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stringclasses 147
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stringclasses 147
values | file_path
stringlengths 7
101
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stringlengths 1
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| start
stringlengths 6
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stringlengths 6
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|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Subharmonic.lean | SubharmonicOn.hartogs | [1043, 1] | [1090, 30] | bound | S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℕ → ℂ → ℝ
s k : Set ℂ
c b : ℝ
fs : ∀ (n : ℕ), SubharmonicOn (f n) s
fb✝ : ∀ (n : ℕ), ∀ z ∈ s, f n z ≤ b
ck : IsCompact k
ks : k ⊆ interior s
bc : c < b
f' : ℕ → ℂ → ℝ
hf' : (fun n z => f n z - b) = f'
g : ℕ → ℂ → ℝ≥0∞
hg : (fun n z => ENNReal.ofReal (-f' n z)) = g
fs' : ∀ (n : ℕ), SubharmonicOn (f' n) s
fn' : ∀ (n : ℕ), ∀ z ∈ interior s, f' n z ≤ 0
gs : ∀ (n : ℕ), SuperharmonicOn (g n) (interior s)
z : ℂ
zs : z ∈ interior s
fc : ∀ d > c, ∀ᶠ (n : ℕ) in atTop, f n z ≤ d
d : ℝ≥0∞
dc : d < ENNReal.ofReal (b - c)
df : d ≠ ⊤
dc' : b - d.toReal > c
n : ℕ
fb : f n z ≤ b - d.toReal
⊢ ENNReal.ofReal (b - f n z) ≥ ENNReal.ofReal (b - (b - d.toReal)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℕ → ℂ → ℝ
s k : Set ℂ
c b : ℝ
fs : ∀ (n : ℕ), SubharmonicOn (f n) s
fb✝ : ∀ (n : ℕ), ∀ z ∈ s, f n z ≤ b
ck : IsCompact k
ks : k ⊆ interior s
bc : c < b
f' : ℕ → ℂ → ℝ
hf' : (fun n z => f n z - b) = f'
g : ℕ → ℂ → ℝ≥0∞
hg : (fun n z => ENNReal.ofReal (-f' n z)) = g
fs' : ∀ (n : ℕ), SubharmonicOn (f' n) s
fn' : ∀ (n : ℕ), ∀ z ∈ interior s, f' n z ≤ 0
gs : ∀ (n : ℕ), SuperharmonicOn (g n) (interior s)
z : ℂ
zs : z ∈ interior s
fc : ∀ d > c, ∀ᶠ (n : ℕ) in atTop, f n z ≤ d
d : ℝ≥0∞
dc : d < ENNReal.ofReal (b - c)
df : d ≠ ⊤
dc' : b - d.toReal > c
n : ℕ
fb : f n z ≤ b - d.toReal
⊢ ENNReal.ofReal (b - f n z) ≥ ENNReal.ofReal (b - (b - d.toReal))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Subharmonic.lean | SubharmonicOn.hartogs | [1043, 1] | [1090, 30] | ring_nf | S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℕ → ℂ → ℝ
s k : Set ℂ
c b : ℝ
fs : ∀ (n : ℕ), SubharmonicOn (f n) s
fb✝ : ∀ (n : ℕ), ∀ z ∈ s, f n z ≤ b
ck : IsCompact k
ks : k ⊆ interior s
bc : c < b
f' : ℕ → ℂ → ℝ
hf' : (fun n z => f n z - b) = f'
g : ℕ → ℂ → ℝ≥0∞
hg : (fun n z => ENNReal.ofReal (-f' n z)) = g
fs' : ∀ (n : ℕ), SubharmonicOn (f' n) s
fn' : ∀ (n : ℕ), ∀ z ∈ interior s, f' n z ≤ 0
gs : ∀ (n : ℕ), SuperharmonicOn (g n) (interior s)
z : ℂ
zs : z ∈ interior s
fc : ∀ d > c, ∀ᶠ (n : ℕ) in atTop, f n z ≤ d
d : ℝ≥0∞
dc : d < ENNReal.ofReal (b - c)
df : d ≠ ⊤
dc' : b - d.toReal > c
n : ℕ
fb : f n z ≤ b - d.toReal
⊢ ENNReal.ofReal (b - (b - d.toReal)) = ENNReal.ofReal d.toReal | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℕ → ℂ → ℝ
s k : Set ℂ
c b : ℝ
fs : ∀ (n : ℕ), SubharmonicOn (f n) s
fb✝ : ∀ (n : ℕ), ∀ z ∈ s, f n z ≤ b
ck : IsCompact k
ks : k ⊆ interior s
bc : c < b
f' : ℕ → ℂ → ℝ
hf' : (fun n z => f n z - b) = f'
g : ℕ → ℂ → ℝ≥0∞
hg : (fun n z => ENNReal.ofReal (-f' n z)) = g
fs' : ∀ (n : ℕ), SubharmonicOn (f' n) s
fn' : ∀ (n : ℕ), ∀ z ∈ interior s, f' n z ≤ 0
gs : ∀ (n : ℕ), SuperharmonicOn (g n) (interior s)
z : ℂ
zs : z ∈ interior s
fc : ∀ d > c, ∀ᶠ (n : ℕ) in atTop, f n z ≤ d
d : ℝ≥0∞
dc : d < ENNReal.ofReal (b - c)
df : d ≠ ⊤
dc' : b - d.toReal > c
n : ℕ
fb : f n z ≤ b - d.toReal
⊢ ENNReal.ofReal (b - (b - d.toReal)) = ENNReal.ofReal d.toReal
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Subharmonic.lean | SubharmonicOn.hartogs | [1043, 1] | [1090, 30] | rw [ENNReal.ofReal_toReal df] | S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℕ → ℂ → ℝ
s k : Set ℂ
c b : ℝ
fs : ∀ (n : ℕ), SubharmonicOn (f n) s
fb✝ : ∀ (n : ℕ), ∀ z ∈ s, f n z ≤ b
ck : IsCompact k
ks : k ⊆ interior s
bc : c < b
f' : ℕ → ℂ → ℝ
hf' : (fun n z => f n z - b) = f'
g : ℕ → ℂ → ℝ≥0∞
hg : (fun n z => ENNReal.ofReal (-f' n z)) = g
fs' : ∀ (n : ℕ), SubharmonicOn (f' n) s
fn' : ∀ (n : ℕ), ∀ z ∈ interior s, f' n z ≤ 0
gs : ∀ (n : ℕ), SuperharmonicOn (g n) (interior s)
z : ℂ
zs : z ∈ interior s
fc : ∀ d > c, ∀ᶠ (n : ℕ) in atTop, f n z ≤ d
d : ℝ≥0∞
dc : d < ENNReal.ofReal (b - c)
df : d ≠ ⊤
dc' : b - d.toReal > c
n : ℕ
fb : f n z ≤ b - d.toReal
⊢ ENNReal.ofReal d.toReal = d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℕ → ℂ → ℝ
s k : Set ℂ
c b : ℝ
fs : ∀ (n : ℕ), SubharmonicOn (f n) s
fb✝ : ∀ (n : ℕ), ∀ z ∈ s, f n z ≤ b
ck : IsCompact k
ks : k ⊆ interior s
bc : c < b
f' : ℕ → ℂ → ℝ
hf' : (fun n z => f n z - b) = f'
g : ℕ → ℂ → ℝ≥0∞
hg : (fun n z => ENNReal.ofReal (-f' n z)) = g
fs' : ∀ (n : ℕ), SubharmonicOn (f' n) s
fn' : ∀ (n : ℕ), ∀ z ∈ interior s, f' n z ≤ 0
gs : ∀ (n : ℕ), SuperharmonicOn (g n) (interior s)
z : ℂ
zs : z ∈ interior s
fc : ∀ d > c, ∀ᶠ (n : ℕ) in atTop, f n z ≤ d
d : ℝ≥0∞
dc : d < ENNReal.ofReal (b - c)
df : d ≠ ⊤
dc' : b - d.toReal > c
n : ℕ
fb : f n z ≤ b - d.toReal
⊢ ENNReal.ofReal d.toReal = d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Subharmonic.lean | SubharmonicOn.hartogs | [1043, 1] | [1090, 30] | rw [ENNReal.ofReal_lt_ofReal_iff (sub_pos.mpr bc)] | S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℕ → ℂ → ℝ
s k : Set ℂ
c b : ℝ
fs : ∀ (n : ℕ), SubharmonicOn (f n) s
fb : ∀ (n : ℕ), ∀ z ∈ s, f n z ≤ b
fc : ∀ z ∈ s, ∀ d > c, ∀ᶠ (n : ℕ) in atTop, f n z ≤ d
ck : IsCompact k
ks : k ⊆ interior s
bc : c < b
f' : ℕ → ℂ → ℝ
hf' : (fun n z => f n z - b) = f'
g : ℕ → ℂ → ℝ≥0∞
hg : (fun n z => ENNReal.ofReal (-f' n z)) = g
fs' : ∀ (n : ℕ), SubharmonicOn (f' n) s
fn' : ∀ (n : ℕ), ∀ z ∈ interior s, f' n z ≤ 0
gs : ∀ (n : ℕ), SuperharmonicOn (g n) (interior s)
gc : ∀ z ∈ interior s, liminf (fun n => g n z) atTop ≥ ENNReal.ofReal (b - c)
ks' : k ⊆ interior (interior s)
h : ∀ d < ENNReal.ofReal (b - c), ∀ᶠ (n : ℕ) in atTop, ∀ z ∈ k, g n z ≥ d
d : ℝ
dc : d > c
⊢ ENNReal.ofReal (b - d) < ENNReal.ofReal (b - c) | S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℕ → ℂ → ℝ
s k : Set ℂ
c b : ℝ
fs : ∀ (n : ℕ), SubharmonicOn (f n) s
fb : ∀ (n : ℕ), ∀ z ∈ s, f n z ≤ b
fc : ∀ z ∈ s, ∀ d > c, ∀ᶠ (n : ℕ) in atTop, f n z ≤ d
ck : IsCompact k
ks : k ⊆ interior s
bc : c < b
f' : ℕ → ℂ → ℝ
hf' : (fun n z => f n z - b) = f'
g : ℕ → ℂ → ℝ≥0∞
hg : (fun n z => ENNReal.ofReal (-f' n z)) = g
fs' : ∀ (n : ℕ), SubharmonicOn (f' n) s
fn' : ∀ (n : ℕ), ∀ z ∈ interior s, f' n z ≤ 0
gs : ∀ (n : ℕ), SuperharmonicOn (g n) (interior s)
gc : ∀ z ∈ interior s, liminf (fun n => g n z) atTop ≥ ENNReal.ofReal (b - c)
ks' : k ⊆ interior (interior s)
h : ∀ d < ENNReal.ofReal (b - c), ∀ᶠ (n : ℕ) in atTop, ∀ z ∈ k, g n z ≥ d
d : ℝ
dc : d > c
⊢ b - d < b - c | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℕ → ℂ → ℝ
s k : Set ℂ
c b : ℝ
fs : ∀ (n : ℕ), SubharmonicOn (f n) s
fb : ∀ (n : ℕ), ∀ z ∈ s, f n z ≤ b
fc : ∀ z ∈ s, ∀ d > c, ∀ᶠ (n : ℕ) in atTop, f n z ≤ d
ck : IsCompact k
ks : k ⊆ interior s
bc : c < b
f' : ℕ → ℂ → ℝ
hf' : (fun n z => f n z - b) = f'
g : ℕ → ℂ → ℝ≥0∞
hg : (fun n z => ENNReal.ofReal (-f' n z)) = g
fs' : ∀ (n : ℕ), SubharmonicOn (f' n) s
fn' : ∀ (n : ℕ), ∀ z ∈ interior s, f' n z ≤ 0
gs : ∀ (n : ℕ), SuperharmonicOn (g n) (interior s)
gc : ∀ z ∈ interior s, liminf (fun n => g n z) atTop ≥ ENNReal.ofReal (b - c)
ks' : k ⊆ interior (interior s)
h : ∀ d < ENNReal.ofReal (b - c), ∀ᶠ (n : ℕ) in atTop, ∀ z ∈ k, g n z ≥ d
d : ℝ
dc : d > c
⊢ ENNReal.ofReal (b - d) < ENNReal.ofReal (b - c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Subharmonic.lean | SubharmonicOn.hartogs | [1043, 1] | [1090, 30] | simpa only [sub_lt_sub_iff_left] | S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℕ → ℂ → ℝ
s k : Set ℂ
c b : ℝ
fs : ∀ (n : ℕ), SubharmonicOn (f n) s
fb : ∀ (n : ℕ), ∀ z ∈ s, f n z ≤ b
fc : ∀ z ∈ s, ∀ d > c, ∀ᶠ (n : ℕ) in atTop, f n z ≤ d
ck : IsCompact k
ks : k ⊆ interior s
bc : c < b
f' : ℕ → ℂ → ℝ
hf' : (fun n z => f n z - b) = f'
g : ℕ → ℂ → ℝ≥0∞
hg : (fun n z => ENNReal.ofReal (-f' n z)) = g
fs' : ∀ (n : ℕ), SubharmonicOn (f' n) s
fn' : ∀ (n : ℕ), ∀ z ∈ interior s, f' n z ≤ 0
gs : ∀ (n : ℕ), SuperharmonicOn (g n) (interior s)
gc : ∀ z ∈ interior s, liminf (fun n => g n z) atTop ≥ ENNReal.ofReal (b - c)
ks' : k ⊆ interior (interior s)
h : ∀ d < ENNReal.ofReal (b - c), ∀ᶠ (n : ℕ) in atTop, ∀ z ∈ k, g n z ≥ d
d : ℝ
dc : d > c
⊢ b - d < b - c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℕ → ℂ → ℝ
s k : Set ℂ
c b : ℝ
fs : ∀ (n : ℕ), SubharmonicOn (f n) s
fb : ∀ (n : ℕ), ∀ z ∈ s, f n z ≤ b
fc : ∀ z ∈ s, ∀ d > c, ∀ᶠ (n : ℕ) in atTop, f n z ≤ d
ck : IsCompact k
ks : k ⊆ interior s
bc : c < b
f' : ℕ → ℂ → ℝ
hf' : (fun n z => f n z - b) = f'
g : ℕ → ℂ → ℝ≥0∞
hg : (fun n z => ENNReal.ofReal (-f' n z)) = g
fs' : ∀ (n : ℕ), SubharmonicOn (f' n) s
fn' : ∀ (n : ℕ), ∀ z ∈ interior s, f' n z ≤ 0
gs : ∀ (n : ℕ), SuperharmonicOn (g n) (interior s)
gc : ∀ z ∈ interior s, liminf (fun n => g n z) atTop ≥ ENNReal.ofReal (b - c)
ks' : k ⊆ interior (interior s)
h : ∀ d < ENNReal.ofReal (b - c), ∀ᶠ (n : ℕ) in atTop, ∀ z ∈ k, g n z ≥ d
d : ℝ
dc : d > c
⊢ b - d < b - c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Grid.lean | Grid.square_dz | [60, 1] | [75, 106] | rw [square] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
⊢ (square lo hi n).dz.1 = (square lo hi n).dz.2 | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
⊢ (let c := ((lo.1 + hi.1) / 2, (lo.2 + hi.2) / 2);
let d := hi - lo;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := ((lo.1 + hi.1) / 2, (lo.2 + hi.2) / 2);
let d := hi - lo;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
⊢ (square lo hi n).dz.1 = (square lo hi n).dz.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Grid.lean | Grid.square_dz | [60, 1] | [75, 106] | generalize ((lo.1 + hi.1) / 2, (lo.2 + hi.2) / 2) = c | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
⊢ (let c := ((lo.1 + hi.1) / 2, (lo.2 + hi.2) / 2);
let d := hi - lo;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := ((lo.1 + hi.1) / 2, (lo.2 + hi.2) / 2);
let d := hi - lo;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2 | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c : ℚ × ℚ
⊢ (let c := c;
let d := hi - lo;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := c;
let d := hi - lo;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
⊢ (let c := ((lo.1 + hi.1) / 2, (lo.2 + hi.2) / 2);
let d := hi - lo;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := ((lo.1 + hi.1) / 2, (lo.2 + hi.2) / 2);
let d := hi - lo;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Grid.lean | Grid.square_dz | [60, 1] | [75, 106] | generalize hi - lo = d | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c : ℚ × ℚ
⊢ (let c := c;
let d := hi - lo;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := c;
let d := hi - lo;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2 | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
⊢ (let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c : ℚ × ℚ
⊢ (let c := c;
let d := hi - lo;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := c;
let d := hi - lo;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Grid.lean | Grid.square_dz | [60, 1] | [75, 106] | generalize hdx : max d.1 d.2 / (n : ℚ) = dx | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
⊢ (let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2 | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
⊢ (let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
⊢ (let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Grid.lean | Grid.square_dz | [60, 1] | [75, 106] | generalize hn1 : max 1 (d.1 / dx).ceil.toNat = n1 | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
⊢ (let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2 | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
⊢ (let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
⊢ (let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Grid.lean | Grid.square_dz | [60, 1] | [75, 106] | generalize hn2 : max 1 (d.2 / dx).ceil.toNat = n2 | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
⊢ (let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2 | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
n2 : ℕ
hn2 : max 1 (d.2 / dx).ceil.toNat = n2
⊢ (let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
⊢ (let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Grid.lean | Grid.square_dz | [60, 1] | [75, 106] | have n1z : (n1 : ℚ) ≠ 0 := by
rw [←hn1]; exact Nat.cast_ne_zero.mpr (ne_of_gt (lt_max_of_lt_left zero_lt_one)) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
n2 : ℕ
hn2 : max 1 (d.2 / dx).ceil.toNat = n2
⊢ (let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2 | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
n2 : ℕ
hn2 : max 1 (d.2 / dx).ceil.toNat = n2
n1z : ↑n1 ≠ 0
⊢ (let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
n2 : ℕ
hn2 : max 1 (d.2 / dx).ceil.toNat = n2
⊢ (let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Grid.lean | Grid.square_dz | [60, 1] | [75, 106] | have n2z : (n2 : ℚ) ≠ 0 := by
rw [←hn2]; exact Nat.cast_ne_zero.mpr (ne_of_gt (lt_max_of_lt_left zero_lt_one)) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
n2 : ℕ
hn2 : max 1 (d.2 / dx).ceil.toNat = n2
n1z : ↑n1 ≠ 0
⊢ (let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2 | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
n2 : ℕ
hn2 : max 1 (d.2 / dx).ceil.toNat = n2
n1z : ↑n1 ≠ 0
n2z : ↑n2 ≠ 0
⊢ (let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
n2 : ℕ
hn2 : max 1 (d.2 / dx).ceil.toNat = n2
n1z : ↑n1 ≠ 0
⊢ (let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Grid.lean | Grid.square_dz | [60, 1] | [75, 106] | simp only [dz, hdx, hn1, hn2, Prod.smul_mk, smul_eq_mul, Prod.fst_add, Prod.fst_sub, add_sub_sub_cancel,
add_div, mul_div_assoc, div_self n1z, mul_one, add_halves', Prod.snd_add, Prod.snd_sub, div_self n2z] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
n2 : ℕ
hn2 : max 1 (d.2 / dx).ceil.toNat = n2
n1z : ↑n1 ≠ 0
n2z : ↑n2 ≠ 0
⊢ (let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
n2 : ℕ
hn2 : max 1 (d.2 / dx).ceil.toNat = n2
n1z : ↑n1 ≠ 0
n2z : ↑n2 ≠ 0
⊢ (let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.1 =
(let c := c;
let d := d;
let dx := max d.1 d.2 / ↑n;
let n := (max 1 (d.1 / dx).ceil.toNat, max 1 (d.2 / dx).ceil.toNat);
let h := (dx / 2) • (↑n.1, ↑n.2);
{ lo := c - h, hi := c + h, n := n }).dz.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Grid.lean | Grid.square_dz | [60, 1] | [75, 106] | rw [←hn1] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
n2 : ℕ
hn2 : max 1 (d.2 / dx).ceil.toNat = n2
⊢ ↑n1 ≠ 0 | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
n2 : ℕ
hn2 : max 1 (d.2 / dx).ceil.toNat = n2
⊢ ↑(max 1 (d.1 / dx).ceil.toNat) ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
n2 : ℕ
hn2 : max 1 (d.2 / dx).ceil.toNat = n2
⊢ ↑n1 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Grid.lean | Grid.square_dz | [60, 1] | [75, 106] | exact Nat.cast_ne_zero.mpr (ne_of_gt (lt_max_of_lt_left zero_lt_one)) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
n2 : ℕ
hn2 : max 1 (d.2 / dx).ceil.toNat = n2
⊢ ↑(max 1 (d.1 / dx).ceil.toNat) ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
n2 : ℕ
hn2 : max 1 (d.2 / dx).ceil.toNat = n2
⊢ ↑(max 1 (d.1 / dx).ceil.toNat) ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Grid.lean | Grid.square_dz | [60, 1] | [75, 106] | rw [←hn2] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
n2 : ℕ
hn2 : max 1 (d.2 / dx).ceil.toNat = n2
n1z : ↑n1 ≠ 0
⊢ ↑n2 ≠ 0 | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
n2 : ℕ
hn2 : max 1 (d.2 / dx).ceil.toNat = n2
n1z : ↑n1 ≠ 0
⊢ ↑(max 1 (d.2 / dx).ceil.toNat) ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
n2 : ℕ
hn2 : max 1 (d.2 / dx).ceil.toNat = n2
n1z : ↑n1 ≠ 0
⊢ ↑n2 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Grid.lean | Grid.square_dz | [60, 1] | [75, 106] | exact Nat.cast_ne_zero.mpr (ne_of_gt (lt_max_of_lt_left zero_lt_one)) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
n2 : ℕ
hn2 : max 1 (d.2 / dx).ceil.toNat = n2
n1z : ↑n1 ≠ 0
⊢ ↑(max 1 (d.2 / dx).ceil.toNat) ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
lo hi : ℚ × ℚ
n : ℕ
c d : ℚ × ℚ
dx : ℚ
hdx : max d.1 d.2 / ↑n = dx
n1 : ℕ
hn1 : max 1 (d.1 / dx).ceil.toNat = n1
n2 : ℕ
hn2 : max 1 (d.2 / dx).ceil.toNat = n2
n1z : ↑n1 ≠ 0
⊢ ↑(max 1 (d.2 / dx).ceil.toNat) ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | mem_analyticGroupoid | [57, 1] | [67, 6] | rfl | 𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E A : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : TopologicalSpace A
I : ModelWithCorners 𝕜 E A
f : PartialHomeomorph A A
⊢ f ∈ analyticGroupoid I ↔
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (↑I ∘ ↑f ∘ ↑I.symm) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I ∘ ↑f ∘ ↑I.symm '' (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ⊆ interior (range ↑I)) ∧
AnalyticOn 𝕜 (↑I ∘ ↑f.symm ∘ ↑I.symm) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I ∘ ↑f.symm ∘ ↑I.symm '' (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ⊆ interior (range ↑I) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E A : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : TopologicalSpace A
I : ModelWithCorners 𝕜 E A
f : PartialHomeomorph A A
⊢ f ∈ analyticGroupoid I ↔
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (↑I ∘ ↑f ∘ ↑I.symm) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I ∘ ↑f ∘ ↑I.symm '' (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ⊆ interior (range ↑I)) ∧
AnalyticOn 𝕜 (↑I ∘ ↑f.symm ∘ ↑I.symm) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I ∘ ↑f.symm ∘ ↑I.symm '' (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ⊆ interior (range ↑I)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | have er : (range fun x : A × B ↦ (I x.1, J x.2)) = range I ×ˢ range J := range_prod_map | 𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
fa : f ∈ analyticGroupoid I
ga : g ∈ analyticGroupoid J
⊢ f.prod g ∈ analyticGroupoid (I.prod J) | 𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
fa : f ∈ analyticGroupoid I
ga : g ∈ analyticGroupoid J
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
⊢ f.prod g ∈ analyticGroupoid (I.prod J) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
fa : f ∈ analyticGroupoid I
ga : g ∈ analyticGroupoid J
⊢ f.prod g ∈ analyticGroupoid (I.prod J)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | have ei : interior (range fun x : A × B ↦ (I x.1, J x.2)) =
interior (range I) ×ˢ interior (range J) := by rw [er, interior_prod_eq] | 𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
fa : f ∈ analyticGroupoid I
ga : g ∈ analyticGroupoid J
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
⊢ f.prod g ∈ analyticGroupoid (I.prod J) | 𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
fa : f ∈ analyticGroupoid I
ga : g ∈ analyticGroupoid J
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
⊢ f.prod g ∈ analyticGroupoid (I.prod J) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
fa : f ∈ analyticGroupoid I
ga : g ∈ analyticGroupoid J
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
⊢ f.prod g ∈ analyticGroupoid (I.prod J)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | simp only [mem_analyticGroupoid, Function.comp, image_subset_iff] at fa ga ⊢ | 𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
fa : f ∈ analyticGroupoid I
ga : g ∈ analyticGroupoid J
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
⊢ f.prod g ∈ analyticGroupoid (I.prod J) | 𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ f.prod g ∈ contDiffGroupoid ⊤ (I.prod J) ∧
(AnalyticOn 𝕜 (fun x => ↑(I.prod J) (↑(f.prod g) (↑(I.prod J).symm x)))
(↑(I.prod J).symm ⁻¹' (f.prod g).source ∩ interior (range ↑(I.prod J))) ∧
↑(I.prod J).symm ⁻¹' (f.prod g).source ∩ interior (range ↑(I.prod J)) ⊆
(fun x => ↑(I.prod J) (↑(f.prod g) (↑(I.prod J).symm x))) ⁻¹' interior (range ↑(I.prod J))) ∧
AnalyticOn 𝕜 (fun x => ↑(I.prod J) (↑(f.prod g).symm (↑(I.prod J).symm x)))
(↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J))) ∧
↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J)) ⊆
(fun x => ↑(I.prod J) (↑(f.prod g).symm (↑(I.prod J).symm x))) ⁻¹' interior (range ↑(I.prod J)) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
fa : f ∈ analyticGroupoid I
ga : g ∈ analyticGroupoid J
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
⊢ f.prod g ∈ analyticGroupoid (I.prod J)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | refine ⟨contDiffGroupoid_prod fa.1 ga.1, ⟨?_, ?_⟩, ⟨?_, ?_⟩⟩ | 𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ f.prod g ∈ contDiffGroupoid ⊤ (I.prod J) ∧
(AnalyticOn 𝕜 (fun x => ↑(I.prod J) (↑(f.prod g) (↑(I.prod J).symm x)))
(↑(I.prod J).symm ⁻¹' (f.prod g).source ∩ interior (range ↑(I.prod J))) ∧
↑(I.prod J).symm ⁻¹' (f.prod g).source ∩ interior (range ↑(I.prod J)) ⊆
(fun x => ↑(I.prod J) (↑(f.prod g) (↑(I.prod J).symm x))) ⁻¹' interior (range ↑(I.prod J))) ∧
AnalyticOn 𝕜 (fun x => ↑(I.prod J) (↑(f.prod g).symm (↑(I.prod J).symm x)))
(↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J))) ∧
↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J)) ⊆
(fun x => ↑(I.prod J) (↑(f.prod g).symm (↑(I.prod J).symm x))) ⁻¹' interior (range ↑(I.prod J)) | case refine_1
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑(I.prod J) (↑(f.prod g) (↑(I.prod J).symm x)))
(↑(I.prod J).symm ⁻¹' (f.prod g).source ∩ interior (range ↑(I.prod J)))
case refine_2
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ ↑(I.prod J).symm ⁻¹' (f.prod g).source ∩ interior (range ↑(I.prod J)) ⊆
(fun x => ↑(I.prod J) (↑(f.prod g) (↑(I.prod J).symm x))) ⁻¹' interior (range ↑(I.prod J))
case refine_3
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑(I.prod J) (↑(f.prod g).symm (↑(I.prod J).symm x)))
(↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J)))
case refine_4
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ ↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J)) ⊆
(fun x => ↑(I.prod J) (↑(f.prod g).symm (↑(I.prod J).symm x))) ⁻¹' interior (range ↑(I.prod J)) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ f.prod g ∈ contDiffGroupoid ⊤ (I.prod J) ∧
(AnalyticOn 𝕜 (fun x => ↑(I.prod J) (↑(f.prod g) (↑(I.prod J).symm x)))
(↑(I.prod J).symm ⁻¹' (f.prod g).source ∩ interior (range ↑(I.prod J))) ∧
↑(I.prod J).symm ⁻¹' (f.prod g).source ∩ interior (range ↑(I.prod J)) ⊆
(fun x => ↑(I.prod J) (↑(f.prod g) (↑(I.prod J).symm x))) ⁻¹' interior (range ↑(I.prod J))) ∧
AnalyticOn 𝕜 (fun x => ↑(I.prod J) (↑(f.prod g).symm (↑(I.prod J).symm x)))
(↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J))) ∧
↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J)) ⊆
(fun x => ↑(I.prod J) (↑(f.prod g).symm (↑(I.prod J).symm x))) ⁻¹' interior (range ↑(I.prod J))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | rw [er, interior_prod_eq] | 𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
fa : f ∈ analyticGroupoid I
ga : g ∈ analyticGroupoid J
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
⊢ interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
fa : f ∈ analyticGroupoid I
ga : g ∈ analyticGroupoid J
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
⊢ interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | apply AnalyticOn.prod | case refine_1
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑(I.prod J) (↑(f.prod g) (↑(I.prod J).symm x)))
(↑(I.prod J).symm ⁻¹' (f.prod g).source ∩ interior (range ↑(I.prod J))) | case refine_1.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑I (↑(f.prod g) (↑(I.prod J).symm x)).1)
(↑(I.prod J).symm ⁻¹' (f.prod g).source ∩ interior (range ↑(I.prod J)))
case refine_1.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑J (↑(f.prod g) (↑(I.prod J).symm x)).2)
(↑(I.prod J).symm ⁻¹' (f.prod g).source ∩ interior (range ↑(I.prod J))) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑(I.prod J) (↑(f.prod g) (↑(I.prod J).symm x)))
(↑(I.prod J).symm ⁻¹' (f.prod g).source ∩ interior (range ↑(I.prod J)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | simp only [ModelWithCorners.symm, modelWithCorners_prod_toPartialEquiv,
PartialEquiv.prod_symm, PartialEquiv.prod_coe, ModelWithCorners.toPartialEquiv_coe_symm,
PartialHomeomorph.prod_apply, PartialHomeomorph.prod_toPartialEquiv,
PartialEquiv.prod_source] | case refine_1.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑I (↑(f.prod g) (↑(I.prod J).symm x)).1)
(↑(I.prod J).symm ⁻¹' (f.prod g).source ∩ interior (range ↑(I.prod J))) | case refine_1.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x.1)))
((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' f.source ×ˢ g.source ∩ interior (range ↑(I.prod J))) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑I (↑(f.prod g) (↑(I.prod J).symm x)).1)
(↑(I.prod J).symm ⁻¹' (f.prod g).source ∩ interior (range ↑(I.prod J)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | refine fa.2.1.1.comp (analyticOn_fst _) ?_ | case refine_1.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x.1)))
((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' f.source ×ˢ g.source ∩ interior (range ↑(I.prod J))) | case refine_1.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ MapsTo (fun x => x.1) ((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' f.source ×ˢ g.source ∩ interior (range ↑(I.prod J)))
(↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x.1)))
((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' f.source ×ˢ g.source ∩ interior (range ↑(I.prod J)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | intro x m | case refine_1.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ MapsTo (fun x => x.1) ((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' f.source ×ˢ g.source ∩ interior (range ↑(I.prod J)))
(↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) | case refine_1.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : x ∈ (fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' f.source ×ˢ g.source ∩ interior (range ↑(I.prod J))
⊢ (fun x => x.1) x ∈ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ MapsTo (fun x => x.1) ((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' f.source ×ˢ g.source ∩ interior (range ↑(I.prod J)))
(↑I.symm ⁻¹' f.source ∩ interior (range ↑I))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | simp only [ModelWithCorners.prod, ModelWithCorners.source_eq, mem_univ, and_self, setOf_true,
PartialEquiv.prod_target, ModelWithCorners.target_eq, ModelWithCorners.mk_coe,
mem_inter_iff, mem_preimage, mem_prod] at m ⊢ | case refine_1.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : x ∈ (fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' f.source ×ˢ g.source ∩ interior (range ↑(I.prod J))
⊢ (fun x => x.1) x ∈ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I) | case refine_1.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : (↑I.symm x.1 ∈ f.source ∧ ↑J.symm x.2 ∈ g.source) ∧ x ∈ interior (range fun x => (↑I x.1, ↑J x.2))
⊢ ↑I.symm x.1 ∈ f.source ∧ x.1 ∈ interior (range ↑I) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : x ∈ (fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' f.source ×ˢ g.source ∩ interior (range ↑(I.prod J))
⊢ (fun x => x.1) x ∈ ↑I.symm ⁻¹' f.source ∩ interior (range ↑I)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | exact ⟨m.1.1, (subset_of_eq ei m.2).1⟩ | case refine_1.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : (↑I.symm x.1 ∈ f.source ∧ ↑J.symm x.2 ∈ g.source) ∧ x ∈ interior (range fun x => (↑I x.1, ↑J x.2))
⊢ ↑I.symm x.1 ∈ f.source ∧ x.1 ∈ interior (range ↑I) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : (↑I.symm x.1 ∈ f.source ∧ ↑J.symm x.2 ∈ g.source) ∧ x ∈ interior (range fun x => (↑I x.1, ↑J x.2))
⊢ ↑I.symm x.1 ∈ f.source ∧ x.1 ∈ interior (range ↑I)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | simp only [ModelWithCorners.symm, modelWithCorners_prod_toPartialEquiv,
PartialEquiv.prod_symm, PartialEquiv.prod_coe, ModelWithCorners.toPartialEquiv_coe_symm,
PartialHomeomorph.prod_apply, PartialHomeomorph.prod_toPartialEquiv,
PartialEquiv.prod_source] | case refine_1.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑J (↑(f.prod g) (↑(I.prod J).symm x)).2)
(↑(I.prod J).symm ⁻¹' (f.prod g).source ∩ interior (range ↑(I.prod J))) | case refine_1.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x.2)))
((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' f.source ×ˢ g.source ∩ interior (range ↑(I.prod J))) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑J (↑(f.prod g) (↑(I.prod J).symm x)).2)
(↑(I.prod J).symm ⁻¹' (f.prod g).source ∩ interior (range ↑(I.prod J)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | refine ga.2.1.1.comp (analyticOn_snd _) ?_ | case refine_1.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x.2)))
((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' f.source ×ˢ g.source ∩ interior (range ↑(I.prod J))) | case refine_1.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ MapsTo (fun x => x.2) ((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' f.source ×ˢ g.source ∩ interior (range ↑(I.prod J)))
(↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x.2)))
((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' f.source ×ˢ g.source ∩ interior (range ↑(I.prod J)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | intro x m | case refine_1.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ MapsTo (fun x => x.2) ((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' f.source ×ˢ g.source ∩ interior (range ↑(I.prod J)))
(↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) | case refine_1.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : x ∈ (fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' f.source ×ˢ g.source ∩ interior (range ↑(I.prod J))
⊢ (fun x => x.2) x ∈ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ MapsTo (fun x => x.2) ((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' f.source ×ˢ g.source ∩ interior (range ↑(I.prod J)))
(↑J.symm ⁻¹' g.source ∩ interior (range ↑J))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | simp only [ModelWithCorners.prod, ModelWithCorners.source_eq, mem_univ, and_self, setOf_true,
PartialEquiv.prod_target, ModelWithCorners.target_eq, ModelWithCorners.mk_coe,
mem_inter_iff, mem_preimage, mem_prod] at m ⊢ | case refine_1.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : x ∈ (fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' f.source ×ˢ g.source ∩ interior (range ↑(I.prod J))
⊢ (fun x => x.2) x ∈ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J) | case refine_1.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : (↑I.symm x.1 ∈ f.source ∧ ↑J.symm x.2 ∈ g.source) ∧ x ∈ interior (range fun x => (↑I x.1, ↑J x.2))
⊢ ↑J.symm x.2 ∈ g.source ∧ x.2 ∈ interior (range ↑J) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : x ∈ (fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' f.source ×ˢ g.source ∩ interior (range ↑(I.prod J))
⊢ (fun x => x.2) x ∈ ↑J.symm ⁻¹' g.source ∩ interior (range ↑J)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | exact ⟨m.1.2, (subset_of_eq ei m.2).2⟩ | case refine_1.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : (↑I.symm x.1 ∈ f.source ∧ ↑J.symm x.2 ∈ g.source) ∧ x ∈ interior (range fun x => (↑I x.1, ↑J x.2))
⊢ ↑J.symm x.2 ∈ g.source ∧ x.2 ∈ interior (range ↑J) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : (↑I.symm x.1 ∈ f.source ∧ ↑J.symm x.2 ∈ g.source) ∧ x ∈ interior (range fun x => (↑I x.1, ↑J x.2))
⊢ ↑J.symm x.2 ∈ g.source ∧ x.2 ∈ interior (range ↑J)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | simp only [ModelWithCorners.prod, ModelWithCorners.source_eq, mem_univ, and_self, setOf_true,
PartialEquiv.prod_target, ModelWithCorners.target_eq, ModelWithCorners.mk_symm,
PartialEquiv.coe_symm_mk, PartialHomeomorph.prod_apply, ModelWithCorners.mk_coe,
PartialHomeomorph.prod_toPartialEquiv, PartialEquiv.prod_source, mk_preimage_prod,
image_subset_iff] | case refine_2
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ ↑(I.prod J).symm ⁻¹' (f.prod g).source ∩ interior (range ↑(I.prod J)) ⊆
(fun x => ↑(I.prod J) (↑(f.prod g) (↑(I.prod J).symm x))) ⁻¹' interior (range ↑(I.prod J)) | case refine_2
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ (fun x => ↑I.symm x.1) ⁻¹' f.source ∩ (fun x => ↑J.symm x.2) ⁻¹' g.source ∩
interior (range fun x => (↑I x.1, ↑J x.2)) ⊆
(fun x => (↑I (↑f (↑I.symm x.1)), ↑J (↑g (↑J.symm x.2)))) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2)) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ ↑(I.prod J).symm ⁻¹' (f.prod g).source ∩ interior (range ↑(I.prod J)) ⊆
(fun x => ↑(I.prod J) (↑(f.prod g) (↑(I.prod J).symm x))) ⁻¹' interior (range ↑(I.prod J))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | intro x ⟨⟨m0,m1⟩,m2⟩ | case refine_2
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ (fun x => ↑I.symm x.1) ⁻¹' f.source ∩ (fun x => ↑J.symm x.2) ⁻¹' g.source ∩
interior (range fun x => (↑I x.1, ↑J x.2)) ⊆
(fun x => (↑I (↑f (↑I.symm x.1)), ↑J (↑g (↑J.symm x.2)))) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2)) | case refine_2
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.source
m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.source
m2 : x ∈ interior (range fun x => (↑I x.1, ↑J x.2))
⊢ x ∈ (fun x => (↑I (↑f (↑I.symm x.1)), ↑J (↑g (↑J.symm x.2)))) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2)) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ (fun x => ↑I.symm x.1) ⁻¹' f.source ∩ (fun x => ↑J.symm x.2) ⁻¹' g.source ∩
interior (range fun x => (↑I x.1, ↑J x.2)) ⊆
(fun x => (↑I (↑f (↑I.symm x.1)), ↑J (↑g (↑J.symm x.2)))) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | replace m2 := subset_of_eq ei m2 | case refine_2
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.source
m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.source
m2 : x ∈ interior (range fun x => (↑I x.1, ↑J x.2))
⊢ x ∈ (fun x => (↑I (↑f (↑I.symm x.1)), ↑J (↑g (↑J.symm x.2)))) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2)) | case refine_2
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.source
m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.source
m2 : x ∈ interior (range ↑I) ×ˢ interior (range ↑J)
⊢ x ∈ (fun x => (↑I (↑f (↑I.symm x.1)), ↑J (↑g (↑J.symm x.2)))) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2)) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.source
m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.source
m2 : x ∈ interior (range fun x => (↑I x.1, ↑J x.2))
⊢ x ∈ (fun x => (↑I (↑f (↑I.symm x.1)), ↑J (↑g (↑J.symm x.2)))) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | exact subset_of_eq ei.symm ⟨fa.2.1.2 ⟨m0,m2.1⟩, ga.2.1.2 ⟨m1,m2.2⟩⟩ | case refine_2
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.source
m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.source
m2 : x ∈ interior (range ↑I) ×ˢ interior (range ↑J)
⊢ x ∈ (fun x => (↑I (↑f (↑I.symm x.1)), ↑J (↑g (↑J.symm x.2)))) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.source
m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.source
m2 : x ∈ interior (range ↑I) ×ˢ interior (range ↑J)
⊢ x ∈ (fun x => (↑I (↑f (↑I.symm x.1)), ↑J (↑g (↑J.symm x.2)))) ⁻¹' interior (range fun x => (↑I x.1, ↑J x.2))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | apply AnalyticOn.prod | case refine_3
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑(I.prod J) (↑(f.prod g).symm (↑(I.prod J).symm x)))
(↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J))) | case refine_3.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑I (↑(f.prod g).symm (↑(I.prod J).symm x)).1)
(↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J)))
case refine_3.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑J (↑(f.prod g).symm (↑(I.prod J).symm x)).2)
(↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J))) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑(I.prod J) (↑(f.prod g).symm (↑(I.prod J).symm x)))
(↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | simp only [ModelWithCorners.symm, modelWithCorners_prod_toPartialEquiv,
PartialEquiv.prod_symm, PartialEquiv.prod_coe, ModelWithCorners.toPartialEquiv_coe_symm,
PartialHomeomorph.prod_apply, PartialHomeomorph.prod_toPartialEquiv,
PartialEquiv.prod_source] | case refine_3.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑I (↑(f.prod g).symm (↑(I.prod J).symm x)).1)
(↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J))) | case refine_3.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1)
((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J))) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑I (↑(f.prod g).symm (↑(I.prod J).symm x)).1)
(↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | refine fa.2.2.1.comp (analyticOn_fst _) ?_ | case refine_3.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1)
((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J))) | case refine_3.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ MapsTo (fun x => x.1)
((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J)))
(↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1)
((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | intro x m | case refine_3.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ MapsTo (fun x => x.1)
((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J)))
(↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) | case refine_3.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : x ∈ (fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J))
⊢ (fun x => x.1) x ∈ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ MapsTo (fun x => x.1)
((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J)))
(↑I.symm ⁻¹' f.target ∩ interior (range ↑I))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | simp only [ModelWithCorners.prod, ModelWithCorners.source_eq, mem_univ, and_self, setOf_true,
PartialEquiv.prod_target, ModelWithCorners.target_eq, ModelWithCorners.mk_coe,
mem_inter_iff, mem_preimage, mem_prod] at m ⊢ | case refine_3.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : x ∈ (fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J))
⊢ (fun x => x.1) x ∈ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I) | case refine_3.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : (↑I.symm x.1 ∈ f.target ∧ ↑J.symm x.2 ∈ g.target) ∧ x ∈ interior (range fun x => (↑I x.1, ↑J x.2))
⊢ ↑I.symm x.1 ∈ f.target ∧ x.1 ∈ interior (range ↑I) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : x ∈ (fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J))
⊢ (fun x => x.1) x ∈ ↑I.symm ⁻¹' f.target ∩ interior (range ↑I)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | exact ⟨m.1.1, (subset_of_eq ei m.2).1⟩ | case refine_3.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : (↑I.symm x.1 ∈ f.target ∧ ↑J.symm x.2 ∈ g.target) ∧ x ∈ interior (range fun x => (↑I x.1, ↑J x.2))
⊢ ↑I.symm x.1 ∈ f.target ∧ x.1 ∈ interior (range ↑I) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3.hf
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : (↑I.symm x.1 ∈ f.target ∧ ↑J.symm x.2 ∈ g.target) ∧ x ∈ interior (range fun x => (↑I x.1, ↑J x.2))
⊢ ↑I.symm x.1 ∈ f.target ∧ x.1 ∈ interior (range ↑I)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | simp only [ModelWithCorners.symm, modelWithCorners_prod_toPartialEquiv,
PartialEquiv.prod_symm, PartialEquiv.prod_coe, ModelWithCorners.toPartialEquiv_coe_symm,
PartialHomeomorph.prod_apply, PartialHomeomorph.prod_toPartialEquiv,
PartialEquiv.prod_source] | case refine_3.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑J (↑(f.prod g).symm (↑(I.prod J).symm x)).2)
(↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J))) | case refine_3.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2)
((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J))) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑J (↑(f.prod g).symm (↑(I.prod J).symm x)).2)
(↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | refine ga.2.2.1.comp (analyticOn_snd _) ?_ | case refine_3.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2)
((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J))) | case refine_3.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ MapsTo (fun x => x.2)
((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J)))
(↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ AnalyticOn 𝕜 (fun x => ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2)
((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | intro x m | case refine_3.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ MapsTo (fun x => x.2)
((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J)))
(↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) | case refine_3.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : x ∈ (fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J))
⊢ (fun x => x.2) x ∈ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ MapsTo (fun x => x.2)
((fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J)))
(↑J.symm ⁻¹' g.target ∩ interior (range ↑J))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | simp only [ModelWithCorners.prod, ModelWithCorners.source_eq, mem_univ, and_self, setOf_true,
PartialEquiv.prod_target, ModelWithCorners.target_eq, ModelWithCorners.mk_coe,
mem_inter_iff, mem_preimage, mem_prod] at m ⊢ | case refine_3.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : x ∈ (fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J))
⊢ (fun x => x.2) x ∈ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J) | case refine_3.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : (↑I.symm x.1 ∈ f.target ∧ ↑J.symm x.2 ∈ g.target) ∧ x ∈ interior (range fun x => (↑I x.1, ↑J x.2))
⊢ ↑J.symm x.2 ∈ g.target ∧ x.2 ∈ interior (range ↑J) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : x ∈ (fun p => (↑I.symm p.1, ↑J.symm p.2)) ⁻¹' (f.prod g.toPartialEquiv).target ∩ interior (range ↑(I.prod J))
⊢ (fun x => x.2) x ∈ ↑J.symm ⁻¹' g.target ∩ interior (range ↑J)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | exact ⟨m.1.2, (subset_of_eq ei m.2).2⟩ | case refine_3.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : (↑I.symm x.1 ∈ f.target ∧ ↑J.symm x.2 ∈ g.target) ∧ x ∈ interior (range fun x => (↑I x.1, ↑J x.2))
⊢ ↑J.symm x.2 ∈ g.target ∧ x.2 ∈ interior (range ↑J) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3.hg
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m : (↑I.symm x.1 ∈ f.target ∧ ↑J.symm x.2 ∈ g.target) ∧ x ∈ interior (range fun x => (↑I x.1, ↑J x.2))
⊢ ↑J.symm x.2 ∈ g.target ∧ x.2 ∈ interior (range ↑J)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | simp only [ModelWithCorners.prod, ModelWithCorners.source_eq, mem_univ, and_self, setOf_true,
PartialEquiv.prod_target, ModelWithCorners.target_eq, ModelWithCorners.mk_symm,
PartialEquiv.coe_symm_mk, PartialHomeomorph.prod_apply, ModelWithCorners.mk_coe,
PartialHomeomorph.prod_toPartialEquiv, PartialEquiv.prod_source, mk_preimage_prod,
image_subset_iff] | case refine_4
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ ↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J)) ⊆
(fun x => ↑(I.prod J) (↑(f.prod g).symm (↑(I.prod J).symm x))) ⁻¹' interior (range ↑(I.prod J)) | case refine_4
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ (fun x => ↑I.symm x.1) ⁻¹' f.target ∩ (fun x => ↑J.symm x.2) ⁻¹' g.target ∩
interior (range fun x => (↑I x.1, ↑J x.2)) ⊆
(fun x =>
(↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1, ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2)) ⁻¹'
interior (range fun x => (↑I x.1, ↑J x.2)) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_4
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ ↑(I.prod J).symm ⁻¹' (f.prod g).target ∩ interior (range ↑(I.prod J)) ⊆
(fun x => ↑(I.prod J) (↑(f.prod g).symm (↑(I.prod J).symm x))) ⁻¹' interior (range ↑(I.prod J))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | intro x ⟨⟨m0,m1⟩,m2⟩ | case refine_4
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ (fun x => ↑I.symm x.1) ⁻¹' f.target ∩ (fun x => ↑J.symm x.2) ⁻¹' g.target ∩
interior (range fun x => (↑I x.1, ↑J x.2)) ⊆
(fun x =>
(↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1, ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2)) ⁻¹'
interior (range fun x => (↑I x.1, ↑J x.2)) | case refine_4
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.target
m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.target
m2 : x ∈ interior (range fun x => (↑I x.1, ↑J x.2))
⊢ x ∈
(fun x =>
(↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1, ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2)) ⁻¹'
interior (range fun x => (↑I x.1, ↑J x.2)) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_4
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
⊢ (fun x => ↑I.symm x.1) ⁻¹' f.target ∩ (fun x => ↑J.symm x.2) ⁻¹' g.target ∩
interior (range fun x => (↑I x.1, ↑J x.2)) ⊆
(fun x =>
(↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1, ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2)) ⁻¹'
interior (range fun x => (↑I x.1, ↑J x.2))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | replace m2 := subset_of_eq ei m2 | case refine_4
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.target
m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.target
m2 : x ∈ interior (range fun x => (↑I x.1, ↑J x.2))
⊢ x ∈
(fun x =>
(↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1, ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2)) ⁻¹'
interior (range fun x => (↑I x.1, ↑J x.2)) | case refine_4
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.target
m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.target
m2 : x ∈ interior (range ↑I) ×ˢ interior (range ↑J)
⊢ x ∈
(fun x =>
(↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1, ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2)) ⁻¹'
interior (range fun x => (↑I x.1, ↑J x.2)) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_4
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.target
m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.target
m2 : x ∈ interior (range fun x => (↑I x.1, ↑J x.2))
⊢ x ∈
(fun x =>
(↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1, ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2)) ⁻¹'
interior (range fun x => (↑I x.1, ↑J x.2))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | exact subset_of_eq ei.symm ⟨fa.2.2.2 ⟨m0,m2.1⟩, ga.2.2.2 ⟨m1,m2.2⟩⟩ | case refine_4
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.target
m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.target
m2 : x ∈ interior (range ↑I) ×ˢ interior (range ↑J)
⊢ x ∈
(fun x =>
(↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1, ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2)) ⁻¹'
interior (range fun x => (↑I x.1, ↑J x.2)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_4
𝕜 : Type
inst✝⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : TopologicalSpace A
F B : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : TopologicalSpace B
I : ModelWithCorners 𝕜 E A
J : ModelWithCorners 𝕜 F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (↑I x.1, ↑J x.2)) = range ↑I ×ˢ range ↑J
ei : interior (range fun x => (↑I x.1, ↑J x.2)) = interior (range ↑I) ×ˢ interior (range ↑J)
fa :
f ∈ contDiffGroupoid ⊤ I ∧
(AnalyticOn 𝕜 (fun x => ↑I (↑f (↑I.symm x))) (↑I.symm ⁻¹' f.source ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.source ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f (↑I.symm x))) ⁻¹' interior (range ↑I)) ∧
AnalyticOn 𝕜 (fun x => ↑I (↑f.symm (↑I.symm x))) (↑I.symm ⁻¹' f.target ∩ interior (range ↑I)) ∧
↑I.symm ⁻¹' f.target ∩ interior (range ↑I) ⊆ (fun x => ↑I (↑f.symm (↑I.symm x))) ⁻¹' interior (range ↑I)
ga :
g ∈ contDiffGroupoid ⊤ J ∧
(AnalyticOn 𝕜 (fun x => ↑J (↑g (↑J.symm x))) (↑J.symm ⁻¹' g.source ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.source ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g (↑J.symm x))) ⁻¹' interior (range ↑J)) ∧
AnalyticOn 𝕜 (fun x => ↑J (↑g.symm (↑J.symm x))) (↑J.symm ⁻¹' g.target ∩ interior (range ↑J)) ∧
↑J.symm ⁻¹' g.target ∩ interior (range ↑J) ⊆ (fun x => ↑J (↑g.symm (↑J.symm x))) ⁻¹' interior (range ↑J)
x : E × F
m0 : x ∈ (fun x => ↑I.symm x.1) ⁻¹' f.target
m1 : x ∈ (fun x => ↑J.symm x.2) ⁻¹' g.target
m2 : x ∈ interior (range ↑I) ×ˢ interior (range ↑J)
⊢ x ∈
(fun x =>
(↑I (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).1, ↑J (↑(f.prod g).symm (↑I.symm x.1, ↑J.symm x.2)).2)) ⁻¹'
interior (range fun x => (↑I x.1, ↑J x.2))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | extChartAt_self_analytic | [208, 1] | [223, 25] | apply AnalyticOn.congr (f := fun z ↦ z) | 𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
M : Type
inst✝ : TopologicalSpace M
f : PartialHomeomorph M E
⊢ AnalyticOn 𝕜 (↑𝓘(𝕜, E) ∘ ↑(f.symm.trans f) ∘ ↑𝓘(𝕜, E).symm) (↑𝓘(𝕜, E) '' (f.symm.trans f).source) | case hs
𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
M : Type
inst✝ : TopologicalSpace M
f : PartialHomeomorph M E
⊢ IsOpen (↑𝓘(𝕜, E) '' (f.symm.trans f).source)
case hf
𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
M : Type
inst✝ : TopologicalSpace M
f : PartialHomeomorph M E
⊢ AnalyticOn 𝕜 (fun z => z) (↑𝓘(𝕜, E) '' (f.symm.trans f).source)
case hg
𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
M : Type
inst✝ : TopologicalSpace M
f : PartialHomeomorph M E
⊢ EqOn (fun z => z) (↑𝓘(𝕜, E) ∘ ↑(f.symm.trans f) ∘ ↑𝓘(𝕜, E).symm) (↑𝓘(𝕜, E) '' (f.symm.trans f).source) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
M : Type
inst✝ : TopologicalSpace M
f : PartialHomeomorph M E
⊢ AnalyticOn 𝕜 (↑𝓘(𝕜, E) ∘ ↑(f.symm.trans f) ∘ ↑𝓘(𝕜, E).symm) (↑𝓘(𝕜, E) '' (f.symm.trans f).source)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | extChartAt_self_analytic | [208, 1] | [223, 25] | simp only [modelWithCornersSelf_coe, id_eq, image_id', PartialHomeomorph.trans_toPartialEquiv,
PartialHomeomorph.symm_toPartialEquiv, PartialEquiv.trans_source, PartialEquiv.symm_source,
PartialHomeomorph.coe_coe_symm] | case hs
𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
M : Type
inst✝ : TopologicalSpace M
f : PartialHomeomorph M E
⊢ IsOpen (↑𝓘(𝕜, E) '' (f.symm.trans f).source) | case hs
𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
M : Type
inst✝ : TopologicalSpace M
f : PartialHomeomorph M E
⊢ IsOpen (f.target ∩ ↑f.symm ⁻¹' f.source) | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
M : Type
inst✝ : TopologicalSpace M
f : PartialHomeomorph M E
⊢ IsOpen (↑𝓘(𝕜, E) '' (f.symm.trans f).source)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | extChartAt_self_analytic | [208, 1] | [223, 25] | exact f.isOpen_inter_preimage_symm f.open_source | case hs
𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
M : Type
inst✝ : TopologicalSpace M
f : PartialHomeomorph M E
⊢ IsOpen (f.target ∩ ↑f.symm ⁻¹' f.source) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
M : Type
inst✝ : TopologicalSpace M
f : PartialHomeomorph M E
⊢ IsOpen (f.target ∩ ↑f.symm ⁻¹' f.source)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | extChartAt_self_analytic | [208, 1] | [223, 25] | exact analyticOn_id _ | case hf
𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
M : Type
inst✝ : TopologicalSpace M
f : PartialHomeomorph M E
⊢ AnalyticOn 𝕜 (fun z => z) (↑𝓘(𝕜, E) '' (f.symm.trans f).source) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
M : Type
inst✝ : TopologicalSpace M
f : PartialHomeomorph M E
⊢ AnalyticOn 𝕜 (fun z => z) (↑𝓘(𝕜, E) '' (f.symm.trans f).source)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | extChartAt_self_analytic | [208, 1] | [223, 25] | intro x m | case hg
𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
M : Type
inst✝ : TopologicalSpace M
f : PartialHomeomorph M E
⊢ EqOn (fun z => z) (↑𝓘(𝕜, E) ∘ ↑(f.symm.trans f) ∘ ↑𝓘(𝕜, E).symm) (↑𝓘(𝕜, E) '' (f.symm.trans f).source) | case hg
𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
M : Type
inst✝ : TopologicalSpace M
f : PartialHomeomorph M E
x : E
m : x ∈ ↑𝓘(𝕜, E) '' (f.symm.trans f).source
⊢ (fun z => z) x = (↑𝓘(𝕜, E) ∘ ↑(f.symm.trans f) ∘ ↑𝓘(𝕜, E).symm) x | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
M : Type
inst✝ : TopologicalSpace M
f : PartialHomeomorph M E
⊢ EqOn (fun z => z) (↑𝓘(𝕜, E) ∘ ↑(f.symm.trans f) ∘ ↑𝓘(𝕜, E).symm) (↑𝓘(𝕜, E) '' (f.symm.trans f).source)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | extChartAt_self_analytic | [208, 1] | [223, 25] | simp only [modelWithCornersSelf_coe, id, image_id', PartialHomeomorph.trans_toPartialEquiv,
PartialHomeomorph.symm_toPartialEquiv, PartialEquiv.trans_source, PartialEquiv.symm_source,
PartialHomeomorph.coe_coe_symm, mem_inter_iff, mem_preimage, Function.comp,
modelWithCornersSelf_coe_symm, PartialHomeomorph.coe_trans] at m ⊢ | case hg
𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
M : Type
inst✝ : TopologicalSpace M
f : PartialHomeomorph M E
x : E
m : x ∈ ↑𝓘(𝕜, E) '' (f.symm.trans f).source
⊢ (fun z => z) x = (↑𝓘(𝕜, E) ∘ ↑(f.symm.trans f) ∘ ↑𝓘(𝕜, E).symm) x | case hg
𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
M : Type
inst✝ : TopologicalSpace M
f : PartialHomeomorph M E
x : E
m : x ∈ f.target ∧ ↑f.symm x ∈ f.source
⊢ x = ↑f (↑f.symm x) | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
M : Type
inst✝ : TopologicalSpace M
f : PartialHomeomorph M E
x : E
m : x ∈ ↑𝓘(𝕜, E) '' (f.symm.trans f).source
⊢ (fun z => z) x = (↑𝓘(𝕜, E) ∘ ↑(f.symm.trans f) ∘ ↑𝓘(𝕜, E).symm) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | extChartAt_self_analytic | [208, 1] | [223, 25] | rw [f.right_inv m.1] | case hg
𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
M : Type
inst✝ : TopologicalSpace M
f : PartialHomeomorph M E
x : E
m : x ∈ f.target ∧ ↑f.symm x ∈ f.source
⊢ x = ↑f (↑f.symm x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
𝕜 : Type
inst✝³ : NontriviallyNormedField 𝕜
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
M : Type
inst✝ : TopologicalSpace M
f : PartialHomeomorph M E
x : E
m : x ∈ f.target ∧ ↑f.symm x ∈ f.source
⊢ x = ↑f (↑f.symm x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | holomorphicAt_iff | [263, 1] | [267, 54] | simp only [HolomorphicAt, ChartedSpace.liftPropAt_iff, extChartAt, PartialHomeomorph.extend_coe,
PartialHomeomorph.extend_coe_symm, Function.comp] | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : M → N
x : M
⊢ HolomorphicAt I J f x ↔
ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : M → N
x : M
⊢ HolomorphicAt I J f x ↔
ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | holomorphic_iff | [270, 1] | [274, 92] | simp only [Holomorphic, holomorphicAt_iff, continuous_iff_continuousAt] | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : M → N
⊢ Holomorphic I J f ↔
Continuous f ∧ ∀ (x : M), AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : M → N
⊢ (∀ (x : M),
ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)) ↔
(∀ (x : M), ContinuousAt f x) ∧
∀ (x : M), AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : M → N
⊢ Holomorphic I J f ↔
Continuous f ∧ ∀ (x : M), AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | holomorphic_iff | [270, 1] | [274, 92] | exact forall_and | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : M → N
⊢ (∀ (x : M),
ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)) ↔
(∀ (x : M), ContinuousAt f x) ∧
∀ (x : M), AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : M → N
⊢ (∀ (x : M),
ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)) ↔
(∀ (x : M), ContinuousAt f x) ∧
∀ (x : M), AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticAt_iff_holomorphicAt | [295, 1] | [300, 32] | simp only [holomorphicAt_iff, extChartAt_eq_refl, PartialEquiv.refl_coe, PartialEquiv.refl_symm,
Function.id_comp, Function.comp_id, id_eq, iff_and_self] | 𝕜 : Type
inst✝³² : NontriviallyNormedField 𝕜
E A : Type
inst✝³¹ : NormedAddCommGroup E
inst✝³⁰ : NormedSpace 𝕜 E
inst✝²⁹ : CompleteSpace E
inst✝²⁸ : TopologicalSpace A
F B : Type
inst✝²⁷ : NormedAddCommGroup F
inst✝²⁶ : NormedSpace 𝕜 F
inst✝²⁵ : CompleteSpace F
inst✝²⁴ : TopologicalSpace B
G C : Type
inst✝²³ : NormedAddCommGroup G
inst✝²² : NormedSpace 𝕜 G
inst✝²¹ : TopologicalSpace C
H D : Type
inst✝²⁰ : NormedAddCommGroup H
inst✝¹⁹ : NormedSpace 𝕜 H
inst✝¹⁸ : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹⁷ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁶ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝¹⁵ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝¹⁴ : TopologicalSpace P
inst✝¹³ : I.Boundaryless
inst✝¹² : ChartedSpace A M
cm : AnalyticManifold I M
inst✝¹¹ : J.Boundaryless
inst✝¹⁰ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝⁹ : K.Boundaryless
inst✝⁸ : ChartedSpace C O
co : AnalyticManifold K O
inst✝⁷ : L.Boundaryless
inst✝⁶ : ChartedSpace D P
cp : AnalyticManifold L P
inst✝⁵ : ChartedSpace A E
inst✝⁴ : AnalyticManifold I E
inst✝³ : ChartedSpace B F
inst✝² : AnalyticManifold J F
inst✝¹ : ExtChartEqRefl I
inst✝ : ExtChartEqRefl J
f : E → F
x : E
⊢ AnalyticAt 𝕜 f x ↔ HolomorphicAt I J f x | 𝕜 : Type
inst✝³² : NontriviallyNormedField 𝕜
E A : Type
inst✝³¹ : NormedAddCommGroup E
inst✝³⁰ : NormedSpace 𝕜 E
inst✝²⁹ : CompleteSpace E
inst✝²⁸ : TopologicalSpace A
F B : Type
inst✝²⁷ : NormedAddCommGroup F
inst✝²⁶ : NormedSpace 𝕜 F
inst✝²⁵ : CompleteSpace F
inst✝²⁴ : TopologicalSpace B
G C : Type
inst✝²³ : NormedAddCommGroup G
inst✝²² : NormedSpace 𝕜 G
inst✝²¹ : TopologicalSpace C
H D : Type
inst✝²⁰ : NormedAddCommGroup H
inst✝¹⁹ : NormedSpace 𝕜 H
inst✝¹⁸ : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹⁷ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁶ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝¹⁵ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝¹⁴ : TopologicalSpace P
inst✝¹³ : I.Boundaryless
inst✝¹² : ChartedSpace A M
cm : AnalyticManifold I M
inst✝¹¹ : J.Boundaryless
inst✝¹⁰ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝⁹ : K.Boundaryless
inst✝⁸ : ChartedSpace C O
co : AnalyticManifold K O
inst✝⁷ : L.Boundaryless
inst✝⁶ : ChartedSpace D P
cp : AnalyticManifold L P
inst✝⁵ : ChartedSpace A E
inst✝⁴ : AnalyticManifold I E
inst✝³ : ChartedSpace B F
inst✝² : AnalyticManifold J F
inst✝¹ : ExtChartEqRefl I
inst✝ : ExtChartEqRefl J
f : E → F
x : E
⊢ AnalyticAt 𝕜 f x → ContinuousAt f x | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝³² : NontriviallyNormedField 𝕜
E A : Type
inst✝³¹ : NormedAddCommGroup E
inst✝³⁰ : NormedSpace 𝕜 E
inst✝²⁹ : CompleteSpace E
inst✝²⁸ : TopologicalSpace A
F B : Type
inst✝²⁷ : NormedAddCommGroup F
inst✝²⁶ : NormedSpace 𝕜 F
inst✝²⁵ : CompleteSpace F
inst✝²⁴ : TopologicalSpace B
G C : Type
inst✝²³ : NormedAddCommGroup G
inst✝²² : NormedSpace 𝕜 G
inst✝²¹ : TopologicalSpace C
H D : Type
inst✝²⁰ : NormedAddCommGroup H
inst✝¹⁹ : NormedSpace 𝕜 H
inst✝¹⁸ : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹⁷ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁶ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝¹⁵ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝¹⁴ : TopologicalSpace P
inst✝¹³ : I.Boundaryless
inst✝¹² : ChartedSpace A M
cm : AnalyticManifold I M
inst✝¹¹ : J.Boundaryless
inst✝¹⁰ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝⁹ : K.Boundaryless
inst✝⁸ : ChartedSpace C O
co : AnalyticManifold K O
inst✝⁷ : L.Boundaryless
inst✝⁶ : ChartedSpace D P
cp : AnalyticManifold L P
inst✝⁵ : ChartedSpace A E
inst✝⁴ : AnalyticManifold I E
inst✝³ : ChartedSpace B F
inst✝² : AnalyticManifold J F
inst✝¹ : ExtChartEqRefl I
inst✝ : ExtChartEqRefl J
f : E → F
x : E
⊢ AnalyticAt 𝕜 f x ↔ HolomorphicAt I J f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticAt_iff_holomorphicAt | [295, 1] | [300, 32] | exact AnalyticAt.continuousAt | 𝕜 : Type
inst✝³² : NontriviallyNormedField 𝕜
E A : Type
inst✝³¹ : NormedAddCommGroup E
inst✝³⁰ : NormedSpace 𝕜 E
inst✝²⁹ : CompleteSpace E
inst✝²⁸ : TopologicalSpace A
F B : Type
inst✝²⁷ : NormedAddCommGroup F
inst✝²⁶ : NormedSpace 𝕜 F
inst✝²⁵ : CompleteSpace F
inst✝²⁴ : TopologicalSpace B
G C : Type
inst✝²³ : NormedAddCommGroup G
inst✝²² : NormedSpace 𝕜 G
inst✝²¹ : TopologicalSpace C
H D : Type
inst✝²⁰ : NormedAddCommGroup H
inst✝¹⁹ : NormedSpace 𝕜 H
inst✝¹⁸ : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹⁷ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁶ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝¹⁵ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝¹⁴ : TopologicalSpace P
inst✝¹³ : I.Boundaryless
inst✝¹² : ChartedSpace A M
cm : AnalyticManifold I M
inst✝¹¹ : J.Boundaryless
inst✝¹⁰ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝⁹ : K.Boundaryless
inst✝⁸ : ChartedSpace C O
co : AnalyticManifold K O
inst✝⁷ : L.Boundaryless
inst✝⁶ : ChartedSpace D P
cp : AnalyticManifold L P
inst✝⁵ : ChartedSpace A E
inst✝⁴ : AnalyticManifold I E
inst✝³ : ChartedSpace B F
inst✝² : AnalyticManifold J F
inst✝¹ : ExtChartEqRefl I
inst✝ : ExtChartEqRefl J
f : E → F
x : E
⊢ AnalyticAt 𝕜 f x → ContinuousAt f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝³² : NontriviallyNormedField 𝕜
E A : Type
inst✝³¹ : NormedAddCommGroup E
inst✝³⁰ : NormedSpace 𝕜 E
inst✝²⁹ : CompleteSpace E
inst✝²⁸ : TopologicalSpace A
F B : Type
inst✝²⁷ : NormedAddCommGroup F
inst✝²⁶ : NormedSpace 𝕜 F
inst✝²⁵ : CompleteSpace F
inst✝²⁴ : TopologicalSpace B
G C : Type
inst✝²³ : NormedAddCommGroup G
inst✝²² : NormedSpace 𝕜 G
inst✝²¹ : TopologicalSpace C
H D : Type
inst✝²⁰ : NormedAddCommGroup H
inst✝¹⁹ : NormedSpace 𝕜 H
inst✝¹⁸ : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹⁷ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁶ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝¹⁵ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝¹⁴ : TopologicalSpace P
inst✝¹³ : I.Boundaryless
inst✝¹² : ChartedSpace A M
cm : AnalyticManifold I M
inst✝¹¹ : J.Boundaryless
inst✝¹⁰ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝⁹ : K.Boundaryless
inst✝⁸ : ChartedSpace C O
co : AnalyticManifold K O
inst✝⁷ : L.Boundaryless
inst✝⁶ : ChartedSpace D P
cp : AnalyticManifold L P
inst✝⁵ : ChartedSpace A E
inst✝⁴ : AnalyticManifold I E
inst✝³ : ChartedSpace B F
inst✝² : AnalyticManifold J F
inst✝¹ : ExtChartEqRefl I
inst✝ : ExtChartEqRefl J
f : E → F
x : E
⊢ AnalyticAt 𝕜 f x → ContinuousAt f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.comp | [321, 1] | [338, 38] | rw [holomorphicAt_iff] at fh gh ⊢ | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
fh : HolomorphicAt J K f (g x)
gh : HolomorphicAt I J g x
⊢ HolomorphicAt I K (fun x => f (g x)) x | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
fh :
ContinuousAt f (g x) ∧
AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x))
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ContinuousAt (fun x => f (g x)) x ∧
AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
fh : HolomorphicAt J K f (g x)
gh : HolomorphicAt I J g x
⊢ HolomorphicAt I K (fun x => f (g x)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.comp | [321, 1] | [338, 38] | use fh.1.comp gh.1 | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
fh :
ContinuousAt f (g x) ∧
AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x))
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ContinuousAt (fun x => f (g x)) x ∧
AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
fh :
ContinuousAt f (g x) ∧
AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x))
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
fh :
ContinuousAt f (g x) ∧
AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x))
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ContinuousAt (fun x => f (g x)) x ∧
AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.comp | [321, 1] | [338, 38] | have e : extChartAt J (g x) (g x) =
(extChartAt J (g x) ∘ g ∘ (extChartAt I x).symm) (extChartAt I x x) := by
simp only [Function.comp_apply, PartialEquiv.left_inv _ (mem_extChartAt_source I x)] | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
fh :
ContinuousAt f (g x) ∧
AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x))
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
fh :
ContinuousAt f (g x) ∧
AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x))
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
e : ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
fh :
ContinuousAt f (g x) ∧
AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x))
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.comp | [321, 1] | [338, 38] | rw [e] at fh | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
fh :
ContinuousAt f (g x) ∧
AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x))
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
e : ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
fh :
ContinuousAt f (g x) ∧
AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm)
((↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x))
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
e : ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
fh :
ContinuousAt f (g x) ∧
AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x))
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
e : ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.comp | [321, 1] | [338, 38] | apply (fh.2.comp gh.2).congr | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
fh :
ContinuousAt f (g x) ∧
AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm)
((↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x))
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
e : ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
fh :
ContinuousAt f (g x) ∧
AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm)
((↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x))
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
e : ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq
((↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) ∘ ↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm)
(↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
fh :
ContinuousAt f (g x) ∧
AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm)
((↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x))
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
e : ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.comp | [321, 1] | [338, 38] | clear e fh | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
fh :
ContinuousAt f (g x) ∧
AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm)
((↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x))
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
e : ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq
((↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) ∘ ↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm)
(↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq
((↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) ∘ ↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm)
(↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
fh :
ContinuousAt f (g x) ∧
AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm)
((↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x))
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
e : ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq
((↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) ∘ ↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm)
(↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.comp | [321, 1] | [338, 38] | simp only [Function.comp] | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq
((↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) ∘ ↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm)
(↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm) | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq
(fun x_1 =>
↑(extChartAt K (f (g x)))
(f (↑(extChartAt J (g x)).symm (↑(extChartAt J (g x)) (g (↑(extChartAt I x).symm x_1))))))
fun x_1 => ↑(extChartAt K (f (g x))) (f (g (↑(extChartAt I x).symm x_1))) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq
((↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) ∘ ↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm)
(↑(extChartAt K (f (g x))) ∘ (fun x => f (g x)) ∘ ↑(extChartAt I x).symm)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.comp | [321, 1] | [338, 38] | refine m.mp (eventually_of_forall fun y m ↦ ?_) | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
m : ∀ᶠ (y : E) in 𝓝 (↑(extChartAt I x) x), g (↑(extChartAt I x).symm y) ∈ (extChartAt J (g x)).source
⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq
(fun x_1 =>
↑(extChartAt K (f (g x)))
(f (↑(extChartAt J (g x)).symm (↑(extChartAt J (g x)) (g (↑(extChartAt I x).symm x_1))))))
fun x_1 => ↑(extChartAt K (f (g x))) (f (g (↑(extChartAt I x).symm x_1))) | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
m✝ : ∀ᶠ (y : E) in 𝓝 (↑(extChartAt I x) x), g (↑(extChartAt I x).symm y) ∈ (extChartAt J (g x)).source
y : E
m : g (↑(extChartAt I x).symm y) ∈ (extChartAt J (g x)).source
⊢ (fun x_1 =>
↑(extChartAt K (f (g x)))
(f (↑(extChartAt J (g x)).symm (↑(extChartAt J (g x)) (g (↑(extChartAt I x).symm x_1))))))
y =
(fun x_1 => ↑(extChartAt K (f (g x))) (f (g (↑(extChartAt I x).symm x_1)))) y | Please generate a tactic in lean4 to solve the state.
STATE:
case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
m : ∀ᶠ (y : E) in 𝓝 (↑(extChartAt I x) x), g (↑(extChartAt I x).symm y) ∈ (extChartAt J (g x)).source
⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq
(fun x_1 =>
↑(extChartAt K (f (g x)))
(f (↑(extChartAt J (g x)).symm (↑(extChartAt J (g x)) (g (↑(extChartAt I x).symm x_1))))))
fun x_1 => ↑(extChartAt K (f (g x))) (f (g (↑(extChartAt I x).symm x_1)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.comp | [321, 1] | [338, 38] | simp_rw [PartialEquiv.left_inv _ m] | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
m✝ : ∀ᶠ (y : E) in 𝓝 (↑(extChartAt I x) x), g (↑(extChartAt I x).symm y) ∈ (extChartAt J (g x)).source
y : E
m : g (↑(extChartAt I x).symm y) ∈ (extChartAt J (g x)).source
⊢ (fun x_1 =>
↑(extChartAt K (f (g x)))
(f (↑(extChartAt J (g x)).symm (↑(extChartAt J (g x)) (g (↑(extChartAt I x).symm x_1))))))
y =
(fun x_1 => ↑(extChartAt K (f (g x))) (f (g (↑(extChartAt I x).symm x_1)))) y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
m✝ : ∀ᶠ (y : E) in 𝓝 (↑(extChartAt I x) x), g (↑(extChartAt I x).symm y) ∈ (extChartAt J (g x)).source
y : E
m : g (↑(extChartAt I x).symm y) ∈ (extChartAt J (g x)).source
⊢ (fun x_1 =>
↑(extChartAt K (f (g x)))
(f (↑(extChartAt J (g x)).symm (↑(extChartAt J (g x)) (g (↑(extChartAt I x).symm x_1))))))
y =
(fun x_1 => ↑(extChartAt K (f (g x))) (f (g (↑(extChartAt I x).symm x_1)))) y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.comp | [321, 1] | [338, 38] | simp only [Function.comp_apply, PartialEquiv.left_inv _ (mem_extChartAt_source I x)] | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
fh :
ContinuousAt f (g x) ∧
AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x))
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
fh :
ContinuousAt f (g x) ∧
AnalyticAt 𝕜 (↑(extChartAt K (f (g x))) ∘ f ∘ ↑(extChartAt J (g x)).symm) (↑(extChartAt J (g x)) (g x))
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ↑(extChartAt J (g x)) (g x) = (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.comp | [321, 1] | [338, 38] | apply ContinuousAt.eventually_mem | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ∀ᶠ (y : E) in 𝓝 (↑(extChartAt I x) x), g (↑(extChartAt I x).symm y) ∈ (extChartAt J (g x)).source | case hf
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ContinuousAt (fun y => g (↑(extChartAt I x).symm y)) (↑(extChartAt I x) x)
case hs
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ (extChartAt J (g x)).source ∈ 𝓝 (g (↑(extChartAt I x).symm (↑(extChartAt I x) x))) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ∀ᶠ (y : E) in 𝓝 (↑(extChartAt I x) x), g (↑(extChartAt I x).symm y) ∈ (extChartAt J (g x)).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.comp | [321, 1] | [338, 38] | apply ContinuousAt.comp | case hf
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ContinuousAt (fun y => g (↑(extChartAt I x).symm y)) (↑(extChartAt I x) x) | case hf.hg
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ContinuousAt g (↑(extChartAt I x).symm (↑(extChartAt I x) x))
case hf.hf
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ContinuousAt (fun y => ↑(extChartAt I x).symm y) (↑(extChartAt I x) x) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ContinuousAt (fun y => g (↑(extChartAt I x).symm y)) (↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.comp | [321, 1] | [338, 38] | rw [PartialEquiv.left_inv _ (mem_extChartAt_source _ _)] | case hf.hg
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ContinuousAt g (↑(extChartAt I x).symm (↑(extChartAt I x) x)) | case hf.hg
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ContinuousAt g x | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hg
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ContinuousAt g (↑(extChartAt I x).symm (↑(extChartAt I x) x))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.comp | [321, 1] | [338, 38] | exact gh.1 | case hf.hg
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ContinuousAt g x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hg
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ContinuousAt g x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.comp | [321, 1] | [338, 38] | exact continuousAt_extChartAt_symm I x | case hf.hf
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ContinuousAt (fun y => ↑(extChartAt I x).symm y) (↑(extChartAt I x) x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ContinuousAt (fun y => ↑(extChartAt I x).symm y) (↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.comp | [321, 1] | [338, 38] | rw [PartialEquiv.left_inv _ (mem_extChartAt_source _ _)] | case hs
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ (extChartAt J (g x)).source ∈ 𝓝 (g (↑(extChartAt I x).symm (↑(extChartAt I x) x))) | case hs
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ (extChartAt J (g x)).source ∈ 𝓝 (g x) | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ (extChartAt J (g x)).source ∈ 𝓝 (g (↑(extChartAt I x).symm (↑(extChartAt I x) x)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.comp | [321, 1] | [338, 38] | exact extChartAt_source_mem_nhds _ _ | case hs
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ (extChartAt J (g x)).source ∈ 𝓝 (g x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt J (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ (extChartAt J (g x)).source ∈ 𝓝 (g x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.comp_of_eq | [345, 1] | [348, 35] | rw [← e] at fh | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
y : N
fh : HolomorphicAt J K f y
gh : HolomorphicAt I J g x
e : g x = y
⊢ HolomorphicAt I K (fun x => f (g x)) x | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
y : N
fh : HolomorphicAt J K f (g x)
gh : HolomorphicAt I J g x
e : g x = y
⊢ HolomorphicAt I K (fun x => f (g x)) x | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
y : N
fh : HolomorphicAt J K f y
gh : HolomorphicAt I J g x
e : g x = y
⊢ HolomorphicAt I K (fun x => f (g x)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.comp_of_eq | [345, 1] | [348, 35] | exact fh.comp gh | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
y : N
fh : HolomorphicAt J K f (g x)
gh : HolomorphicAt I J g x
e : g x = y
⊢ HolomorphicAt I K (fun x => f (g x)) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : N → O
g : M → N
x : M
y : N
fh : HolomorphicAt J K f (g x)
gh : HolomorphicAt I J g x
e : g x = y
⊢ HolomorphicAt I K (fun x => f (g x)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.prod | [351, 1] | [355, 76] | rw [holomorphicAt_iff] at fh gh ⊢ | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : M → N
g : M → O
x : M
fh : HolomorphicAt I J f x
gh : HolomorphicAt I K g x
⊢ HolomorphicAt I (J.prod K) (fun x => (f x, g x)) x | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : M → N
g : M → O
x : M
fh : ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ContinuousAt (fun x => (f x, g x)) x ∧
AnalyticAt 𝕜 (↑(extChartAt (J.prod K) (f x, g x)) ∘ (fun x => (f x, g x)) ∘ ↑(extChartAt I x).symm)
(↑(extChartAt I x) x) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : M → N
g : M → O
x : M
fh : HolomorphicAt I J f x
gh : HolomorphicAt I K g x
⊢ HolomorphicAt I (J.prod K) (fun x => (f x, g x)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.prod | [351, 1] | [355, 76] | use fh.1.prod gh.1 | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : M → N
g : M → O
x : M
fh : ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ContinuousAt (fun x => (f x, g x)) x ∧
AnalyticAt 𝕜 (↑(extChartAt (J.prod K) (f x, g x)) ∘ (fun x => (f x, g x)) ∘ ↑(extChartAt I x).symm)
(↑(extChartAt I x) x) | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : M → N
g : M → O
x : M
fh : ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ AnalyticAt 𝕜 (↑(extChartAt (J.prod K) (f x, g x)) ∘ (fun x => (f x, g x)) ∘ ↑(extChartAt I x).symm)
(↑(extChartAt I x) x) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : M → N
g : M → O
x : M
fh : ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ ContinuousAt (fun x => (f x, g x)) x ∧
AnalyticAt 𝕜 (↑(extChartAt (J.prod K) (f x, g x)) ∘ (fun x => (f x, g x)) ∘ ↑(extChartAt I x).symm)
(↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.prod | [351, 1] | [355, 76] | refine (fh.2.prod gh.2).congr (eventually_of_forall fun y ↦ ?_) | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : M → N
g : M → O
x : M
fh : ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ AnalyticAt 𝕜 (↑(extChartAt (J.prod K) (f x, g x)) ∘ (fun x => (f x, g x)) ∘ ↑(extChartAt I x).symm)
(↑(extChartAt I x) x) | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : M → N
g : M → O
x : M
fh : ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
y : E
⊢ (fun x_1 =>
((↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) x_1,
(↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) x_1))
y =
(↑(extChartAt (J.prod K) (f x, g x)) ∘ (fun x => (f x, g x)) ∘ ↑(extChartAt I x).symm) y | Please generate a tactic in lean4 to solve the state.
STATE:
case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : M → N
g : M → O
x : M
fh : ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
⊢ AnalyticAt 𝕜 (↑(extChartAt (J.prod K) (f x, g x)) ∘ (fun x => (f x, g x)) ∘ ↑(extChartAt I x).symm)
(↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.prod | [351, 1] | [355, 76] | simp only [extChartAt_prod, Function.comp, PartialEquiv.prod_coe] | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : M → N
g : M → O
x : M
fh : ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
y : E
⊢ (fun x_1 =>
((↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) x_1,
(↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) x_1))
y =
(↑(extChartAt (J.prod K) (f x, g x)) ∘ (fun x => (f x, g x)) ∘ ↑(extChartAt I x).symm) y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
f : M → N
g : M → O
x : M
fh : ContinuousAt f x ∧ AnalyticAt 𝕜 (↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
gh : ContinuousAt g x ∧ AnalyticAt 𝕜 (↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
y : E
⊢ (fun x_1 =>
((↑(extChartAt J (f x)) ∘ f ∘ ↑(extChartAt I x).symm) x_1,
(↑(extChartAt K (g x)) ∘ g ∘ ↑(extChartAt I x).symm) x_1))
y =
(↑(extChartAt (J.prod K) (f x, g x)) ∘ (fun x => (f x, g x)) ∘ ↑(extChartAt I x).symm) y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.comp₂_of_eq | [368, 1] | [371, 91] | rw [← e] at ha | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
h : N → O → P
f : M → N
g : M → O
x : M
y : N × O
ha : HolomorphicAt (J.prod K) L (uncurry h) y
fa : HolomorphicAt I J f x
ga : HolomorphicAt I K g x
e : (f x, g x) = y
⊢ HolomorphicAt I L (fun x => h (f x) (g x)) x | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
h : N → O → P
f : M → N
g : M → O
x : M
y : N × O
ha : HolomorphicAt (J.prod K) L (uncurry h) (f x, g x)
fa : HolomorphicAt I J f x
ga : HolomorphicAt I K g x
e : (f x, g x) = y
⊢ HolomorphicAt I L (fun x => h (f x) (g x)) x | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
h : N → O → P
f : M → N
g : M → O
x : M
y : N × O
ha : HolomorphicAt (J.prod K) L (uncurry h) y
fa : HolomorphicAt I J f x
ga : HolomorphicAt I K g x
e : (f x, g x) = y
⊢ HolomorphicAt I L (fun x => h (f x) (g x)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | HolomorphicAt.comp₂_of_eq | [368, 1] | [371, 91] | exact ha.comp₂ fa ga | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
h : N → O → P
f : M → N
g : M → O
x : M
y : N × O
ha : HolomorphicAt (J.prod K) L (uncurry h) (f x, g x)
fa : HolomorphicAt I J f x
ga : HolomorphicAt I K g x
e : (f x, g x) = y
⊢ HolomorphicAt I L (fun x => h (f x) (g x)) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
h : N → O → P
f : M → N
g : M → O
x : M
y : N × O
ha : HolomorphicAt (J.prod K) L (uncurry h) (f x, g x)
fa : HolomorphicAt I J f x
ga : HolomorphicAt I K g x
e : (f x, g x) = y
⊢ HolomorphicAt I L (fun x => h (f x) (g x)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | holomorphicAt_id | [374, 1] | [378, 60] | rw [holomorphicAt_iff] | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
⊢ HolomorphicAt I I (fun x => x) x | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
⊢ ContinuousAt (fun x => x) x ∧
AnalyticAt 𝕜 (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
⊢ HolomorphicAt I I (fun x => x) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | holomorphicAt_id | [374, 1] | [378, 60] | use continuousAt_id | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
⊢ ContinuousAt (fun x => x) x ∧
AnalyticAt 𝕜 (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
⊢ AnalyticAt 𝕜 (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
⊢ ContinuousAt (fun x => x) x ∧
AnalyticAt 𝕜 (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | holomorphicAt_id | [374, 1] | [378, 60] | apply (analyticAt_id _ _).congr | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
⊢ AnalyticAt 𝕜 (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq id (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
⊢ AnalyticAt 𝕜 (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | holomorphicAt_id | [374, 1] | [378, 60] | filter_upwards [((isOpen_extChartAt_target I x).eventually_mem (mem_extChartAt_target I x))] | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq id (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) | case h
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
⊢ ∀ a ∈ (extChartAt I x).target, id a = (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) a | Please generate a tactic in lean4 to solve the state.
STATE:
case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
⊢ (𝓝 (↑(extChartAt I x) x)).EventuallyEq id (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | holomorphicAt_id | [374, 1] | [378, 60] | intro y m | case h
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
⊢ ∀ a ∈ (extChartAt I x).target, id a = (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) a | case h
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
y : E
m : y ∈ (extChartAt I x).target
⊢ id y = (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
⊢ ∀ a ∈ (extChartAt I x).target, id a = (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | holomorphicAt_id | [374, 1] | [378, 60] | simp only [Function.comp, PartialEquiv.right_inv _ m, id] | case h
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
y : E
m : y ∈ (extChartAt I x).target
⊢ id y = (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
y : E
m : y ∈ (extChartAt I x).target
⊢ id y = (↑(extChartAt I x) ∘ (fun x => x) ∘ ↑(extChartAt I x).symm) y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | holomorphicAt_const | [384, 1] | [386, 70] | rw [holomorphicAt_iff] | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
c : N
⊢ HolomorphicAt I J (fun x => c) x | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
c : N
⊢ ContinuousAt (fun x => c) x ∧
AnalyticAt 𝕜 (↑(extChartAt J c) ∘ (fun x => c) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
c : N
⊢ HolomorphicAt I J (fun x => c) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | holomorphicAt_const | [384, 1] | [386, 70] | use continuousAt_const | 𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
c : N
⊢ ContinuousAt (fun x => c) x ∧
AnalyticAt 𝕜 (↑(extChartAt J c) ∘ (fun x => c) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
c : N
⊢ AnalyticAt 𝕜 (↑(extChartAt J c) ∘ (fun x => c) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
c : N
⊢ ContinuousAt (fun x => c) x ∧
AnalyticAt 𝕜 (↑(extChartAt J c) ∘ (fun x => c) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | holomorphicAt_const | [384, 1] | [386, 70] | simp only [Prod.fst_comp_mk, Function.comp] | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
c : N
⊢ AnalyticAt 𝕜 (↑(extChartAt J c) ∘ (fun x => c) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x) | case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
c : N
⊢ AnalyticAt 𝕜 (fun x => ↑(extChartAt J c) c) (↑(extChartAt I x) x) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
𝕜 : Type
inst✝²⁶ : NontriviallyNormedField 𝕜
E A : Type
inst✝²⁵ : NormedAddCommGroup E
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : CompleteSpace E
inst✝²² : TopologicalSpace A
F B : Type
inst✝²¹ : NormedAddCommGroup F
inst✝²⁰ : NormedSpace 𝕜 F
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : TopologicalSpace B
G C : Type
inst✝¹⁷ : NormedAddCommGroup G
inst✝¹⁶ : NormedSpace 𝕜 G
inst✝¹⁵ : TopologicalSpace C
H D : Type
inst✝¹⁴ : NormedAddCommGroup H
inst✝¹³ : NormedSpace 𝕜 H
inst✝¹² : TopologicalSpace D
M : Type
I : ModelWithCorners 𝕜 E A
inst✝¹¹ : TopologicalSpace M
N : Type
J : ModelWithCorners 𝕜 F B
inst✝¹⁰ : TopologicalSpace N
O : Type
K : ModelWithCorners 𝕜 G C
inst✝⁹ : TopologicalSpace O
P : Type
L : ModelWithCorners 𝕜 H D
inst✝⁸ : TopologicalSpace P
inst✝⁷ : I.Boundaryless
inst✝⁶ : ChartedSpace A M
cm : AnalyticManifold I M
inst✝⁵ : J.Boundaryless
inst✝⁴ : ChartedSpace B N
cn : AnalyticManifold J N
inst✝³ : K.Boundaryless
inst✝² : ChartedSpace C O
co : AnalyticManifold K O
inst✝¹ : L.Boundaryless
inst✝ : ChartedSpace D P
cp : AnalyticManifold L P
x : M
c : N
⊢ AnalyticAt 𝕜 (↑(extChartAt J c) ∘ (fun x => c) ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) x)
TACTIC:
|
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