url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Interval.mem_approx_quantize | [189, 1] | [222, 53] | apply lo_le (by native_decide) | α β : Type
x' : ℝ
x : Interval
c : UInt8
hc : x.untrusted_quantize = c
lo : Floating
hl : c.unquantize.sub half_color_error.lo true = lo
hi : Floating
hh : c.unquantize.add half_color_error.lo false = hi
n : ¬(lo = nan ∨ hi = nan)
n0 : ¬lo = nan
n1 : ¬hi = nan
mn : c.unquantize ≠ nan
g : lo.val ≤ x.lo.val ∧ x.hi.val ≤ ... | α β : Type
x' : ℝ
x : Interval
c : UInt8
hc : x.untrusted_quantize = c
lo : Floating
hl : c.unquantize.sub half_color_error.lo true = lo
hi : Floating
hh : c.unquantize.add half_color_error.lo false = hi
n : ¬(lo = nan ∨ hi = nan)
n0 : ¬lo = nan
n1 : ¬hi = nan
mn : c.unquantize ≠ nan
g : lo.val ≤ x.lo.val ∧ x.hi.val ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
x' : ℝ
x : Interval
c : UInt8
hc : x.untrusted_quantize = c
lo : Floating
hl : c.unquantize.sub half_color_error.lo true = lo
hi : Floating
hh : c.unquantize.add half_color_error.lo false = hi
n : ¬(lo = nan ∨ hi = nan)
n0 : ¬lo = nan
n1 : ¬hi = na... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Interval.mem_approx_quantize | [189, 1] | [222, 53] | have e : (color_error : ℝ) / 2 = (color_error / 2 : ℚ) := by
simp only [Rat.cast_div, Rat.cast_ofNat] | α β : Type
x' : ℝ
x : Interval
c : UInt8
hc : x.untrusted_quantize = c
lo : Floating
hl : c.unquantize.sub half_color_error.lo true = lo
hi : Floating
hh : c.unquantize.add half_color_error.lo false = hi
n : ¬(lo = nan ∨ hi = nan)
n0 : ¬lo = nan
n1 : ¬hi = nan
mn : c.unquantize ≠ nan
g : lo.val ≤ x.lo.val ∧ x.hi.val ≤ ... | α β : Type
x' : ℝ
x : Interval
c : UInt8
hc : x.untrusted_quantize = c
lo : Floating
hl : c.unquantize.sub half_color_error.lo true = lo
hi : Floating
hh : c.unquantize.add half_color_error.lo false = hi
n : ¬(lo = nan ∨ hi = nan)
n0 : ¬lo = nan
n1 : ¬hi = nan
mn : c.unquantize ≠ nan
g : lo.val ≤ x.lo.val ∧ x.hi.val ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
x' : ℝ
x : Interval
c : UInt8
hc : x.untrusted_quantize = c
lo : Floating
hl : c.unquantize.sub half_color_error.lo true = lo
hi : Floating
hh : c.unquantize.add half_color_error.lo false = hi
n : ¬(lo = nan ∨ hi = nan)
n0 : ¬lo = nan
n1 : ¬hi = na... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Interval.mem_approx_quantize | [189, 1] | [222, 53] | rw [e] | α β : Type
x' : ℝ
x : Interval
c : UInt8
hc : x.untrusted_quantize = c
lo : Floating
hl : c.unquantize.sub half_color_error.lo true = lo
hi : Floating
hh : c.unquantize.add half_color_error.lo false = hi
n : ¬(lo = nan ∨ hi = nan)
n0 : ¬lo = nan
n1 : ¬hi = nan
mn : c.unquantize ≠ nan
g : lo.val ≤ x.lo.val ∧ x.hi.val ≤ ... | α β : Type
x' : ℝ
x : Interval
c : UInt8
hc : x.untrusted_quantize = c
lo : Floating
hl : c.unquantize.sub half_color_error.lo true = lo
hi : Floating
hh : c.unquantize.add half_color_error.lo false = hi
n : ¬(lo = nan ∨ hi = nan)
n0 : ¬lo = nan
n1 : ¬hi = nan
mn : c.unquantize ≠ nan
g : lo.val ≤ x.lo.val ∧ x.hi.val ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
x' : ℝ
x : Interval
c : UInt8
hc : x.untrusted_quantize = c
lo : Floating
hl : c.unquantize.sub half_color_error.lo true = lo
hi : Floating
hh : c.unquantize.add half_color_error.lo false = hi
n : ¬(lo = nan ∨ hi = nan)
n0 : ¬lo = nan
n1 : ¬hi = na... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Interval.mem_approx_quantize | [189, 1] | [222, 53] | mono | α β : Type
x' : ℝ
x : Interval
c : UInt8
hc : x.untrusted_quantize = c
lo : Floating
hl : c.unquantize.sub half_color_error.lo true = lo
hi : Floating
hh : c.unquantize.add half_color_error.lo false = hi
n : ¬(lo = nan ∨ hi = nan)
n0 : ¬lo = nan
n1 : ¬hi = nan
mn : c.unquantize ≠ nan
g : lo.val ≤ x.lo.val ∧ x.hi.val ≤ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
x' : ℝ
x : Interval
c : UInt8
hc : x.untrusted_quantize = c
lo : Floating
hl : c.unquantize.sub half_color_error.lo true = lo
hi : Floating
hh : c.unquantize.add half_color_error.lo false = hi
n : ¬(lo = nan ∨ hi = nan)
n0 : ¬lo = nan
n1 : ¬hi = na... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Interval.mem_approx_quantize | [189, 1] | [222, 53] | native_decide | α β : Type
x' : ℝ
x : Interval
c : UInt8
hc : x.untrusted_quantize = c
lo : Floating
hl : c.unquantize.sub half_color_error.lo true = lo
hi : Floating
hh : c.unquantize.add half_color_error.lo false = hi
n : ¬(lo = nan ∨ hi = nan)
n0 : ¬lo = nan
n1 : ¬hi = nan
mn : c.unquantize ≠ nan
g : lo.val ≤ x.lo.val ∧ x.hi.val ≤ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
x' : ℝ
x : Interval
c : UInt8
hc : x.untrusted_quantize = c
lo : Floating
hl : c.unquantize.sub half_color_error.lo true = lo
hi : Floating
hh : c.unquantize.add half_color_error.lo false = hi
n : ¬(lo = nan ∨ hi = nan)
n0 : ¬lo = nan
n1 : ¬hi = na... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Interval.mem_approx_quantize | [189, 1] | [222, 53] | simp only [Rat.cast_div, Rat.cast_ofNat] | α β : Type
x' : ℝ
x : Interval
c : UInt8
hc : x.untrusted_quantize = c
lo : Floating
hl : c.unquantize.sub half_color_error.lo true = lo
hi : Floating
hh : c.unquantize.add half_color_error.lo false = hi
n : ¬(lo = nan ∨ hi = nan)
n0 : ¬lo = nan
n1 : ¬hi = nan
mn : c.unquantize ≠ nan
g : lo.val ≤ x.lo.val ∧ x.hi.val ≤ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
x' : ℝ
x : Interval
c : UInt8
hc : x.untrusted_quantize = c
lo : Floating
hl : c.unquantize.sub half_color_error.lo true = lo
hi : Floating
hh : c.unquantize.add half_color_error.lo false = hi
n : ¬(lo = nan ∨ hi = nan)
n0 : ¬lo = nan
n1 : ¬hi = na... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Interval.mem_approx_quantize | [189, 1] | [222, 53] | linarith | α β : Type
x' : ℝ
x : Interval
c : UInt8
hc : x.untrusted_quantize = c
lo : Floating
hl : c.unquantize.sub half_color_error.lo true = lo
hi : Floating
hh : c.unquantize.add half_color_error.lo false = hi
n : ¬(lo = nan ∨ hi = nan)
mn : c.unquantize ≠ nan
xn : x ≠ nan
xm : x.lo.val ≤ x' ∧ x' ≤ x.hi.val
e : half_color_er... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
x' : ℝ
x : Interval
c : UInt8
hc : x.untrusted_quantize = c
lo : Floating
hl : c.unquantize.sub half_color_error.lo true = lo
hi : Floating
hh : c.unquantize.add half_color_error.lo false = hi
n : ¬(lo = nan ∨ hi = nan)
mn : c.unquantize ≠ nan
xn :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Interval.mem_approx_quantize | [189, 1] | [222, 53] | linarith | α β : Type
x' : ℝ
x : Interval
c : UInt8
hc : x.untrusted_quantize = c
lo : Floating
hl : c.unquantize.sub half_color_error.lo true = lo
hi : Floating
hh : c.unquantize.add half_color_error.lo false = hi
n : ¬(lo = nan ∨ hi = nan)
mn : c.unquantize ≠ nan
xn : x ≠ nan
xm : x.lo.val ≤ x' ∧ x' ≤ x.hi.val
e : half_color_er... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
x' : ℝ
x : Interval
c : UInt8
hc : x.untrusted_quantize = c
lo : Floating
hl : c.unquantize.sub half_color_error.lo true = lo
hi : Floating
hh : c.unquantize.add half_color_error.lo false = hi
n : ¬(lo = nan ∨ hi = nan)
mn : c.unquantize ≠ nan
xn :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Interval.mem_approx_quantize | [189, 1] | [222, 53] | simp only [g, ↓reduceIte, approx_none, mem_univ] | case neg
α β : Type
x' : ℝ
x : Interval
xm : x' ∈ approx x
c : UInt8
hc : x.untrusted_quantize = c
lo : Floating
hl : c.unquantize.sub half_color_error.lo true = lo
hi : Floating
hh : c.unquantize.add half_color_error.lo false = hi
n : ¬(lo = nan ∨ hi = nan)
n0 : ¬lo = nan
n1 : ¬hi = nan
mn : c.unquantize ≠ nan
g : ¬(l... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α β : Type
x' : ℝ
x : Interval
xm : x' ∈ approx x
c : UInt8
hc : x.untrusted_quantize = c
lo : Floating
hl : c.unquantize.sub half_color_error.lo true = lo
hi : Floating
hh : c.unquantize.add half_color_error.lo false = hi
n : ¬(lo = nan ∨ hi = nan)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_quantize | [230, 1] | [255, 66] | rw [quantize] | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
⊢ c' ∈ approx c.quantize | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
⊢ c' ∈
approx
(match (c.r.quantize, c.g.quantize, c.b.quantize, c.a.quantize) with
| (some r, some g, some b, some a) => { r := r, g := g, b := b, a := a }
| x => nan) | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
⊢ c' ∈ approx c.quantize
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_quantize | [230, 1] | [255, 66] | generalize hr : c.r.quantize = r | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
⊢ c' ∈
approx
(match (c.r.quantize, c.g.quantize, c.b.quantize, c.a.quantize) with
| (some r, some g, some b, some a) => { r := r, g := g, b := b, a := a }
| x => nan) | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r : Option UInt8
hr : c.r.quantize = r
⊢ c' ∈
approx
(match (r, c.g.quantize, c.b.quantize, c.a.quantize) with
| (some r, some g, some b, some a) => { r := r, g := g, b := b, a := a }
| x => nan) | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
⊢ c' ∈
approx
(match (c.r.quantize, c.g.quantize, c.b.quantize, c.a.quantize) with
| (some r, some g, some b, some a) => { r := r, g := g, b := b, a := a }
| x => nan)
TACTIC:... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_quantize | [230, 1] | [255, 66] | generalize hg : c.g.quantize = g | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r : Option UInt8
hr : c.r.quantize = r
⊢ c' ∈
approx
(match (r, c.g.quantize, c.b.quantize, c.a.quantize) with
| (some r, some g, some b, some a) => { r := r, g := g, b := b, a := a }
| x => nan) | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r : Option UInt8
hr : c.r.quantize = r
g : Option UInt8
hg : c.g.quantize = g
⊢ c' ∈
approx
(match (r, g, c.b.quantize, c.a.quantize) with
| (some r, some g, some b, some a) => { r := r, g := g, b := b, a := a }
| x => nan) | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r : Option UInt8
hr : c.r.quantize = r
⊢ c' ∈
approx
(match (r, c.g.quantize, c.b.quantize, c.a.quantize) with
| (some r, some g, some b, some a) => { r := r, g := g, b := b, a := a... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_quantize | [230, 1] | [255, 66] | generalize hb : c.b.quantize = b | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r : Option UInt8
hr : c.r.quantize = r
g : Option UInt8
hg : c.g.quantize = g
⊢ c' ∈
approx
(match (r, g, c.b.quantize, c.a.quantize) with
| (some r, some g, some b, some a) => { r := r, g := g, b := b, a := a }
| x => nan) | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r : Option UInt8
hr : c.r.quantize = r
g : Option UInt8
hg : c.g.quantize = g
b : Option UInt8
hb : c.b.quantize = b
⊢ c' ∈
approx
(match (r, g, b, c.a.quantize) with
| (some r, some g, some b, some a) => { r := r, g := g, b := b, a := a }
... | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r : Option UInt8
hr : c.r.quantize = r
g : Option UInt8
hg : c.g.quantize = g
⊢ c' ∈
approx
(match (r, g, c.b.quantize, c.a.quantize) with
| (some r, some g, some b, some a) => { r ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_quantize | [230, 1] | [255, 66] | generalize ha : c.a.quantize = a | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r : Option UInt8
hr : c.r.quantize = r
g : Option UInt8
hg : c.g.quantize = g
b : Option UInt8
hb : c.b.quantize = b
⊢ c' ∈
approx
(match (r, g, b, c.a.quantize) with
| (some r, some g, some b, some a) => { r := r, g := g, b := b, a := a }
... | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r : Option UInt8
hr : c.r.quantize = r
g : Option UInt8
hg : c.g.quantize = g
b : Option UInt8
hb : c.b.quantize = b
a : Option UInt8
ha : c.a.quantize = a
⊢ c' ∈
approx
(match (r, g, b, a) with
| (some r, some g, some b, some a) => { r := r,... | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r : Option UInt8
hr : c.r.quantize = r
g : Option UInt8
hg : c.g.quantize = g
b : Option UInt8
hb : c.b.quantize = b
⊢ c' ∈
approx
(match (r, g, b, c.a.quantize) with
| (some r, som... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_quantize | [230, 1] | [255, 66] | match r with
| none => simp only [approx_nan, mem_univ]
| some r =>
match g with
| none => simp only [approx_nan, mem_univ]
| some g =>
match b with
| none => simp only [approx_nan, mem_univ]
| some b =>
match a with
| none => simp only [approx_nan, mem_univ]
| some a =>
simp only [approx,... | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r : Option UInt8
hr : c.r.quantize = r
g : Option UInt8
hg : c.g.quantize = g
b : Option UInt8
hb : c.b.quantize = b
a : Option UInt8
ha : c.a.quantize = a
⊢ c' ∈
approx
(match (r, g, b, a) with
| (some r, some g, some b, some a) => { r := r,... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r : Option UInt8
hr : c.r.quantize = r
g : Option UInt8
hg : c.g.quantize = g
b : Option UInt8
hb : c.b.quantize = b
a : Option UInt8
ha : c.a.quantize = a
⊢ c' ∈
approx
(match (r, g, b, ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_quantize | [230, 1] | [255, 66] | simp only [approx_nan, mem_univ] | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r g : Option UInt8
hg : c.g.quantize = g
b : Option UInt8
hb : c.b.quantize = b
a : Option UInt8
ha : c.a.quantize = a
hr : c.r.quantize = none
⊢ c' ∈
approx
(match (none, g, b, a) with
| (some r, some g, some b, some a) => { r := r, g := g, ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r g : Option UInt8
hg : c.g.quantize = g
b : Option UInt8
hb : c.b.quantize = b
a : Option UInt8
ha : c.a.quantize = a
hr : c.r.quantize = none
⊢ c' ∈
approx
(match (none, g, b, a) with
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_quantize | [230, 1] | [255, 66] | match g with
| none => simp only [approx_nan, mem_univ]
| some g =>
match b with
| none => simp only [approx_nan, mem_univ]
| some b =>
match a with
| none => simp only [approx_nan, mem_univ]
| some a =>
simp only [approx, mem_ite_univ_left]
intro _
simp only [approx] at cm; simp only [approx', mem_se... | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r✝ g : Option UInt8
hg : c.g.quantize = g
b : Option UInt8
hb : c.b.quantize = b
a : Option UInt8
ha : c.a.quantize = a
r : UInt8
hr : c.r.quantize = some r
⊢ c' ∈
approx
(match (some r, g, b, a) with
| (some r, some g, some b, some a) => { r... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r✝ g : Option UInt8
hg : c.g.quantize = g
b : Option UInt8
hb : c.b.quantize = b
a : Option UInt8
ha : c.a.quantize = a
r : UInt8
hr : c.r.quantize = some r
⊢ c' ∈
approx
(match (some r, ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_quantize | [230, 1] | [255, 66] | simp only [approx_nan, mem_univ] | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r✝ g b : Option UInt8
hb : c.b.quantize = b
a : Option UInt8
ha : c.a.quantize = a
r : UInt8
hr : c.r.quantize = some r
hg : c.g.quantize = none
⊢ c' ∈
approx
(match (some r, none, b, a) with
| (some r, some g, some b, some a) => { r := r, g ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r✝ g b : Option UInt8
hb : c.b.quantize = b
a : Option UInt8
ha : c.a.quantize = a
r : UInt8
hr : c.r.quantize = some r
hg : c.g.quantize = none
⊢ c' ∈
approx
(match (some r, none, b, a) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_quantize | [230, 1] | [255, 66] | match b with
| none => simp only [approx_nan, mem_univ]
| some b =>
match a with
| none => simp only [approx_nan, mem_univ]
| some a =>
simp only [approx, mem_ite_univ_left]
intro _
simp only [approx] at cm; simp only [approx', mem_setOf_eq] at cm
rcases cm with ⟨m0, m1, m2, m3⟩
simp only [app... | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r✝ g✝ b : Option UInt8
hb : c.b.quantize = b
a : Option UInt8
ha : c.a.quantize = a
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
⊢ c' ∈
approx
(match (some r, some g, b, a) with
| (some r, some g, some b, some a) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r✝ g✝ b : Option UInt8
hb : c.b.quantize = b
a : Option UInt8
ha : c.a.quantize = a
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
⊢ c' ∈
approx
(match (some r,... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_quantize | [230, 1] | [255, 66] | simp only [approx_nan, mem_univ] | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r✝ g✝ b a : Option UInt8
ha : c.a.quantize = a
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
hb : c.b.quantize = none
⊢ c' ∈
approx
(match (some r, some g, none, a) with
| (some r, some g, some b, some a) => { r :=... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r✝ g✝ b a : Option UInt8
ha : c.a.quantize = a
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
hb : c.b.quantize = none
⊢ c' ∈
approx
(match (some r, some g, non... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_quantize | [230, 1] | [255, 66] | match a with
| none => simp only [approx_nan, mem_univ]
| some a =>
simp only [approx, mem_ite_univ_left]
intro _
simp only [approx] at cm; simp only [approx', mem_setOf_eq] at cm
rcases cm with ⟨m0, m1, m2, m3⟩
simp only [approx', mem_setOf_eq, ←approx_some, ←hr, ←hg, ←hb, ←ha,
Interval.mem_approx_quanti... | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r✝ g✝ b✝ a : Option UInt8
ha : c.a.quantize = a
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
b : UInt8
hb : c.b.quantize = some b
⊢ c' ∈
approx
(match (some r, some g, some b, a) with
| (some r, some g, some b, so... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r✝ g✝ b✝ a : Option UInt8
ha : c.a.quantize = a
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
b : UInt8
hb : c.b.quantize = some b
⊢ c' ∈
approx
(match (some r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_quantize | [230, 1] | [255, 66] | simp only [approx_nan, mem_univ] | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r✝ g✝ b✝ a : Option UInt8
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
b : UInt8
hb : c.b.quantize = some b
ha : c.a.quantize = none
⊢ c' ∈
approx
(match (some r, some g, some b, none) with
| (some r, some g, some... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r✝ g✝ b✝ a : Option UInt8
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
b : UInt8
hb : c.b.quantize = some b
ha : c.a.quantize = none
⊢ c' ∈
approx
(match (som... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_quantize | [230, 1] | [255, 66] | simp only [approx, mem_ite_univ_left] | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r✝ g✝ b✝ a✝ : Option UInt8
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
b : UInt8
hb : c.b.quantize = some b
a : UInt8
ha : c.a.quantize = some a
⊢ c' ∈
approx
(match (some r, some g, some b, some a) with
| (some ... | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r✝ g✝ b✝ a✝ : Option UInt8
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
b : UInt8
hb : c.b.quantize = some b
a : UInt8
ha : c.a.quantize = some a
⊢ ¬{ r := r, g := g, b := b, a := a } = nan → c' ∈ { r := r, g := g, b := b, a := a... | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r✝ g✝ b✝ a✝ : Option UInt8
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
b : UInt8
hb : c.b.quantize = some b
a : UInt8
ha : c.a.quantize = some a
⊢ c' ∈
approx
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_quantize | [230, 1] | [255, 66] | intro _ | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r✝ g✝ b✝ a✝ : Option UInt8
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
b : UInt8
hb : c.b.quantize = some b
a : UInt8
ha : c.a.quantize = some a
⊢ ¬{ r := r, g := g, b := b, a := a } = nan → c' ∈ { r := r, g := g, b := b, a := a... | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r✝ g✝ b✝ a✝¹ : Option UInt8
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
b : UInt8
hb : c.b.quantize = some b
a : UInt8
ha : c.a.quantize = some a
a✝ : ¬{ r := r, g := g, b := b, a := a } = nan
⊢ c' ∈ { r := r, g := g, b := b, a ... | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r✝ g✝ b✝ a✝ : Option UInt8
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
b : UInt8
hb : c.b.quantize = some b
a : UInt8
ha : c.a.quantize = some a
⊢ ¬{ r := r, g := g, b... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_quantize | [230, 1] | [255, 66] | simp only [approx] at cm | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r✝ g✝ b✝ a✝¹ : Option UInt8
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
b : UInt8
hb : c.b.quantize = some b
a : UInt8
ha : c.a.quantize = some a
a✝ : ¬{ r := r, g := g, b := b, a := a } = nan
⊢ c' ∈ { r := r, g := g, b := b, a ... | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ c.approx'
r✝ g✝ b✝ a✝¹ : Option UInt8
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
b : UInt8
hb : c.b.quantize = some b
a : UInt8
ha : c.a.quantize = some a
a✝ : ¬{ r := r, g := g, b := b, a := a } = nan
⊢ c' ∈ { r := r, g := g, b := b, a... | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ approx c
r✝ g✝ b✝ a✝¹ : Option UInt8
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
b : UInt8
hb : c.b.quantize = some b
a : UInt8
ha : c.a.quantize = some a
a✝ : ¬{ r := r, g := ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_quantize | [230, 1] | [255, 66] | simp only [approx', mem_setOf_eq] at cm | α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ c.approx'
r✝ g✝ b✝ a✝¹ : Option UInt8
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
b : UInt8
hb : c.b.quantize = some b
a : UInt8
ha : c.a.quantize = some a
a✝ : ¬{ r := r, g := g, b := b, a := a } = nan
⊢ c' ∈ { r := r, g := g, b := b, a... | α β : Type
c' : Color ℝ
c : Color Interval
cm : c'.r ∈ approx c.r ∧ c'.g ∈ approx c.g ∧ c'.b ∈ approx c.b ∧ c'.a ∈ approx c.a
r✝ g✝ b✝ a✝¹ : Option UInt8
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
b : UInt8
hb : c.b.quantize = some b
a : UInt8
ha : c.a.quantize = some a
a✝ : ¬{ r := r, g ... | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
c' : Color ℝ
c : Color Interval
cm : c' ∈ c.approx'
r✝ g✝ b✝ a✝¹ : Option UInt8
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
b : UInt8
hb : c.b.quantize = some b
a : UInt8
ha : c.a.quantize = some a
a✝ : ¬{ r := r, g :=... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_quantize | [230, 1] | [255, 66] | rcases cm with ⟨m0, m1, m2, m3⟩ | α β : Type
c' : Color ℝ
c : Color Interval
cm : c'.r ∈ approx c.r ∧ c'.g ∈ approx c.g ∧ c'.b ∈ approx c.b ∧ c'.a ∈ approx c.a
r✝ g✝ b✝ a✝¹ : Option UInt8
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
b : UInt8
hb : c.b.quantize = some b
a : UInt8
ha : c.a.quantize = some a
a✝ : ¬{ r := r, g ... | case intro.intro.intro
α β : Type
c' : Color ℝ
c : Color Interval
r✝ g✝ b✝ a✝¹ : Option UInt8
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
b : UInt8
hb : c.b.quantize = some b
a : UInt8
ha : c.a.quantize = some a
a✝ : ¬{ r := r, g := g, b := b, a := a } = nan
m0 : c'.r ∈ approx c.r
m1 : c'.... | Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
c' : Color ℝ
c : Color Interval
cm : c'.r ∈ approx c.r ∧ c'.g ∈ approx c.g ∧ c'.b ∈ approx c.b ∧ c'.a ∈ approx c.a
r✝ g✝ b✝ a✝¹ : Option UInt8
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
b : UInt8
hb : c.b.quantize = s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_quantize | [230, 1] | [255, 66] | simp only [approx', mem_setOf_eq, ←approx_some, ←hr, ←hg, ←hb, ←ha,
Interval.mem_approx_quantize, m0, m1, m2, m3, true_and] | case intro.intro.intro
α β : Type
c' : Color ℝ
c : Color Interval
r✝ g✝ b✝ a✝¹ : Option UInt8
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
b : UInt8
hb : c.b.quantize = some b
a : UInt8
ha : c.a.quantize = some a
a✝ : ¬{ r := r, g := g, b := b, a := a } = nan
m0 : c'.r ∈ approx c.r
m1 : c'.... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α β : Type
c' : Color ℝ
c : Color Interval
r✝ g✝ b✝ a✝¹ : Option UInt8
r : UInt8
hr : c.r.quantize = some r
g : UInt8
hg : c.g.quantize = some g
b : UInt8
hb : c.b.quantize = some b
a : UInt8
ha : c.a.quantize = some a
a✝ : ¬{ r := r, g... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | mem_approx_lerp | [327, 1] | [333, 24] | rw [lerp, lerp] | α β α' β' : Type
inst✝⁹ : Approx α α'
inst✝⁸ : Approx β β'
inst✝⁷ : Add β
inst✝⁶ : Sub β
inst✝⁵ : AddGroup β'
inst✝⁴ : SMul α β
inst✝³ : SMul α' β'
inst✝² : ApproxAdd β β'
inst✝¹ : ApproxSub β β'
inst✝ : ApproxSMul α β α' β'
t' : α'
x' y' : β'
t : α
x y : β
tm : t' ∈ approx t
xm : x' ∈ approx x
ym : y' ∈ approx y
⊢ ler... | α β α' β' : Type
inst✝⁹ : Approx α α'
inst✝⁸ : Approx β β'
inst✝⁷ : Add β
inst✝⁶ : Sub β
inst✝⁵ : AddGroup β'
inst✝⁴ : SMul α β
inst✝³ : SMul α' β'
inst✝² : ApproxAdd β β'
inst✝¹ : ApproxSub β β'
inst✝ : ApproxSMul α β α' β'
t' : α'
x' y' : β'
t : α
x y : β
tm : t' ∈ approx t
xm : x' ∈ approx x
ym : y' ∈ approx y
⊢ x' ... | Please generate a tactic in lean4 to solve the state.
STATE:
α β α' β' : Type
inst✝⁹ : Approx α α'
inst✝⁸ : Approx β β'
inst✝⁷ : Add β
inst✝⁶ : Sub β
inst✝⁵ : AddGroup β'
inst✝⁴ : SMul α β
inst✝³ : SMul α' β'
inst✝² : ApproxAdd β β'
inst✝¹ : ApproxSub β β'
inst✝ : ApproxSMul α β α' β'
t' : α'
x' y' : β'
t : α
x y : β
t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | mem_approx_lerp | [327, 1] | [333, 24] | mono | α β α' β' : Type
inst✝⁹ : Approx α α'
inst✝⁸ : Approx β β'
inst✝⁷ : Add β
inst✝⁶ : Sub β
inst✝⁵ : AddGroup β'
inst✝⁴ : SMul α β
inst✝³ : SMul α' β'
inst✝² : ApproxAdd β β'
inst✝¹ : ApproxSub β β'
inst✝ : ApproxSMul α β α' β'
t' : α'
x' y' : β'
t : α
x y : β
tm : t' ∈ approx t
xm : x' ∈ approx x
ym : y' ∈ approx y
⊢ x' ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α β α' β' : Type
inst✝⁹ : Approx α α'
inst✝⁸ : Approx β β'
inst✝⁷ : Add β
inst✝⁶ : Sub β
inst✝⁵ : AddGroup β'
inst✝⁴ : SMul α β
inst✝³ : SMul α' β'
inst✝² : ApproxAdd β β'
inst✝¹ : ApproxSub β β'
inst✝ : ApproxSMul α β α' β'
t' : α'
x' y' : β'
t : α
x y : β
t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_ofRat | [335, 1] | [340, 40] | simp only [coe, approx, ofRat, map] | α β α' β' : Type
inst✝¹ : Approx α α'
inst✝ : Approx β β'
c : Color ℚ
⊢ ↑c ∈ approx c.ofRat | α β α' β' : Type
inst✝¹ : Approx α α'
inst✝ : Approx β β'
c : Color ℚ
⊢ { r := ↑c.r, g := ↑c.g, b := ↑c.b, a := ↑c.a } ∈
{ r := Interval.ofRat c.r, g := Interval.ofRat c.g, b := Interval.ofRat c.b, a := Interval.ofRat c.a }.approx' | Please generate a tactic in lean4 to solve the state.
STATE:
α β α' β' : Type
inst✝¹ : Approx α α'
inst✝ : Approx β β'
c : Color ℚ
⊢ ↑c ∈ approx c.ofRat
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_ofRat | [335, 1] | [340, 40] | simp only [approx', mem_setOf_eq] | α β α' β' : Type
inst✝¹ : Approx α α'
inst✝ : Approx β β'
c : Color ℚ
⊢ { r := ↑c.r, g := ↑c.g, b := ↑c.b, a := ↑c.a } ∈
{ r := Interval.ofRat c.r, g := Interval.ofRat c.g, b := Interval.ofRat c.b, a := Interval.ofRat c.a }.approx' | α β α' β' : Type
inst✝¹ : Approx α α'
inst✝ : Approx β β'
c : Color ℚ
⊢ ↑c.r ∈ approx (Interval.ofRat c.r) ∧
↑c.g ∈ approx (Interval.ofRat c.g) ∧ ↑c.b ∈ approx (Interval.ofRat c.b) ∧ ↑c.a ∈ approx (Interval.ofRat c.a) | Please generate a tactic in lean4 to solve the state.
STATE:
α β α' β' : Type
inst✝¹ : Approx α α'
inst✝ : Approx β β'
c : Color ℚ
⊢ { r := ↑c.r, g := ↑c.g, b := ↑c.b, a := ↑c.a } ∈
{ r := Interval.ofRat c.r, g := Interval.ofRat c.g, b := Interval.ofRat c.b, a := Interval.ofRat c.a }.approx'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_ofRat | [335, 1] | [340, 40] | refine ⟨?_, ?_, ?_, ?_⟩ | α β α' β' : Type
inst✝¹ : Approx α α'
inst✝ : Approx β β'
c : Color ℚ
⊢ ↑c.r ∈ approx (Interval.ofRat c.r) ∧
↑c.g ∈ approx (Interval.ofRat c.g) ∧ ↑c.b ∈ approx (Interval.ofRat c.b) ∧ ↑c.a ∈ approx (Interval.ofRat c.a) | case refine_1
α β α' β' : Type
inst✝¹ : Approx α α'
inst✝ : Approx β β'
c : Color ℚ
⊢ ↑c.r ∈ approx (Interval.ofRat c.r)
case refine_2
α β α' β' : Type
inst✝¹ : Approx α α'
inst✝ : Approx β β'
c : Color ℚ
⊢ ↑c.g ∈ approx (Interval.ofRat c.g)
case refine_3
α β α' β' : Type
inst✝¹ : Approx α α'
inst✝ : Approx β β'
c : ... | Please generate a tactic in lean4 to solve the state.
STATE:
α β α' β' : Type
inst✝¹ : Approx α α'
inst✝ : Approx β β'
c : Color ℚ
⊢ ↑c.r ∈ approx (Interval.ofRat c.r) ∧
↑c.g ∈ approx (Interval.ofRat c.g) ∧ ↑c.b ∈ approx (Interval.ofRat c.b) ∧ ↑c.a ∈ approx (Interval.ofRat c.a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_ofRat | [335, 1] | [340, 40] | all_goals apply Interval.approx_ofRat | case refine_1
α β α' β' : Type
inst✝¹ : Approx α α'
inst✝ : Approx β β'
c : Color ℚ
⊢ ↑c.r ∈ approx (Interval.ofRat c.r)
case refine_2
α β α' β' : Type
inst✝¹ : Approx α α'
inst✝ : Approx β β'
c : Color ℚ
⊢ ↑c.g ∈ approx (Interval.ofRat c.g)
case refine_3
α β α' β' : Type
inst✝¹ : Approx α α'
inst✝ : Approx β β'
c : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
α β α' β' : Type
inst✝¹ : Approx α α'
inst✝ : Approx β β'
c : Color ℚ
⊢ ↑c.r ∈ approx (Interval.ofRat c.r)
case refine_2
α β α' β' : Type
inst✝¹ : Approx α α'
inst✝ : Approx β β'
c : Color ℚ
⊢ ↑c.g ∈ approx (Interval.ofRat c.g)
case refine_3
α... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Color.lean | Color.mem_approx_ofRat | [335, 1] | [340, 40] | apply Interval.approx_ofRat | case refine_4
α β α' β' : Type
inst✝¹ : Approx α α'
inst✝ : Approx β β'
c : Color ℚ
⊢ ↑c.a ∈ approx (Interval.ofRat c.a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_4
α β α' β' : Type
inst✝¹ : Approx α α'
inst✝ : Approx β β'
c : Color ℚ
⊢ ↑c.a ∈ approx (Interval.ofRat c.a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | constructor | E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
⊢ Diffe... | case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | intro d | case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s... | case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | apply osgood o d.continuousOn | case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s... | case mp.fa0
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOp... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | intro z0 z1 zs | case mp.fa0
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOp... | case mp.fa0
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOp... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.fa0
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGrou... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | rcases Metric.isOpen_iff.mp o (z0, z1) zs with ⟨r, rp, rs⟩ | case mp.fa0
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOp... | case mp.fa0.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpac... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.fa0
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGrou... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | have d0 : DifferentiableOn ℂ (fun z0 ↦ f (z0, z1)) (ball z0 r) := by
apply DifferentiableOn.comp d
exact DifferentiableOn.prod differentiableOn_id (differentiableOn_const _)
intro z0 z0s; apply rs; simp at z0s ⊢; assumption | case mp.fa0.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpac... | case mp.fa0.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpac... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.fa0.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : Norme... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | exact (analyticOn_iff_differentiableOn isOpen_ball).mpr d0 z0 (Metric.mem_ball_self rp) | case mp.fa0.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpac... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.fa0.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : Norme... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | apply DifferentiableOn.comp d | E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : Dif... | case hf
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | exact DifferentiableOn.prod differentiableOn_id (differentiableOn_const _) | case hf
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s... | case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | intro z0 z0s | case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s... | case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s... | Please generate a tactic in lean4 to solve the state.
STATE:
case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | apply rs | case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s... | case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen... | Please generate a tactic in lean4 to solve the state.
STATE:
case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | simp at z0s ⊢ | case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen... | case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen... | Please generate a tactic in lean4 to solve the state.
STATE:
case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | assumption | case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | intro z0 z1 zs | case mp.fa1
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOp... | case mp.fa1
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOp... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.fa1
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGrou... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | rcases Metric.isOpen_iff.mp o (z0, z1) zs with ⟨r, rp, rs⟩ | case mp.fa1
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOp... | case mp.fa1.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpac... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.fa1
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGrou... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | have d1 : DifferentiableOn ℂ (fun z1 ↦ f (z0, z1)) (ball z1 r) := by
apply DifferentiableOn.comp d
exact DifferentiableOn.prod (differentiableOn_const _) differentiableOn_id
intro z1 z1s; apply rs; simp at z1s ⊢; assumption | case mp.fa1.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpac... | case mp.fa1.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpac... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.fa1.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : Norme... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | exact (analyticOn_iff_differentiableOn isOpen_ball).mpr d1 z1 (Metric.mem_ball_self rp) | case mp.fa1.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpac... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.fa1.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : Norme... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | apply DifferentiableOn.comp d | E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
d : Dif... | case hf
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | exact DifferentiableOn.prod (differentiableOn_const _) differentiableOn_id | case hf
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s... | case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | intro z1 z1s | case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s... | case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s... | Please generate a tactic in lean4 to solve the state.
STATE:
case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | apply rs | case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s... | case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen... | Please generate a tactic in lean4 to solve the state.
STATE:
case st
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | simp at z1s ⊢ | case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen... | case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen... | Please generate a tactic in lean4 to solve the state.
STATE:
case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | assumption | case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case st.a
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | differentiable_iff_analytic2 | [36, 1] | [55, 37] | exact fun a ↦ a.differentiableOn | case mpr
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | contDiffAt_iff_analytic_at2 | [58, 1] | [66, 45] | constructor | E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
⊢ ContDi... | case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
... | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : Norme... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | contDiffAt_iff_analytic_at2 | [58, 1] | [66, 45] | intro d | case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
... | case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | contDiffAt_iff_analytic_at2 | [58, 1] | [66, 45] | rcases d.contDiffOn n1 with ⟨u, un, d⟩ | case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
... | case mp.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | contDiffAt_iff_analytic_at2 | [58, 1] | [66, 45] | rcases mem_nhds_iff.mp un with ⟨v, uv, vo, vx⟩ | case mp.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞... | case mp.intro.intro.intro.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : Comp... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGr... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | contDiffAt_iff_analytic_at2 | [58, 1] | [66, 45] | refine (differentiable_iff_analytic2 vo).mp ?_ _ vx | case mp.intro.intro.intro.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : Comp... | case mp.intro.intro.intro.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : Comp... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝²... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | contDiffAt_iff_analytic_at2 | [58, 1] | [66, 45] | exact (d.mono uv).differentiableOn (by norm_num) | case mp.intro.intro.intro.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : Comp... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro.intro
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝²... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | contDiffAt_iff_analytic_at2 | [58, 1] | [66, 45] | norm_num | E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n
d✝ : Con... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : Norme... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | contDiffAt_iff_analytic_at2 | [58, 1] | [66, 45] | intro a | case mpr
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n... | case mpr
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | contDiffAt_iff_analytic_at2 | [58, 1] | [66, 45] | exact a.contDiffAt.of_le le_top | case mpr
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | have m1 : ∀ t : ℝ, t ≤ 1 → t * abs z < 1 :=
fun _ m ↦ Right.mul_lt_one_of_le_of_lt_of_nonneg m z1 (Complex.abs.nonneg _) | z : ℂ
z1 : abs z < 1
⊢ abs (1 + z).log ≤ -(1 - abs z).log | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
⊢ abs (1 + z).log ≤ -(1 - abs z).log | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
⊢ abs (1 + z).log ≤ -(1 - abs z).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | have dr : ∀ t : ℝ, t ∈ uIcc 0 1 →
HasDerivAt (fun t : ℝ ↦ -Real.log (1 - t * abs z)) (- (-abs z / (1 - t * abs z))) t := by
intro t m
simp only [ge_iff_le, zero_le_one, uIcc_of_le, mem_Icc] at m
exact (((hasDerivAt_mul_const _).const_sub _).log ((sub_pos.mpr (m1 _ m.2)).ne')).neg | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
⊢ abs (1 + z).log ≤ -(1 - abs z).log | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
⊢ abs (1 + z).log ≤ -(1 - abs z).log | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
⊢ abs (1 + z).log ≤ -(1 - abs z).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | have ic : IntervalIntegrable (fun t ↦ z / (1 + t*z)) MeasureTheory.volume 0 1 := by
apply ContinuousOn.intervalIntegrable_of_Icc zero_le_one
apply continuousOn_const.div (Continuous.continuousOn (by continuity))
intro t ⟨t0,t1⟩
rw [←Complex.abs.ne_zero_iff]
apply ne_of_gt
calc abs (1 + t*z)
_ ≥ Complex.... | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
⊢ abs (1 + z).log ≤ -(1 - abs z).log | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
⊢ abs (1 + z).log... | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
⊢ abs (1 + z).log ≤ -(1 - abs z).... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | simp only [neg_div, neg_neg] at dr | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
⊢ abs (1 + z).log... | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
⊢ abs (1 + z).log ≤ -... | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
ic : IntervalIntegrable (fun t =>... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | have ir : IntervalIntegrable (fun t ↦ abs z / (1 - t * abs z)) MeasureTheory.volume 0 1 := by
apply ContinuousOn.intervalIntegrable_of_Icc zero_le_one
apply continuousOn_const.div (Continuous.continuousOn (by continuity))
intro t ⟨_,t1⟩; exact ne_of_gt (sub_pos.mpr (m1 _ t1)) | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
⊢ abs (1 + z).log ≤ -... | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
ir : IntervalIntegrab... | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | have fc := intervalIntegral.integral_eq_sub_of_hasDerivAt dc ic | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
ir : IntervalIntegrab... | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
ir : IntervalIntegrab... | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | have fr := intervalIntegral.integral_eq_sub_of_hasDerivAt dr ir | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
ir : IntervalIntegrab... | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
ir : IntervalIntegrab... | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | simp only [Complex.ofReal_one, one_mul, Complex.ofReal_zero, zero_mul, add_zero, Complex.log_one,
sub_zero, Real.log_one, neg_zero] at fc fr | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
ir : IntervalIntegrab... | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
ir : IntervalIntegrab... | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | rw [←fc, ←fr, ←Complex.norm_eq_abs] | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
ir : IntervalIntegrab... | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
ir : IntervalIntegrab... | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | clear dc dr fc fr | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (abs z / (1 - t * abs z)) t
ir : IntervalIntegrab... | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
ir : IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
⊢ ‖∫ (y : ℝ) in 0 ..1, z / (1 + ↑y * z)‖ ≤ ∫ (y : ℝ) in 0 ..1, abs z / (1 - y * abs z) | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | apply le_trans (intervalIntegral.norm_integral_le_integral_norm zero_le_one) ?_ | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
ir : IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
⊢ ‖∫ (y : ℝ) in 0 ..1, z / (1 + ↑y * z)‖ ≤ ∫ (y : ℝ) in 0 ..1, abs z / (1 - y * abs z) | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
ir : IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
⊢ ∫ (x : ℝ) in 0 ..1, ‖z / (1 + ↑x * z)‖ ≤ ∫ (y : ℝ) in 0 ..1, abs z / (1 - y * abs z) | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
ir : IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
⊢ ‖∫ (y : ℝ) in 0 ..1, z / (1 + ↑y * z)‖ ≤ ∫ (y :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | apply intervalIntegral.integral_mono_on zero_le_one ic.norm ir | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
ir : IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
⊢ ∫ (x : ℝ) in 0 ..1, ‖z / (1 + ↑x * z)‖ ≤ ∫ (y : ℝ) in 0 ..1, abs z / (1 - y * abs z) | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
ir : IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
⊢ ∀ x ∈ Icc 0 1, ‖z / (1 + ↑x * z)‖ ≤ abs z / (1 - x * abs z) | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
ir : IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
⊢ ∫ (x : ℝ) in 0 ..1, ‖z / (1 + ↑x * z)‖ ≤ ∫ (y :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | intro t ⟨t0,t1⟩ | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
ir : IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
⊢ ∀ x ∈ Icc 0 1, ‖z / (1 + ↑x * z)‖ ≤ abs z / (1 - x * abs z) | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
ir : IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
t : ℝ
t0 : 0 ≤ t
t1 : t ≤ 1
⊢ ‖z / (1 + ↑t * z)‖ ≤ abs z / (1 - t * abs z) | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
ir : IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
⊢ ∀ x ∈ Icc 0 1, ‖z / (1 + ↑x * z)‖ ≤ abs z / (1 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | simp only [norm_div, Complex.norm_eq_abs] | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
ir : IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
t : ℝ
t0 : 0 ≤ t
t1 : t ≤ 1
⊢ ‖z / (1 + ↑t * z)‖ ≤ abs z / (1 - t * abs z) | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
ir : IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
t : ℝ
t0 : 0 ≤ t
t1 : t ≤ 1
⊢ abs z / abs (1 + ↑t * z) ≤ abs z / (1 - t * abs z) | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
ir : IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
t : ℝ
t0 : 0 ≤ t
t1 : t ≤ 1
⊢ ‖z / (1 + ↑t * z)‖ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | apply div_le_div_of_nonneg_left (Complex.abs.nonneg _) (sub_pos.mpr (m1 _ t1)) ?_ | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
ir : IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
t : ℝ
t0 : 0 ≤ t
t1 : t ≤ 1
⊢ abs z / abs (1 + ↑t * z) ≤ abs z / (1 - t * abs z) | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
ir : IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
t : ℝ
t0 : 0 ≤ t
t1 : t ≤ 1
⊢ 1 - t * abs z ≤ abs (1 + ↑t * z) | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
ir : IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
t : ℝ
t0 : 0 ≤ t
t1 : t ≤ 1
⊢ abs z / abs (1 + ↑t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | calc abs (1 + t * z)
_ ≥ Complex.abs 1 - abs (t * z) := Complex.abs.le_add _ _
_ = 1 - t * abs z := by
simp only [map_one, map_mul, Complex.abs_ofReal, _root_.abs_of_nonneg t0] | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
ir : IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
t : ℝ
t0 : 0 ≤ t
t1 : t ≤ 1
⊢ 1 - t * abs z ≤ abs (1 + ↑t * z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
ic : IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1
ir : IntervalIntegrable (fun t => abs z / (1 - t * abs z)) MeasureTheory.volume 0 1
t : ℝ
t0 : 0 ≤ t
t1 : t ≤ 1
⊢ 1 - t * abs z ≤ abs... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | intro t m | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
⊢ ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : t ∈ uIcc 0 1
⊢ HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
⊢ ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | apply HasDerivAt.clog_real | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : t ∈ uIcc 0 1
⊢ HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t | case h₁
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : t ∈ uIcc 0 1
⊢ HasDerivAt (fun t => 1 + ↑t * z) z t
case h₂
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : t ∈ uIcc 0 1
⊢ 1 + ↑t * z ∈ slitPlane | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : t ∈ uIcc 0 1
⊢ HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | exact ((hasDerivAt_mul_const _).const_add _).comp_ofReal | case h₁
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : t ∈ uIcc 0 1
⊢ HasDerivAt (fun t => 1 + ↑t * z) z t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : t ∈ uIcc 0 1
⊢ HasDerivAt (fun t => 1 + ↑t * z) z t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | apply Complex.mem_slitPlane_of_norm_lt_one | case h₂
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : t ∈ uIcc 0 1
⊢ 1 + ↑t * z ∈ slitPlane | case h₂.hz
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : t ∈ uIcc 0 1
⊢ ‖↑t * z‖ < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : t ∈ uIcc 0 1
⊢ 1 + ↑t * z ∈ slitPlane
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal] | case h₂.hz
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : t ∈ uIcc 0 1
⊢ ‖↑t * z‖ < 1 | case h₂.hz
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : t ∈ uIcc 0 1
⊢ |t| * abs z < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.hz
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : t ∈ uIcc 0 1
⊢ ‖↑t * z‖ < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | simp only [ge_iff_le, zero_le_one, uIcc_of_le, mem_Icc] at m | case h₂.hz
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : t ∈ uIcc 0 1
⊢ |t| * abs z < 1 | case h₂.hz
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : 0 ≤ t ∧ t ≤ 1
⊢ |t| * abs z < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.hz
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : t ∈ uIcc 0 1
⊢ |t| * abs z < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | apply m1 | case h₂.hz
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : 0 ≤ t ∧ t ≤ 1
⊢ |t| * abs z < 1 | case h₂.hz.a
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : 0 ≤ t ∧ t ≤ 1
⊢ |t| ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.hz
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : 0 ≤ t ∧ t ≤ 1
⊢ |t| * abs z < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | simp only [abs_le] | case h₂.hz.a
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : 0 ≤ t ∧ t ≤ 1
⊢ |t| ≤ 1 | case h₂.hz.a
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : 0 ≤ t ∧ t ≤ 1
⊢ -1 ≤ t ∧ t ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.hz.a
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : 0 ≤ t ∧ t ≤ 1
⊢ |t| ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | exact ⟨by linarith, m.2⟩ | case h₂.hz.a
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : 0 ≤ t ∧ t ≤ 1
⊢ -1 ≤ t ∧ t ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.hz.a
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : 0 ≤ t ∧ t ≤ 1
⊢ -1 ≤ t ∧ t ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | linarith | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : 0 ≤ t ∧ t ≤ 1
⊢ -1 ≤ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
t : ℝ
m : 0 ≤ t ∧ t ≤ 1
⊢ -1 ≤ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | intro t m | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
⊢ ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
t : ℝ
m : t ∈ uIcc 0 1
⊢ HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
⊢ ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | simp only [ge_iff_le, zero_le_one, uIcc_of_le, mem_Icc] at m | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
t : ℝ
m : t ∈ uIcc 0 1
⊢ HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
t : ℝ
m : 0 ≤ t ∧ t ≤ 1
⊢ HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
t : ℝ
m : t ∈ uIcc 0 1
⊢ HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | exact (((hasDerivAt_mul_const _).const_sub _).log ((sub_pos.mpr (m1 _ m.2)).ne')).neg | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
t : ℝ
m : 0 ≤ t ∧ t ≤ 1
⊢ HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
t : ℝ
m : 0 ≤ t ∧ t ≤ 1
⊢ HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | apply ContinuousOn.intervalIntegrable_of_Icc zero_le_one | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
⊢ IntervalIntegrable (fun t => z / (1 + ↑t * z)) MeasureTheory.volume 0 1 | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
⊢ ContinuousOn (fun t => z / (1 + ↑t * z)) (Icc 0 1) | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
⊢ IntervalIntegrable (fun t => z ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | apply continuousOn_const.div (Continuous.continuousOn (by continuity)) | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
⊢ ContinuousOn (fun t => z / (1 + ↑t * z)) (Icc 0 1) | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
⊢ ∀ x ∈ Icc 0 1, 1 + ↑x * z ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
⊢ ContinuousOn (fun t => z / (1 +... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Log1p.lean | Complex.abs_log_one_add_le | [14, 1] | [65, 80] | intro t ⟨t0,t1⟩ | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
⊢ ∀ x ∈ Icc 0 1, 1 + ↑x * z ≠ 0 | z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
t : ℝ
t0 : 0 ≤ t
t1 : t ≤ 1
⊢ 1 + ↑t * z ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
z1 : abs z < 1
m1 : ∀ t ≤ 1, t * abs z < 1
dc : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => (1 + ↑t * z).log) (z / (1 + ↑t * z)) t
dr : ∀ t ∈ uIcc 0 1, HasDerivAt (fun t => -(1 - t * abs z).log) (-(-abs z / (1 - t * abs z))) t
⊢ ∀ x ∈ Icc 0 1, 1 + ↑x * z ≠ 0
T... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.