (i.n + j.n)) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.large
β’ s.potential c' βz' β
approx
(match Exit.large with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | generalize hn : i.n + j.n = n | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.large
β’ s.potential c' βz' β approx (j.z.potential_large.iter_sqrt (i.n + j.n)) | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.large
n : β
hn : i.n + j.n = n
β’ s.potential c' βz' β approx (j.z.potential_large.iter_sqrt n) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.large
β’ s.potential c' βz' β approx (j.z.potential_large.iter_sqrt (i.n + j.n))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | apply Interval.mem_approx_iter_sqrt' s.potential_nonneg | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.large
n : β
hn : i.n + j.n = n
β’ s.potential c' βz' β approx (j.z.potential_large.iter_sqrt n) | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.large
n : β
hn : i.n + j.n = n
β’ s.potential c' βz' ^ 2 ^ n β approx j.z.potential_large | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.large
n : β
hn : i.n + j.n = n
β’ s.potential c' βz' β approx (j.z.potential_large.iter_sqrt n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [βs.potential_eqn_iter, f_f'_iter, βhj] at hje β’ | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.large
n : β
hn : i.n + j.n = n
β’ s.potential c' βz' ^ 2 ^ n β approx j.z.potential_large | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
β’ s.potential c' β((f' 2 c')^[n] z') β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.large
n : β
hn : i.n + j.n = n
β’ s.potential c' βz' ^ 2 ^ n β approx j.z.potential_large
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | generalize hw' : (f' 2 c')^[n] z' = w' | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
β’ s.potential c' β((f' 2 c')^[n] z') β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
β’ s.potential c' β((f' 2 c')^[n] z') β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | have izm : (f' 2 c')^[i.n] z' β approx i.z := by rw [βhi]; exact mem_approx_iterate cm zm _ | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | have jl := iterate_large cm izm hje | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl :
((r.mul r true).max (cs.max 36)).val <
Complex.abs ((f' 2 c')^[(iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n] ((f' 2 c')^[i.n] z')) ^ 2
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | have jrn := ne_nan_of_iterate (hje.trans_ne (by decide)) | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl :
((r.mul r true).max (cs.max 36)).val <
Complex.abs ((f' 2 c')^[(iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n] ((f' 2 c')^[i.n] z')) ^ 2
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl :
((r.mul r true).max (cs.max 36)).val <
Complex.abs ((f' 2 c')^[(iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n] ((f' 2 c')^[i.n] z')) ^ 2
jrn : (r.mul r true).max (cs.max 36) β nan
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl :
((r.mul r true).max (cs.max 36)).val <
Complex.abs ((f' 2 c')^[(iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n] ((f' 2 c')^[i.n] z')) ^ 2
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [hj, β Function.iterate_add_apply, add_comm _ i.n, hn, hw'] at jl | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl :
((r.mul r true).max (cs.max 36)).val <
Complex.abs ((f' 2 c')^[(iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n] ((f' 2 c')^[i.n] z')) ^ 2
jrn : (r.mul r true).max (cs.max 36) β nan
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jrn : (r.mul r true).max (cs.max 36) β nan
jl : ((r.mul r true).max (cs.max 36)).val < Complex.abs w' ^ 2
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl :
((r.mul r true).max (cs.max 36)).val <
Complex.abs ((f' 2 c')^[(iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n] ((f' 2 c')^[i.n] z')) ^ 2
jrn : (r.mul r true).max (cs.max 36) β nan
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [ne_eq, Floating.max_eq_nan, not_or] at jrn | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jrn : (r.mul r true).max (cs.max 36) β nan
jl : ((r.mul r true).max (cs.max 36)).val < Complex.abs w' ^ 2
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : ((r.mul r true).max (cs.max 36)).val < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jrn : (r.mul r true).max (cs.max 36) β nan
jl : ((r.mul r true).max (cs.max 36)).val < Complex.abs w' ^ 2
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [Floating.val_max jrn.1 (Floating.max_ne_nan.mpr jrn.2),
Floating.val_max jrn.2.1 jrn.2.2, max_lt_iff, max_lt_iff, Floating.val_ofNat,
Nat.cast_eq_ofNat] at jl | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : ((r.mul r true).max (cs.max 36)).val < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : ((r.mul r true).max (cs.max 36)).val < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | apply approx_potential_large | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large.cz
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ Complex.abs c' β€ Complex.abs w'
case neg.large.large.z6
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ 6 β€ Complex.abs w'
case neg.large.large.zm
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ w' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [βhi] | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
β’ (f' 2 c')^[i.n] z' β approx i.z | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
β’ (f' 2 c')^[(iterate c z (cs.max 9) nβ).n] z' β approx (iterate c z (cs.max 9) nβ).z | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
β’ (f' 2 c')^[i.n] z' β approx i.z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | exact mem_approx_iterate cm zm _ | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
β’ (f' 2 c')^[(iterate c z (cs.max 9) nβ).n] z' β approx (iterate c z (cs.max 9) nβ).z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
β’ (f' 2 c')^[(iterate c z (cs.max 9) nβ).n] z' β approx (iterate c z (cs.max 9) nβ).z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | decide | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl :
((r.mul r true).max (cs.max 36)).val <
Complex.abs ((f' 2 c')^[(iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n] ((f' 2 c')^[i.n] z')) ^ 2
β’ Exit.large β Exit.nan | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl :
((r.mul r true).max (cs.max 36)).val <
Complex.abs ((f' 2 c')^[(iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n] ((f' 2 c')^[i.n] z')) ^ 2
β’ Exit.large β Exit.nan
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | refine le_trans ?_ (le_trans (Real.sqrt_le_sqrt jl.2.1.le) ?_) | case neg.large.large.cz
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ Complex.abs c' β€ Complex.abs w' | case neg.large.large.cz.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ Complex.abs c' β€ cs.val.sqrt
case neg.large.large.cz.refine_2
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ (Complex.abs w' ^ 2).sqrt β€ Complex.abs w' | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large.cz
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ Complex.abs c' β€ Complex.abs w'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [β hcs, Interval.hi_eq_nan] at csn β’ | case neg.large.large.cz.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ Complex.abs c' β€ cs.val.sqrt | case neg.large.large.cz.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
csn : Β¬c.normSq = nan
β’ Complex.abs c' β€ c.normSq.hi.val.sqrt | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large.cz.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ Complex.abs c' β€ cs.val.sqrt
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | exact abs_le_sqrt_normSq cm csn | case neg.large.large.cz.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
csn : Β¬c.normSq = nan
β’ Complex.abs c' β€ c.normSq.hi.val.sqrt | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large.cz.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
csn : Β¬c.normSq = nan
β’ Complex.abs c' β€ c.normSq.hi.val.sqrt
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [apply_nonneg, Real.sqrt_sq, le_refl] | case neg.large.large.cz.refine_2
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ (Complex.abs w' ^ 2).sqrt β€ Complex.abs w' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large.cz.refine_2
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ (Complex.abs w' ^ 2).sqrt β€ Complex.abs w'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | refine le_trans ?_ (le_trans (Real.sqrt_le_sqrt jl.2.2.le) ?_) | case neg.large.large.z6
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ 6 β€ Complex.abs w' | case neg.large.large.z6.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ 6 β€ β36
case neg.large.large.z6.refine_2
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ (Complex.abs w' ^ 2).sqrt β€ Complex.abs w' | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large.z6
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ 6 β€ Complex.abs w'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | have e : (36 : β) = 6 ^ 2 := by norm_num | case neg.large.large.z6.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ 6 β€ β36 | case neg.large.large.z6.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
e : 36 = 6 ^ 2
β’ 6 β€ β36 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large.z6.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ 6 β€ β36
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [e, Real.sqrt_sq (by norm_num)] | case neg.large.large.z6.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
e : 36 = 6 ^ 2
β’ 6 β€ β36 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large.z6.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
e : 36 = 6 ^ 2
β’ 6 β€ β36
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | norm_num | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ 36 = 6 ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ 36 = 6 ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | norm_num | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
e : 36 = 6 ^ 2
β’ 0 β€ 6 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
e : 36 = 6 ^ 2
β’ 0 β€ 6
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [apply_nonneg, Real.sqrt_sq, le_refl] | case neg.large.large.z6.refine_2
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ (Complex.abs w' ^ 2).sqrt β€ Complex.abs w' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large.z6.refine_2
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ (Complex.abs w' ^ 2).sqrt β€ Complex.abs w'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [βhw', βhn, add_comm _ j.n, Function.iterate_add_apply, βhj] | case neg.large.large.zm
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ w' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z | case neg.large.large.zm
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ (f' 2 c')^[(iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n] ((f' 2 c')^[i.n] z') β
approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large.zm
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ w' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | exact mem_approx_iterate cm izm _ | case neg.large.large.zm
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ (f' 2 c')^[(iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n] ((f' 2 c')^[i.n] z') β
approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large.zm
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ (f' 2 c')^[(iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n] ((f' 2 c')^[i.n] z') β
approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [Interval.approx_nan, mem_univ] | case neg.large.nan
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.nan
β’ s.potential c' βz' β
approx
(match Exit.nan with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.nan
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.nan
β’ s.potential c' βz' β
approx
(match Exit.nan with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [Interval.approx_nan, mem_univ] | case neg.nan
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.nan
β’ s.potential c' βz' β
approx
(match Exit.nan with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.nan
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : Β¬cs = nan
hie : i.exit = Exit.nan
β’ s.potential c' βz' β
approx
(match Exit.nan with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential' | [214, 1] | [217, 83] | simp only [_root_.potential, RiemannSphere.fill_coe, mem_approx_potential cm cm] | c' : β
c : Box
cm : c' β approx c
n : β
r : Floating
β’ _root_.potential 2 βc' β approx (c.potential c n r).1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c' : β
c : Box
cm : c' β approx c
n : β
r : Floating
β’ _root_.potential 2 βc' β approx (c.potential c n r).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_multibrotExt | [30, 1] | [33, 40] | rw [β ray_surj d] | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ IsPathConnected (multibrotExt d) | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ IsPathConnected (ray d '' ball 0 1) | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ IsPathConnected (multibrotExt d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_multibrotExt | [30, 1] | [33, 40] | apply IsPathConnected.image_of_continuousOn | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ IsPathConnected (ray d '' ball 0 1) | case sc
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ IsPathConnected (ball 0 1)
case fc
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ ContinuousOn (ray d) (ball 0 1) | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ IsPathConnected (ray d '' ball 0 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_multibrotExt | [30, 1] | [33, 40] | exact (convex_ball _ _).isPathConnected (Metric.nonempty_ball.mpr one_pos) | case sc
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ IsPathConnected (ball 0 1)
case fc
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ ContinuousOn (ray d) (ball 0 1) | case fc
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ ContinuousOn (ray d) (ball 0 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case sc
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ IsPathConnected (ball 0 1)
case fc
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ ContinuousOn (ray d) (ball 0 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_multibrotExt | [30, 1] | [33, 40] | exact (rayHolomorphic d).continuousOn | case fc
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ ContinuousOn (ray d) (ball 0 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case fc
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ ContinuousOn (ray d) (ball 0 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | rw [e] | c : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
e : potential d β»ΒΉ' {p} = ray d '' sphere 0 p
β’ IsPathConnected (potential d β»ΒΉ' {p}) | c : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
e : potential d β»ΒΉ' {p} = ray d '' sphere 0 p
β’ IsPathConnected (ray d '' sphere 0 p) | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
e : potential d β»ΒΉ' {p} = ray d '' sphere 0 p
β’ IsPathConnected (potential d β»ΒΉ' {p})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | apply (isPathConnected_sphere p0).image_of_continuousOn | c : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
e : potential d β»ΒΉ' {p} = ray d '' sphere 0 p
β’ IsPathConnected (ray d '' sphere 0 p) | c : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
e : potential d β»ΒΉ' {p} = ray d '' sphere 0 p
β’ ContinuousOn (ray d) (sphere 0 p) | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
e : potential d β»ΒΉ' {p} = ray d '' sphere 0 p
β’ IsPathConnected (ray d '' sphere 0 p)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | exact (rayHolomorphic d).continuousOn.mono (Metric.sphere_subset_ball p1) | c : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
e : potential d β»ΒΉ' {p} = ray d '' sphere 0 p
β’ ContinuousOn (ray d) (sphere 0 p) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
e : potential d β»ΒΉ' {p} = ray d '' sphere 0 p
β’ ContinuousOn (ray d) (sphere 0 p)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | apply Set.ext | c : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
β’ potential d β»ΒΉ' {p} = ray d '' sphere 0 p | case h
c : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
β’ β (x : π), x β potential d β»ΒΉ' {p} β x β ray d '' sphere 0 p | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
β’ potential d β»ΒΉ' {p} = ray d '' sphere 0 p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | intro c | case h
c : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
β’ β (x : π), x β potential d β»ΒΉ' {p} β x β ray d '' sphere 0 p | case h
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
β’ c β potential d β»ΒΉ' {p} β c β ray d '' sphere 0 p | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
β’ β (x : π), x β potential d β»ΒΉ' {p} β x β ray d '' sphere 0 p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | simp only [mem_preimage, mem_singleton_iff, β abs_bottcher, mem_image, mem_sphere,
Complex.dist_eq, sub_zero] | case h
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
β’ c β potential d β»ΒΉ' {p} β c β ray d '' sphere 0 p | case h
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
β’ Complex.abs (bottcher d c) = p β β x, Complex.abs x = p β§ ray d x = c | Please generate a tactic in lean4 to solve the state.
STATE:
case h
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
β’ c β potential d β»ΒΉ' {p} β c β ray d '' sphere 0 p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | constructor | case h
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
β’ Complex.abs (bottcher d c) = p β β x, Complex.abs x = p β§ ray d x = c | case h.mp
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
β’ Complex.abs (bottcher d c) = p β β x, Complex.abs x = p β§ ray d x = c
case h.mpr
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
β’ (β x, Complex.abs x = p β§ ray d x = c) β Complex.abs (bottcher d c) = p | Please generate a tactic in lean4 to solve the state.
STATE:
case h
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
β’ Complex.abs (bottcher d c) = p β β x, Complex.abs x = p β§ ray d x = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | intro h | case h.mp
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
β’ Complex.abs (bottcher d c) = p β β x, Complex.abs x = p β§ ray d x = c | case h.mp
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
h : Complex.abs (bottcher d c) = p
β’ β x, Complex.abs x = p β§ ray d x = c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
β’ Complex.abs (bottcher d c) = p β β x, Complex.abs x = p β§ ray d x = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | use bottcher d c | case h.mp
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
h : Complex.abs (bottcher d c) = p
β’ β x, Complex.abs x = p β§ ray d x = c | case h
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
h : Complex.abs (bottcher d c) = p
β’ Complex.abs (bottcher d c) = p β§ ray d (bottcher d c) = c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
h : Complex.abs (bottcher d c) = p
β’ β x, Complex.abs x = p β§ ray d x = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | use h | case h
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
h : Complex.abs (bottcher d c) = p
β’ Complex.abs (bottcher d c) = p β§ ray d (bottcher d c) = c | case right
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
h : Complex.abs (bottcher d c) = p
β’ ray d (bottcher d c) = c | Please generate a tactic in lean4 to solve the state.
STATE:
case h
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
h : Complex.abs (bottcher d c) = p
β’ Complex.abs (bottcher d c) = p β§ ray d (bottcher d c) = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | rw [ray_bottcher] | case right
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
h : Complex.abs (bottcher d c) = p
β’ ray d (bottcher d c) = c | case right
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
h : Complex.abs (bottcher d c) = p
β’ c β multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
case right
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
h : Complex.abs (bottcher d c) = p
β’ ray d (bottcher d c) = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | rw [β potential_lt_one, β abs_bottcher, h] | case right
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
h : Complex.abs (bottcher d c) = p
β’ c β multibrotExt d | case right
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
h : Complex.abs (bottcher d c) = p
β’ p < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
h : Complex.abs (bottcher d c) = p
β’ c β multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | exact p1 | case right
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
h : Complex.abs (bottcher d c) = p
β’ p < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
h : Complex.abs (bottcher d c) = p
β’ p < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | intro β¨e, ep, ecβ© | case h.mpr
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
β’ (β x, Complex.abs x = p β§ ray d x = c) β Complex.abs (bottcher d c) = p | case h.mpr
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
e : β
ep : Complex.abs e = p
ec : ray d e = c
β’ Complex.abs (bottcher d c) = p | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
β’ (β x, Complex.abs x = p β§ ray d x = c) β Complex.abs (bottcher d c) = p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | rw [β ec, bottcher_ray] | case h.mpr
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
e : β
ep : Complex.abs e = p
ec : ray d e = c
β’ Complex.abs (bottcher d c) = p | case h.mpr
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
e : β
ep : Complex.abs e = p
ec : ray d e = c
β’ Complex.abs e = p
case h.mpr
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
e : β
ep : Complex.abs e = p
ec : ray d e = c
β’ e β ball 0 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
cβ : β
d : β
instβ : Fact (2 β€ d)
p : β
p0 : 0 β€ p
p1 : p < 1
c : π
e : β
ep : Complex.abs e = p
ec : ray d e = c
β’ Complex.abs (bottcher d c) = p
TACTIC:
|
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