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/Tactic/BoundTests.lean | test_sq | [54, 1] | [54, 62] | bound | a b c x y : β
z : β
nβ : β
n : x β₯ 0
h : x β€ y
β’ x ^ 2 β€ y ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : β
z : β
nβ : β
n : x β₯ 0
h : x β€ y
β’ x ^ 2 β€ y ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_sq_ge | [55, 1] | [55, 65] | bound | a b c x y : β
z : β
nβ : β
n : x β₯ 0
h : x β€ y
β’ y ^ 2 β₯ x ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : β
z : β
nβ : β
n : x β₯ 0
h : x β€ y
β’ y ^ 2 β₯ x ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_mul_left | [56, 1] | [56, 72] | bound | a b c x y : β
z : β
nβ : β
n : a β₯ 0
h : x β€ y
β’ a * x β€ a * y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : β
z : β
nβ : β
n : a β₯ 0
h : x β€ y
β’ a * x β€ a * y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_mul_right | [57, 1] | [57, 73] | bound | a b c x y : β
z : β
nβ : β
n : a β₯ 0
h : x β€ y
β’ x * a β€ y * a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : β
z : β
nβ : β
n : a β₯ 0
h : x β€ y
β’ x * a β€ y * a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_mul_both | [58, 1] | [58, 100] | bound | a b c x y : β
z : β
n : β
bp : b β₯ 0
xp : x β₯ 0
ab : a β€ b
xy : x β€ y
β’ a * x β€ b * y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : β
z : β
n : β
bp : b β₯ 0
xp : x β₯ 0
ab : a β€ b
xy : x β€ y
β’ a * x β€ b * y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_abs_mul | [59, 1] | [59, 67] | bound | a b c x y : β
z : β
n : β
h : x β€ y
β’ Complex.abs z * x β€ Complex.abs z * y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : β
z : β
n : β
h : x β€ y
β’ Complex.abs z * x β€ Complex.abs z * y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_add_left | [60, 1] | [60, 60] | bound | a b c x y : β
z : β
n : β
h : x β€ y
β’ a + x β€ a + y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : β
z : β
n : β
h : x β€ y
β’ a + x β€ a + y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_add_right | [61, 1] | [61, 61] | bound | a b c x y : β
z : β
n : β
h : x β€ y
β’ x + a β€ y + a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : β
z : β
n : β
h : x β€ y
β’ x + a β€ y + a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_add_both | [62, 1] | [62, 74] | bound | a b c x y : β
z : β
n : β
ab : a β€ b
xy : x β€ y
β’ a + x β€ b + y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : β
z : β
n : β
ab : a β€ b
xy : x β€ y
β’ a + x β€ b + y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_sub_left | [63, 1] | [63, 60] | bound | a b c x y : β
z : β
n : β
h : x β₯ y
β’ a - x β€ a - y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : β
z : β
n : β
h : x β₯ y
β’ a - x β€ a - y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_sub_right | [64, 1] | [64, 61] | bound | a b c x y : β
z : β
n : β
h : x β€ y
β’ x - a β€ y - a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : β
z : β
n : β
h : x β€ y
β’ x - a β€ y - a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_sub_both | [65, 1] | [65, 74] | bound | a b c x y : β
z : β
n : β
ab : a β€ b
xy : x β₯ y
β’ a - x β€ b - y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : β
z : β
n : β
ab : a β€ b
xy : x β₯ y
β’ a - x β€ b - y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_sub_pos | [66, 1] | [66, 55] | bound | a b c x y : β
z : β
n : β
h : x < y
β’ y - x > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : β
z : β
n : β
h : x < y
β’ y - x > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_le_of_lt | [67, 1] | [67, 52] | bound | a b c x y : β
z : β
n : β
h : x > 0
β’ x β₯ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : β
z : β
n : β
h : x > 0
β’ x β₯ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_extra | [68, 1] | [68, 76] | bound [h n] | a b c x y : β
z : β
n : β
f : β β β
h : β (n : β), f n β₯ 0
β’ f n β₯ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : β
z : β
n : β
f : β β β
h : β (n : β), f n β₯ 0
β’ f n β₯ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_1_4 | [69, 1] | [69, 41] | bound | a b c x y : β
z : β
n : β
β’ 1 < 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : β
z : β
n : β
β’ 1 < 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_2_4 | [70, 1] | [70, 41] | bound | a b c x y : β
z : β
n : β
β’ 2 < 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : β
z : β
n : β
β’ 2 < 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_div_left | [71, 1] | [71, 73] | bound | a b c x y : β
z : β
n : β
hc : c β₯ 0
h : a β€ b
β’ a / c β€ b / c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : β
z : β
n : β
hc : c β₯ 0
h : a β€ b
β’ a / c β€ b / c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_div_right | [72, 1] | [72, 87] | bound | a b c x y : β
z : β
n : β
ha : a β₯ 0
hc : c > 0
h : b β₯ c
β’ a / b β€ a / c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : β
z : β
n : β
ha : a β₯ 0
hc : c > 0
h : b β₯ c
β’ a / b β€ a / c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_coe | [73, 1] | [73, 65] | bound | a b c xβ yβ : β
z : β
n : β
x y : ββ₯0
h : x < y
β’ βx < βy | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c xβ yβ : β
z : β
n : β
x y : ββ₯0
h : x < y
β’ βx < βy
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_dist | [74, 1] | [74, 61] | bound | a b c x y : β
z : β
n : β
β’ dist a c β€ dist a b + dist b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : β
z : β
n : β
β’ dist a c β€ dist a b + dist b c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_log | [75, 1] | [75, 78] | bound | a b c xβ yβ : β
z : β
n : β
x y : β
x0 : 0 < x
h : x β€ y
β’ x.log β€ y.log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c xβ yβ : β
z : β
n : β
x y : β
x0 : 0 < x
h : x β€ y
β’ x.log β€ y.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_le_max_of_le_left | [81, 1] | [81, 67] | bound | a b c : β
n m : β
h : a β€ b
β’ a β€ max b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : β
n m : β
h : a β€ b
β’ a β€ max b c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_le_max_of_le_right | [82, 1] | [82, 68] | bound | a b c : β
n m : β
h : a β€ c
β’ a β€ max b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : β
n m : β
h : a β€ c
β’ a β€ max b c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_lt_max_of_lt_left | [83, 1] | [83, 67] | bound | a b c : β
n m : β
h : a < b
β’ a < max b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : β
n m : β
h : a < b
β’ a < max b c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_lt_max_of_lt_right | [84, 1] | [84, 68] | bound | a b c : β
n m : β
h : a < c
β’ a < max b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : β
n m : β
h : a < c
β’ a < max b c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_min_le_of_left_le | [85, 1] | [85, 67] | bound | a b c : β
n m : β
h : a β€ c
β’ min a b β€ c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : β
n m : β
h : a β€ c
β’ min a b β€ c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_min_le_of_right_le | [86, 1] | [86, 68] | bound | a b c : β
n m : β
h : b β€ c
β’ min a b β€ c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : β
n m : β
h : b β€ c
β’ min a b β€ c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_min_lt_of_left_lt | [87, 1] | [87, 67] | bound | a b c : β
n m : β
h : a < c
β’ min a b < c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : β
n m : β
h : a < c
β’ min a b < c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_min_lt_of_right_lt | [88, 1] | [88, 68] | bound | a b c : β
n m : β
h : b < c
β’ min a b < c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : β
n m : β
h : b < c
β’ min a b < c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_pow_le_pow_right | [89, 1] | [89, 77] | bound | a b c : β
n m : β
a1 : 1 β€ a
h : m β€ n
β’ a ^ m β€ a ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : β
n m : β
a1 : 1 β€ a
h : m β€ n
β’ a ^ m β€ a ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_pow_le_pow_of_le_one | [90, 1] | [90, 94] | bound | a b c : β
n m : β
a0 : 0 β€ a
a1 : a β€ 1
h : n β€ m
β’ a ^ m β€ a ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : β
n m : β
a0 : 0 β€ a
a1 : a β€ 1
h : n β€ m
β’ a ^ m β€ a ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_rpow_le_rpow_of_exponent_le | [91, 1] | [91, 88] | bound | a b c : β
n m : β
a1 : 1 β€ a
h : b β€ c
β’ a ^ b β€ a ^ c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : β
n m : β
a1 : 1 β€ a
h : b β€ c
β’ a ^ b β€ a ^ c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_rpow_le_rpow_of_exponent_ge | [92, 1] | [92, 101] | bound | a b c : β
n m : β
a0 : 0 < a
a1 : a β€ 1
h : c β€ b
β’ a ^ b β€ a ^ c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : β
n m : β
a0 : 0 < a
a1 : a β€ 1
h : c β€ b
β’ a ^ b β€ a ^ c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_with_context | [96, 1] | [101, 8] | rw [Metric.isOpen_iff] at o | s : Set β
o : IsOpen s
z : β
h : z β s
β’ β r, r > 0 | s : Set β
o : β x β s, β Ξ΅ > 0, Metric.ball x Ξ΅ β s
z : β
h : z β s
β’ β r, r > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
s : Set β
o : IsOpen s
z : β
h : z β s
β’ β r, r > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_with_context | [96, 1] | [101, 8] | rcases o z h with β¨t, tp, bsβ© | s : Set β
o : β x β s, β Ξ΅ > 0, Metric.ball x Ξ΅ β s
z : β
h : z β s
β’ β r, r > 0 | case intro.intro
s : Set β
o : β x β s, β Ξ΅ > 0, Metric.ball x Ξ΅ β s
z : β
h : z β s
t : β
tp : t > 0
bs : Metric.ball z t β s
β’ β r, r > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
s : Set β
o : β x β s, β Ξ΅ > 0, Metric.ball x Ξ΅ β s
z : β
h : z β s
β’ β r, r > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_with_context | [96, 1] | [101, 8] | exists t/2 | case intro.intro
s : Set β
o : β x β s, β Ξ΅ > 0, Metric.ball x Ξ΅ β s
z : β
h : z β s
t : β
tp : t > 0
bs : Metric.ball z t β s
β’ β r, r > 0 | case intro.intro
s : Set β
o : β x β s, β Ξ΅ > 0, Metric.ball x Ξ΅ β s
z : β
h : z β s
t : β
tp : t > 0
bs : Metric.ball z t β s
β’ t / 2 > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s : Set β
o : β x β s, β Ξ΅ > 0, Metric.ball x Ξ΅ β s
z : β
h : z β s
t : β
tp : t > 0
bs : Metric.ball z t β s
β’ β r, r > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_with_context | [96, 1] | [101, 8] | clear o h bs z s | case intro.intro
s : Set β
o : β x β s, β Ξ΅ > 0, Metric.ball x Ξ΅ β s
z : β
h : z β s
t : β
tp : t > 0
bs : Metric.ball z t β s
β’ t / 2 > 0 | case intro.intro
t : β
tp : t > 0
β’ t / 2 > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s : Set β
o : β x β s, β Ξ΅ > 0, Metric.ball x Ξ΅ β s
z : β
h : z β s
t : β
tp : t > 0
bs : Metric.ball z t β s
β’ t / 2 > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_with_context | [96, 1] | [101, 8] | bound | case intro.intro
t : β
tp : t > 0
β’ t / 2 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
t : β
tp : t > 0
β’ t / 2 > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_try_elab | [104, 1] | [110, 50] | calc abs (f z) = abs (f w - (f w - f z)) := by ring_nf
_ β€ abs (f w) + abs (f w - f z) := by bound
_ β€ c * abs w + e := by bound [h w wr, sc wz] | f : β β β
z w : β
s r c e : β
sc : β {w : β}, Complex.abs (w - z) < s β Complex.abs (f w - f z) < e
wz : Complex.abs (w - z) < s
wr : Complex.abs w < r
h : β (z : β), Complex.abs z < r β Complex.abs (f z) β€ c * Complex.abs z
β’ Complex.abs (f z) β€ c * Complex.abs w + e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
z w : β
s r c e : β
sc : β {w : β}, Complex.abs (w - z) < s β Complex.abs (f w - f z) < e
wz : Complex.abs (w - z) < s
wr : Complex.abs w < r
h : β (z : β), Complex.abs z < r β Complex.abs (f z) β€ c * Complex.abs z
β’ Complex.abs (f z) β€ c * Complex.abs w + e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_try_elab | [104, 1] | [110, 50] | ring_nf | f : β β β
z w : β
s r c e : β
sc : β {w : β}, Complex.abs (w - z) < s β Complex.abs (f w - f z) < e
wz : Complex.abs (w - z) < s
wr : Complex.abs w < r
h : β (z : β), Complex.abs z < r β Complex.abs (f z) β€ c * Complex.abs z
β’ Complex.abs (f z) = Complex.abs (f w - (f w - f z)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
z w : β
s r c e : β
sc : β {w : β}, Complex.abs (w - z) < s β Complex.abs (f w - f z) < e
wz : Complex.abs (w - z) < s
wr : Complex.abs w < r
h : β (z : β), Complex.abs z < r β Complex.abs (f z) β€ c * Complex.abs z
β’ Complex.abs (f z) = Complex.abs (f w - (f w - f z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_try_elab | [104, 1] | [110, 50] | bound | f : β β β
z w : β
s r c e : β
sc : β {w : β}, Complex.abs (w - z) < s β Complex.abs (f w - f z) < e
wz : Complex.abs (w - z) < s
wr : Complex.abs w < r
h : β (z : β), Complex.abs z < r β Complex.abs (f z) β€ c * Complex.abs z
β’ Complex.abs (f w - (f w - f z)) β€ Complex.abs (f w) + Complex.abs (f w - f z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
z w : β
s r c e : β
sc : β {w : β}, Complex.abs (w - z) < s β Complex.abs (f w - f z) < e
wz : Complex.abs (w - z) < s
wr : Complex.abs w < r
h : β (z : β), Complex.abs z < r β Complex.abs (f z) β€ c * Complex.abs z
β’ Complex.abs (f w - (f w - f z)) β€ Complex.abs (f w) + Complex.abs (f w - f z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_try_elab | [104, 1] | [110, 50] | bound [h w wr, sc wz] | f : β β β
z w : β
s r c e : β
sc : β {w : β}, Complex.abs (w - z) < s β Complex.abs (f w - f z) < e
wz : Complex.abs (w - z) < s
wr : Complex.abs w < r
h : β (z : β), Complex.abs z < r β Complex.abs (f z) β€ c * Complex.abs z
β’ Complex.abs (f w) + Complex.abs (f w - f z) β€ c * Complex.abs w + e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
z w : β
s r c e : β
sc : β {w : β}, Complex.abs (w - z) < s β Complex.abs (f w - f z) < e
wz : Complex.abs (w - z) < s
wr : Complex.abs w < r
h : β (z : β), Complex.abs z < r β Complex.abs (f z) β€ c * Complex.abs z
β’ Complex.abs (f w) + Complex.abs (f w - f z) β€ c * Complex.abs w + e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_fun_inference | [113, 1] | [115, 8] | bound | Ξ± : Type
s : Finset Ξ±
f g : Ξ± β β
β’ βs.sum fun x => f x + g xβ β€ s.sum fun x => βf x + g xβ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
s : Finset Ξ±
f g : Ξ± β β
β’ βs.sum fun x => f x + g xβ β€ s.sum fun x => βf x + g xβ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_whnf | [118, 1] | [119, 14] | bound [h.1] | x y : β
h : x < y β§ True
β’ x β€ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : β
h : x < y β§ True
β’ x β€ y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_unknown_identifier | [122, 1] | [123, 24] | intro n | f : β β β
le : β (n : β), f n β€ βn
β’ β (n : β), f n β€ βn | f : β β β
le : β (n : β), f n β€ βn
n : β
β’ f n β€ βn | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
le : β (n : β), f n β€ βn
β’ β (n : β), f n β€ βn
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_unknown_identifier | [122, 1] | [123, 24] | bound [le n] | f : β β β
le : β (n : β), f n β€ βn
n : β
β’ f n β€ βn | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
le : β (n : β), f n β€ βn
n : β
β’ f n β€ βn
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | le_sqr_add | [126, 1] | [132, 30] | calc Complex.abs (z^2 + c)
_ β₯ Complex.abs (z^2) - abs c := by bound
_ β₯ Complex.abs (z^2) - abs z := by bound
_ β₯ (abs z - 1) * abs z := by rw [mul_comm, mul_sub_one, βpow_two, βComplex.abs.map_pow]
_ β₯ 2 * abs z := by bound | c z : β
cz : Complex.abs c β€ Complex.abs z
z3 : 3 β€ Complex.abs z
β’ 2 * Complex.abs z β€ Complex.abs (z ^ 2 + c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : β
cz : Complex.abs c β€ Complex.abs z
z3 : 3 β€ Complex.abs z
β’ 2 * Complex.abs z β€ Complex.abs (z ^ 2 + c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | le_sqr_add | [126, 1] | [132, 30] | bound | c z : β
cz : Complex.abs c β€ Complex.abs z
z3 : 3 β€ Complex.abs z
β’ Complex.abs (z ^ 2 + c) β₯ Complex.abs (z ^ 2) - Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : β
cz : Complex.abs c β€ Complex.abs z
z3 : 3 β€ Complex.abs z
β’ Complex.abs (z ^ 2 + c) β₯ Complex.abs (z ^ 2) - Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | le_sqr_add | [126, 1] | [132, 30] | bound | c z : β
cz : Complex.abs c β€ Complex.abs z
z3 : 3 β€ Complex.abs z
β’ Complex.abs (z ^ 2) - Complex.abs c β₯ Complex.abs (z ^ 2) - Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : β
cz : Complex.abs c β€ Complex.abs z
z3 : 3 β€ Complex.abs z
β’ Complex.abs (z ^ 2) - Complex.abs c β₯ Complex.abs (z ^ 2) - Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | le_sqr_add | [126, 1] | [132, 30] | rw [mul_comm, mul_sub_one, βpow_two, βComplex.abs.map_pow] | c z : β
cz : Complex.abs c β€ Complex.abs z
z3 : 3 β€ Complex.abs z
β’ Complex.abs (z ^ 2) - Complex.abs z β₯ (Complex.abs z - 1) * Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : β
cz : Complex.abs c β€ Complex.abs z
z3 : 3 β€ Complex.abs z
β’ Complex.abs (z ^ 2) - Complex.abs z β₯ (Complex.abs z - 1) * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | le_sqr_add | [126, 1] | [132, 30] | bound | c z : β
cz : Complex.abs c β€ Complex.abs z
z3 : 3 β€ Complex.abs z
β’ (Complex.abs z - 1) * Complex.abs z β₯ 2 * Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : β
cz : Complex.abs c β€ Complex.abs z
z3 : 3 β€ Complex.abs z
β’ (Complex.abs z - 1) * Complex.abs z β₯ 2 * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | induction' n with n h generalizing k z z' | c z : Box
rs : Floating
c' z' : β
rs' : β
cm : c' β approx c
zm : z' β approx z
rsn : rs β nan
rsm : rs' β€ rs.val
k n : β
β’ let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2) | case zero
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
z : Box
z' : β
zm : z' β approx z
k : β
β’ let i := iterate' c z rs k 0;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
β’ let i := iterate' c z rs k (n + 1);
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : Box
rs : Floating
c' z' : β
rs' : β
cm : c' β approx c
zm : z' β approx z
rsn : rs β nan
rsm : rs' β€ rs.val
k n : β
β’ let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | simpa only [iterate', le_refl, ge_iff_le, tsub_eq_zero_of_le, Function.iterate_zero, id_eq,
IsEmpty.forall_iff, and_true, true_and] | case zero
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
z : Box
z' : β
zm : z' β approx z
k : β
β’ let i := iterate' c z rs k 0;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
z : Box
z' : β
zm : z' β approx z
k : β
β’ let i := iterate' c z rs k 0;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | simp only [iterate', Floating.val_lt_val] | case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
β’ let i := iterate' c z rs k (n + 1);
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2) | case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
β’ k β€
(if z.re.sqr.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (z.re.sqr.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := z.re.sqr - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n β§
(f' 2
c')^[(if z.re.sqr.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (z.re.sqr.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := z.re.sqr - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z' β
approx
(if z.re.sqr.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (z.re.sqr.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := z.re.sqr - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).z β§
((if z.re.sqr.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (z.re.sqr.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := z.re.sqr - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z.re.sqr.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (z.re.sqr.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else
iterate' c ({ re := z.re.sqr - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z') ^
2) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
β’ let i := iterate' c z rs k (n + 1);
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | generalize hzr2 : z.re.sqr = zr2 | case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
β’ k β€
(if z.re.sqr.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (z.re.sqr.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := z.re.sqr - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n β§
(f' 2
c')^[(if z.re.sqr.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (z.re.sqr.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := z.re.sqr - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z' β
approx
(if z.re.sqr.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (z.re.sqr.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := z.re.sqr - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).z β§
((if z.re.sqr.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (z.re.sqr.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := z.re.sqr - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z.re.sqr.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (z.re.sqr.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else
iterate' c ({ re := z.re.sqr - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z') ^
2) | case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
β’ k β€
(if zr2.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n β§
(f' 2
c')^[(if zr2.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z' β
approx
(if zr2.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).z β§
((if zr2.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if zr2.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z') ^
2) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
β’ k β€
(if z.re.sqr.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (z.re.sqr.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := z.re.sqr - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n β§
(f' 2
c')^[(if z.re.sqr.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (z.re.sqr.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := z.re.sqr - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z' β
approx
(if z.re.sqr.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (z.re.sqr.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := z.re.sqr - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).z β§
((if z.re.sqr.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (z.re.sqr.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := z.re.sqr - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z.re.sqr.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (z.re.sqr.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else
iterate' c ({ re := z.re.sqr - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z') ^
2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | generalize hzi2 : z.im.sqr = zi2 | case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
β’ k β€
(if zr2.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n β§
(f' 2
c')^[(if zr2.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z' β
approx
(if zr2.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).z β§
((if zr2.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if zr2.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z') ^
2) | case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
β’ k β€
(if zr2.lo.add zi2.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add zi2.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n β§
(f' 2
c')^[(if zr2.lo.add zi2.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add zi2.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z' β
approx
(if zr2.lo.add zi2.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add zi2.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).z β§
((if zr2.lo.add zi2.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add zi2.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if zr2.lo.add zi2.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add zi2.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z') ^
2) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
β’ k β€
(if zr2.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n β§
(f' 2
c')^[(if zr2.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z' β
approx
(if zr2.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).z β§
((if zr2.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if zr2.lo.add z.im.sqr.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add z.im.sqr.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - z.im.sqr, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z') ^
2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | generalize hz2 : zr2.lo.add zi2.lo false = z2 | case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
β’ k β€
(if zr2.lo.add zi2.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add zi2.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n β§
(f' 2
c')^[(if zr2.lo.add zi2.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add zi2.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z' β
approx
(if zr2.lo.add zi2.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add zi2.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).z β§
((if zr2.lo.add zi2.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add zi2.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if zr2.lo.add zi2.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add zi2.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z') ^
2) | case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z') ^
2) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
β’ k β€
(if zr2.lo.add zi2.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add zi2.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n β§
(f' 2
c')^[(if zr2.lo.add zi2.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add zi2.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z' β
approx
(if zr2.lo.add zi2.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add zi2.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).z β§
((if zr2.lo.add zi2.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add zi2.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if zr2.lo.add zi2.lo false = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < (zr2.lo.add zi2.lo false).val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z') ^
2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | generalize hw : (β¨zr2 - zi2, (z.re * z.im).scaleB 1β© : Box) + c = w | case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z') ^
2) | case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c ({ re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c) rs (k + 1) n).n -
k]
z') ^
2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | have we : w = z.sqr + c := by rw [βhw, Box.sqr, hzr2, hzi2] | case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2) | case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | have wa : f' 2 c' z' β approx w := by rw [we, f']; mono | case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2) | case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
wa : f' 2 c' z' β approx w
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | generalize hw' : f' 2 c' z' = w' at wa | case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
wa : f' 2 c' z' β approx w
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2) | case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
wa : f' 2 c' z' β approx w
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | by_cases z2n : z2 = nan | case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2) | case pos
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : z2 = nan
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2)
case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | by_cases rz : rs.val < z2.val | case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2) | case pos
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : rs.val < z2.val
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2)
case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | rw [βhw, Box.sqr, hzr2, hzi2] | c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
β’ w = z.sqr + c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
β’ w = z.sqr + c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | rw [we, f'] | c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
β’ f' 2 c' z' β approx w | c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
β’ z' ^ 2 + c' β approx (z.sqr + c) | Please generate a tactic in lean4 to solve the state.
STATE:
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
β’ f' 2 c' z' β approx w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | mono | c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
β’ z' ^ 2 + c' β approx (z.sqr + c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
β’ z' ^ 2 + c' β approx (z.sqr + c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | simpa only [z2n, ite_true, le_refl, ge_iff_le, tsub_eq_zero_of_le, Function.iterate_zero,
id_eq, IsEmpty.forall_iff, and_true, true_and] | case pos
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : z2 = nan
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : z2 = nan
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | simp only [z2n, rz, ite_true, le_refl, ge_iff_le, tsub_eq_zero_of_le, Function.iterate_zero,
id_eq, zm, forall_true_left, true_and, Complex.abs_def,
Real.sq_sqrt (Complex.normSq_nonneg _), if_false] | case pos
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : rs.val < z2.val
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2) | case pos
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : rs.val < z2.val
β’ rs' < Complex.normSq z' | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : rs.val < z2.val
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | refine lt_of_le_of_lt rsm (lt_of_lt_of_le rz ?_) | case pos
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : rs.val < z2.val
β’ rs' < Complex.normSq z' | case pos
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : rs.val < z2.val
β’ z2.val β€ Complex.normSq z' | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : rs.val < z2.val
β’ rs' < Complex.normSq z'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | simp only [Complex.normSq_apply, βsq, βhz2, βhzr2, βhzi2] at z2n β’ | case pos
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : rs.val < z2.val
β’ z2.val β€ Complex.normSq z' | case pos
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
rz : rs.val < z2.val
z2n : Β¬z.re.sqr.lo.add z.im.sqr.lo false = nan
β’ (z.re.sqr.lo.add z.im.sqr.lo false).val β€ z'.re ^ 2 + z'.im ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : rs.val < z2.val
β’ z2.val β€ Complex.normSq z'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | rcases Floating.ne_nan_of_add z2n with β¨nr, niβ© | case pos
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
rz : rs.val < z2.val
z2n : Β¬z.re.sqr.lo.add z.im.sqr.lo false = nan
β’ (z.re.sqr.lo.add z.im.sqr.lo false).val β€ z'.re ^ 2 + z'.im ^ 2 | case pos.intro
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
rz : rs.val < z2.val
z2n : Β¬z.re.sqr.lo.add z.im.sqr.lo false = nan
nr : z.re.sqr.lo β nan
ni : z.im.sqr.lo β nan
β’ (z.re.sqr.lo.add z.im.sqr.lo false).val β€ z'.re ^ 2 + z'.im ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
rz : rs.val < z2.val
z2n : Β¬z.re.sqr.lo.add z.im.sqr.lo false = nan
β’ (z.re.sqr.lo.add z.im.sqr.lo false).val β€ z'.re ^ 2 + z'.im ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | simp only [ne_eq, Interval.lo_eq_nan] at nr ni | case pos.intro
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
rz : rs.val < z2.val
z2n : Β¬z.re.sqr.lo.add z.im.sqr.lo false = nan
nr : z.re.sqr.lo β nan
ni : z.im.sqr.lo β nan
β’ (z.re.sqr.lo.add z.im.sqr.lo false).val β€ z'.re ^ 2 + z'.im ^ 2 | case pos.intro
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
rz : rs.val < z2.val
z2n : Β¬z.re.sqr.lo.add z.im.sqr.lo false = nan
nr : Β¬z.re.sqr = nan
ni : Β¬z.im.sqr = nan
β’ (z.re.sqr.lo.add z.im.sqr.lo false).val β€ z'.re ^ 2 + z'.im ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
rz : rs.val < z2.val
z2n : Β¬z.re.sqr.lo.add z.im.sqr.lo false = nan
nr : z.re.sqr.lo β nan
ni : z.im.sqr.lo β nan
β’ (z.re.sqr.lo.add z.im.sqr.lo false).val β€ z'.re ^ 2 + z'.im ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | refine le_trans (Floating.add_le z2n) (add_le_add ?_ ?_) | case pos.intro
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
rz : rs.val < z2.val
z2n : Β¬z.re.sqr.lo.add z.im.sqr.lo false = nan
nr : Β¬z.re.sqr = nan
ni : Β¬z.im.sqr = nan
β’ (z.re.sqr.lo.add z.im.sqr.lo false).val β€ z'.re ^ 2 + z'.im ^ 2 | case pos.intro.refine_1
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
rz : rs.val < z2.val
z2n : Β¬z.re.sqr.lo.add z.im.sqr.lo false = nan
nr : Β¬z.re.sqr = nan
ni : Β¬z.im.sqr = nan
β’ z.re.sqr.lo.val β€ z'.re ^ 2
case pos.intro.refine_2
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
rz : rs.val < z2.val
z2n : Β¬z.re.sqr.lo.add z.im.sqr.lo false = nan
nr : Β¬z.re.sqr = nan
ni : Β¬z.im.sqr = nan
β’ z.im.sqr.lo.val β€ z'.im ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
rz : rs.val < z2.val
z2n : Β¬z.re.sqr.lo.add z.im.sqr.lo false = nan
nr : Β¬z.re.sqr = nan
ni : Β¬z.im.sqr = nan
β’ (z.re.sqr.lo.add z.im.sqr.lo false).val β€ z'.re ^ 2 + z'.im ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | apply Interval.lo_le nr | case pos.intro.refine_1
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
rz : rs.val < z2.val
z2n : Β¬z.re.sqr.lo.add z.im.sqr.lo false = nan
nr : Β¬z.re.sqr = nan
ni : Β¬z.im.sqr = nan
β’ z.re.sqr.lo.val β€ z'.re ^ 2 | case pos.intro.refine_1
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
rz : rs.val < z2.val
z2n : Β¬z.re.sqr.lo.add z.im.sqr.lo false = nan
nr : Β¬z.re.sqr = nan
ni : Β¬z.im.sqr = nan
β’ z'.re ^ 2 β approx z.re.sqr | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.refine_1
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
rz : rs.val < z2.val
z2n : Β¬z.re.sqr.lo.add z.im.sqr.lo false = nan
nr : Β¬z.re.sqr = nan
ni : Β¬z.im.sqr = nan
β’ z.re.sqr.lo.val β€ z'.re ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | mono | case pos.intro.refine_1
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
rz : rs.val < z2.val
z2n : Β¬z.re.sqr.lo.add z.im.sqr.lo false = nan
nr : Β¬z.re.sqr = nan
ni : Β¬z.im.sqr = nan
β’ z'.re ^ 2 β approx z.re.sqr | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.refine_1
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
rz : rs.val < z2.val
z2n : Β¬z.re.sqr.lo.add z.im.sqr.lo false = nan
nr : Β¬z.re.sqr = nan
ni : Β¬z.im.sqr = nan
β’ z'.re ^ 2 β approx z.re.sqr
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | apply Interval.lo_le ni | case pos.intro.refine_2
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
rz : rs.val < z2.val
z2n : Β¬z.re.sqr.lo.add z.im.sqr.lo false = nan
nr : Β¬z.re.sqr = nan
ni : Β¬z.im.sqr = nan
β’ z.im.sqr.lo.val β€ z'.im ^ 2 | case pos.intro.refine_2
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
rz : rs.val < z2.val
z2n : Β¬z.re.sqr.lo.add z.im.sqr.lo false = nan
nr : Β¬z.re.sqr = nan
ni : Β¬z.im.sqr = nan
β’ z'.im ^ 2 β approx z.im.sqr | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.refine_2
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
rz : rs.val < z2.val
z2n : Β¬z.re.sqr.lo.add z.im.sqr.lo false = nan
nr : Β¬z.re.sqr = nan
ni : Β¬z.im.sqr = nan
β’ z.im.sqr.lo.val β€ z'.im ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | mono | case pos.intro.refine_2
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
rz : rs.val < z2.val
z2n : Β¬z.re.sqr.lo.add z.im.sqr.lo false = nan
nr : Β¬z.re.sqr = nan
ni : Β¬z.im.sqr = nan
β’ z'.im ^ 2 β approx z.im.sqr | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.refine_2
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
rz : rs.val < z2.val
z2n : Β¬z.re.sqr.lo.add z.im.sqr.lo false = nan
nr : Β¬z.re.sqr = nan
ni : Β¬z.im.sqr = nan
β’ z'.im ^ 2 β approx z.im.sqr
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | simp only [rz, ite_false, z2n] | case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2) | case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
β’ k β€ (iterate' c w rs (k + 1) n).n β§
(f' 2 c')^[(iterate' c w rs (k + 1) n).n - k] z' β approx (iterate' c w rs (k + 1) n).z β§
((iterate' c w rs (k + 1) n).exit = Exit.large β
rs' < Complex.abs ((f' 2 c')^[(iterate' c w rs (k + 1) n).n - k] z') ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
β’ k β€
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n β§
(f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).n -
k]
z' β
approx
(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).z β§
((if z2 = nan then { z := z, n := k, exit := Exit.nan }
else if rs.val < z2.val then { z := z, n := k, exit := Exit.large } else iterate' c w rs (k + 1) n).exit =
Exit.large β
rs' <
Complex.abs
((f' 2
c')^[(if z2 = nan then { z := z, n := k, exit := Exit.nan }
else
if rs.val < z2.val then { z := z, n := k, exit := Exit.large }
else iterate' c w rs (k + 1) n).n -
k]
z') ^
2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | generalize hi : iterate' c w rs (k + 1) n = i | case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
β’ k β€ (iterate' c w rs (k + 1) n).n β§
(f' 2 c')^[(iterate' c w rs (k + 1) n).n - k] z' β approx (iterate' c w rs (k + 1) n).z β§
((iterate' c w rs (k + 1) n).exit = Exit.large β
rs' < Complex.abs ((f' 2 c')^[(iterate' c w rs (k + 1) n).n - k] z') ^ 2) | case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
β’ k β€ i.n β§ (f' 2 c')^[i.n - k] z' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - k] z') ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
β’ k β€ (iterate' c w rs (k + 1) n).n β§
(f' 2 c')^[(iterate' c w rs (k + 1) n).n - k] z' β approx (iterate' c w rs (k + 1) n).z β§
((iterate' c w rs (k + 1) n).exit = Exit.large β
rs' < Complex.abs ((f' 2 c')^[(iterate' c w rs (k + 1) n).n - k] z') ^ 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | specialize h wa (k+1) | case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
β’ k β€ i.n β§ (f' 2 c')^[i.n - k] z' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - k] z') ^ 2) | case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
h :
let i := iterate' c w rs (k + 1) n;
let w' := (f' 2 c')^[i.n - (k + 1)] w';
k + 1 β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
β’ k β€ i.n β§ (f' 2 c')^[i.n - k] z' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - k] z') ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
h :
β {z : Box} {z' : β},
z' β approx z β
β (k : β),
let i := iterate' c z rs k n;
let w' := (f' 2 c')^[i.n - k] z';
k β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
β’ k β€ i.n β§ (f' 2 c')^[i.n - k] z' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - k] z') ^ 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | simp only [hi] at h | case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
h :
let i := iterate' c w rs (k + 1) n;
let w' := (f' 2 c')^[i.n - (k + 1)] w';
k + 1 β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
β’ k β€ i.n β§ (f' 2 c')^[i.n - k] z' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - k] z') ^ 2) | case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
h :
k + 1 β€ i.n β§
(f' 2 c')^[i.n - (k + 1)] w' β approx i.z β§
(i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - (k + 1)] w') ^ 2)
β’ k β€ i.n β§ (f' 2 c')^[i.n - k] z' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - k] z') ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
h :
let i := iterate' c w rs (k + 1) n;
let w' := (f' 2 c')^[i.n - (k + 1)] w';
k + 1 β€ i.n β§ w' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs w' ^ 2)
β’ k β€ i.n β§ (f' 2 c')^[i.n - k] z' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - k] z') ^ 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | refine β¨by omega, ?_β© | case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
h :
k + 1 β€ i.n β§
(f' 2 c')^[i.n - (k + 1)] w' β approx i.z β§
(i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - (k + 1)] w') ^ 2)
β’ k β€ i.n β§ (f' 2 c')^[i.n - k] z' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - k] z') ^ 2) | case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
h :
k + 1 β€ i.n β§
(f' 2 c')^[i.n - (k + 1)] w' β approx i.z β§
(i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - (k + 1)] w') ^ 2)
β’ (f' 2 c')^[i.n - k] z' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - k] z') ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
h :
k + 1 β€ i.n β§
(f' 2 c')^[i.n - (k + 1)] w' β approx i.z β§
(i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - (k + 1)] w') ^ 2)
β’ k β€ i.n β§ (f' 2 c')^[i.n - k] z' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - k] z') ^ 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | have ie : i.n - k = (i.n - (k + 1)) + 1 := by omega | case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
h :
k + 1 β€ i.n β§
(f' 2 c')^[i.n - (k + 1)] w' β approx i.z β§
(i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - (k + 1)] w') ^ 2)
β’ (f' 2 c')^[i.n - k] z' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - k] z') ^ 2) | case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
h :
k + 1 β€ i.n β§
(f' 2 c')^[i.n - (k + 1)] w' β approx i.z β§
(i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - (k + 1)] w') ^ 2)
ie : i.n - k = i.n - (k + 1) + 1
β’ (f' 2 c')^[i.n - k] z' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - k] z') ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
h :
k + 1 β€ i.n β§
(f' 2 c')^[i.n - (k + 1)] w' β approx i.z β§
(i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - (k + 1)] w') ^ 2)
β’ (f' 2 c')^[i.n - k] z' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - k] z') ^ 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | rw [ie, Function.iterate_succ_apply, hw'] | case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
h :
k + 1 β€ i.n β§
(f' 2 c')^[i.n - (k + 1)] w' β approx i.z β§
(i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - (k + 1)] w') ^ 2)
ie : i.n - k = i.n - (k + 1) + 1
β’ (f' 2 c')^[i.n - k] z' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - k] z') ^ 2) | case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
h :
k + 1 β€ i.n β§
(f' 2 c')^[i.n - (k + 1)] w' β approx i.z β§
(i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - (k + 1)] w') ^ 2)
ie : i.n - k = i.n - (k + 1) + 1
β’ (f' 2 c')^[i.n - (k + 1)] w' β approx i.z β§
(i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - (k + 1)] w') ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
h :
k + 1 β€ i.n β§
(f' 2 c')^[i.n - (k + 1)] w' β approx i.z β§
(i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - (k + 1)] w') ^ 2)
ie : i.n - k = i.n - (k + 1) + 1
β’ (f' 2 c')^[i.n - k] z' β approx i.z β§ (i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - k] z') ^ 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | exact h.2 | case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
h :
k + 1 β€ i.n β§
(f' 2 c')^[i.n - (k + 1)] w' β approx i.z β§
(i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - (k + 1)] w') ^ 2)
ie : i.n - k = i.n - (k + 1) + 1
β’ (f' 2 c')^[i.n - (k + 1)] w' β approx i.z β§
(i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - (k + 1)] w') ^ 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
h :
k + 1 β€ i.n β§
(f' 2 c')^[i.n - (k + 1)] w' β approx i.z β§
(i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - (k + 1)] w') ^ 2)
ie : i.n - k = i.n - (k + 1) + 1
β’ (f' 2 c')^[i.n - (k + 1)] w' β approx i.z β§
(i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - (k + 1)] w') ^ 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | omega | c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
h :
k + 1 β€ i.n β§
(f' 2 c')^[i.n - (k + 1)] w' β approx i.z β§
(i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - (k + 1)] w') ^ 2)
β’ k β€ i.n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
h :
k + 1 β€ i.n β§
(f' 2 c')^[i.n - (k + 1)] w' β approx i.z β§
(i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - (k + 1)] w') ^ 2)
β’ k β€ i.n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate'_correct | [53, 1] | [91, 16] | omega | c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
h :
k + 1 β€ i.n β§
(f' 2 c')^[i.n - (k + 1)] w' β approx i.z β§
(i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - (k + 1)] w') ^ 2)
β’ i.n - k = i.n - (k + 1) + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : Box
rs : Floating
c' : β
rs' : β
cm : c' β approx c
rsn : rs β nan
rsm : rs' β€ rs.val
n : β
z : Box
z' : β
zm : z' β approx z
k : β
zr2 : Interval
hzr2 : z.re.sqr = zr2
zi2 : Interval
hzi2 : z.im.sqr = zi2
z2 : Floating
hz2 : zr2.lo.add zi2.lo false = z2
w : Box
hw : { re := zr2 - zi2, im := (z.re * z.im).scaleB 1 } + c = w
we : w = z.sqr + c
w' : β
hw' : f' 2 c' z' = w'
wa : w' β approx w
z2n : Β¬z2 = nan
rz : Β¬rs.val < z2.val
i : Iter
hi : iterate' c w rs (k + 1) n = i
h :
k + 1 β€ i.n β§
(f' 2 c')^[i.n - (k + 1)] w' β approx i.z β§
(i.exit = Exit.large β rs' < Complex.abs ((f' 2 c')^[i.n - (k + 1)] w') ^ 2)
β’ i.n - k = i.n - (k + 1) + 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | mem_approx_iterate | [93, 1] | [101, 59] | rw [iterate] | c z : Box
rs : Floating
c' z' : β
cm : c' β approx c
zm : z' β approx z
n : β
β’ (f' 2 c')^[(iterate c z rs n).n] z' β approx (iterate c z rs n).z | c z : Box
rs : Floating
c' z' : β
cm : c' β approx c
zm : z' β approx z
n : β
β’ (f' 2 c')^[(if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).n] z' β
approx (if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).z | Please generate a tactic in lean4 to solve the state.
STATE:
c z : Box
rs : Floating
c' z' : β
cm : c' β approx c
zm : z' β approx z
n : β
β’ (f' 2 c')^[(iterate c z rs n).n] z' β approx (iterate c z rs n).z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | mem_approx_iterate | [93, 1] | [101, 59] | by_cases rsn : rs = nan | c z : Box
rs : Floating
c' z' : β
cm : c' β approx c
zm : z' β approx z
n : β
β’ (f' 2 c')^[(if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).n] z' β
approx (if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).z | case pos
c z : Box
rs : Floating
c' z' : β
cm : c' β approx c
zm : z' β approx z
n : β
rsn : rs = nan
β’ (f' 2 c')^[(if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).n] z' β
approx (if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).z
case neg
c z : Box
rs : Floating
c' z' : β
cm : c' β approx c
zm : z' β approx z
n : β
rsn : Β¬rs = nan
β’ (f' 2 c')^[(if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).n] z' β
approx (if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).z | Please generate a tactic in lean4 to solve the state.
STATE:
c z : Box
rs : Floating
c' z' : β
cm : c' β approx c
zm : z' β approx z
n : β
β’ (f' 2 c')^[(if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).n] z' β
approx (if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | mem_approx_iterate | [93, 1] | [101, 59] | simpa only [rsn, ite_true, Function.iterate_zero, id_eq] | case pos
c z : Box
rs : Floating
c' z' : β
cm : c' β approx c
zm : z' β approx z
n : β
rsn : rs = nan
β’ (f' 2 c')^[(if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).n] z' β
approx (if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c z : Box
rs : Floating
c' z' : β
cm : c' β approx c
zm : z' β approx z
n : β
rsn : rs = nan
β’ (f' 2 c')^[(if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).n] z' β
approx (if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | mem_approx_iterate | [93, 1] | [101, 59] | simp only [rsn, ite_false] | case neg
c z : Box
rs : Floating
c' z' : β
cm : c' β approx c
zm : z' β approx z
n : β
rsn : Β¬rs = nan
β’ (f' 2 c')^[(if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).n] z' β
approx (if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).z | case neg
c z : Box
rs : Floating
c' z' : β
cm : c' β approx c
zm : z' β approx z
n : β
rsn : Β¬rs = nan
β’ (f' 2 c')^[(iterate' c z rs 0 n).n] z' β approx (iterate' c z rs 0 n).z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c z : Box
rs : Floating
c' z' : β
cm : c' β approx c
zm : z' β approx z
n : β
rsn : Β¬rs = nan
β’ (f' 2 c')^[(if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).n] z' β
approx (if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | mem_approx_iterate | [93, 1] | [101, 59] | exact (iterate'_correct cm zm rsn (le_refl _) 0 n).2.1 | case neg
c z : Box
rs : Floating
c' z' : β
cm : c' β approx c
zm : z' β approx z
n : β
rsn : Β¬rs = nan
β’ (f' 2 c')^[(iterate' c z rs 0 n).n] z' β approx (iterate' c z rs 0 n).z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c z : Box
rs : Floating
c' z' : β
cm : c' β approx c
zm : z' β approx z
n : β
rsn : Β¬rs = nan
β’ (f' 2 c')^[(iterate' c z rs 0 n).n] z' β approx (iterate' c z rs 0 n).z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate_large | [103, 1] | [111, 95] | rw [iterate] at l β’ | c z : Box
rs : Floating
n : β
c' z' : β
cm : c' β approx c
zm : z' β approx z
l : (iterate c z rs n).exit = Exit.large
β’ rs.val < Complex.abs ((f' 2 c')^[(iterate c z rs n).n] z') ^ 2 | c z : Box
rs : Floating
n : β
c' z' : β
cm : c' β approx c
zm : z' β approx z
l : (if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).exit = Exit.large
β’ rs.val <
Complex.abs ((f' 2 c')^[(if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).n] z') ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : Box
rs : Floating
n : β
c' z' : β
cm : c' β approx c
zm : z' β approx z
l : (iterate c z rs n).exit = Exit.large
β’ rs.val < Complex.abs ((f' 2 c')^[(iterate c z rs n).n] z') ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate_large | [103, 1] | [111, 95] | by_cases rsn : rs = nan | c z : Box
rs : Floating
n : β
c' z' : β
cm : c' β approx c
zm : z' β approx z
l : (if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).exit = Exit.large
β’ rs.val <
Complex.abs ((f' 2 c')^[(if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).n] z') ^ 2 | case pos
c z : Box
rs : Floating
n : β
c' z' : β
cm : c' β approx c
zm : z' β approx z
l : (if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).exit = Exit.large
rsn : rs = nan
β’ rs.val <
Complex.abs ((f' 2 c')^[(if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).n] z') ^ 2
case neg
c z : Box
rs : Floating
n : β
c' z' : β
cm : c' β approx c
zm : z' β approx z
l : (if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).exit = Exit.large
rsn : Β¬rs = nan
β’ rs.val <
Complex.abs ((f' 2 c')^[(if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).n] z') ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : Box
rs : Floating
n : β
c' z' : β
cm : c' β approx c
zm : z' β approx z
l : (if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).exit = Exit.large
β’ rs.val <
Complex.abs ((f' 2 c')^[(if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).n] z') ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate_large | [103, 1] | [111, 95] | simp only [rsn, βreduceIte] at l | case pos
c z : Box
rs : Floating
n : β
c' z' : β
cm : c' β approx c
zm : z' β approx z
l : (if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).exit = Exit.large
rsn : rs = nan
β’ rs.val <
Complex.abs ((f' 2 c')^[(if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).n] z') ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c z : Box
rs : Floating
n : β
c' z' : β
cm : c' β approx c
zm : z' β approx z
l : (if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).exit = Exit.large
rsn : rs = nan
β’ rs.val <
Complex.abs ((f' 2 c')^[(if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).n] z') ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate_large | [103, 1] | [111, 95] | simp only [rsn, ite_false] at l β’ | case neg
c z : Box
rs : Floating
n : β
c' z' : β
cm : c' β approx c
zm : z' β approx z
l : (if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).exit = Exit.large
rsn : Β¬rs = nan
β’ rs.val <
Complex.abs ((f' 2 c')^[(if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).n] z') ^ 2 | case neg
c z : Box
rs : Floating
n : β
c' z' : β
cm : c' β approx c
zm : z' β approx z
rsn : Β¬rs = nan
l : (iterate' c z rs 0 n).exit = Exit.large
β’ rs.val < Complex.abs ((f' 2 c')^[(iterate' c z rs 0 n).n] z') ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c z : Box
rs : Floating
n : β
c' z' : β
cm : c' β approx c
zm : z' β approx z
l : (if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).exit = Exit.large
rsn : Β¬rs = nan
β’ rs.val <
Complex.abs ((f' 2 c')^[(if rs = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z rs 0 n).n] z') ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate_large | [103, 1] | [111, 95] | simpa only [not_lt, Nat.sub_zero] using (iterate'_correct cm zm rsn (le_refl _) 0 n).2.2 l | case neg
c z : Box
rs : Floating
n : β
c' z' : β
cm : c' β approx c
zm : z' β approx z
rsn : Β¬rs = nan
l : (iterate' c z rs 0 n).exit = Exit.large
β’ rs.val < Complex.abs ((f' 2 c')^[(iterate' c z rs 0 n).n] z') ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c z : Box
rs : Floating
n : β
c' z' : β
cm : c' β approx c
zm : z' β approx z
rsn : Β¬rs = nan
l : (iterate' c z rs 0 n).exit = Exit.large
β’ rs.val < Complex.abs ((f' 2 c')^[(iterate' c z rs 0 n).n] z') ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate_nan | [113, 1] | [115, 60] | rw [iterate] | c z : Box
n : β
β’ (iterate c z nan n).exit = Exit.nan | c z : Box
n : β
β’ (if nan = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z nan 0 n).exit = Exit.nan | Please generate a tactic in lean4 to solve the state.
STATE:
c z : Box
n : β
β’ (iterate c z nan n).exit = Exit.nan
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | iterate_nan | [113, 1] | [115, 60] | simp only [ite_true, Function.iterate_zero] | c z : Box
n : β
β’ (if nan = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z nan 0 n).exit = Exit.nan | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : Box
n : β
β’ (if nan = nan then { z := z, n := 0, exit := Exit.nan } else iterate' c z nan 0 n).exit = Exit.nan
TACTIC:
|
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