url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Analytic.lean | leadingCoeff_const_smul | [354, 1] | [368, 31] | clear hn | case neg
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
hn : orderAt f c = n
⊢ (Function.swap dslope c)^[n] (fun z => a • f z) c = a • (Function.swap dslope c)^[n] f c | case neg
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
⊢ (Function.swap dslope c)^[n] (fun z => a • f z) c = a • (Function.swap dslope c)^[n] f c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
hn : orderAt f c = n
⊢ (Function.swap dslope c)^[n] (fun z => a • f z) c = a • (Function.swap dslope c)^[n] f c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Analytic.lean | leadingCoeff_const_smul | [354, 1] | [368, 31] | have e : ((Function.swap dslope c)^[n] fun z : 𝕜 ↦ a • f z) =
a • (Function.swap dslope c)^[n] f := by
induction' n with n h; funext; simp only [Function.iterate_zero_apply, Pi.smul_apply]
generalize hg : (Function.swap dslope c)^[n] f = g
simp only [Function.iterate_succ_apply', h, hg]
funext x; simp only [Function.swap]
by_cases cx : x = c
simp only [cx, dslope_same, Pi.smul_apply, Pi.smul_def, deriv_const_smul']
simp only [dslope_of_ne _ cx, Pi.smul_apply, slope, vsub_eq_sub, ← smul_sub, smul_comm _ a] | case neg
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
⊢ (Function.swap dslope c)^[n] (fun z => a • f z) c = a • (Function.swap dslope c)^[n] f c | case neg
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
e : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
⊢ (Function.swap dslope c)^[n] (fun z => a • f z) c = a • (Function.swap dslope c)^[n] f c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
⊢ (Function.swap dslope c)^[n] (fun z => a • f z) c = a • (Function.swap dslope c)^[n] f c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Analytic.lean | leadingCoeff_const_smul | [354, 1] | [368, 31] | simp only [e, Pi.smul_apply] | case neg
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
e : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
⊢ (Function.swap dslope c)^[n] (fun z => a • f z) c = a • (Function.swap dslope c)^[n] f c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
e : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
⊢ (Function.swap dslope c)^[n] (fun z => a • f z) c = a • (Function.swap dslope c)^[n] f c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Analytic.lean | leadingCoeff_const_smul | [354, 1] | [368, 31] | induction' n with n h | 𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
⊢ ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f | case zero
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
⊢ ((Function.swap dslope c)^[0] fun z => a • f z) = a • (Function.swap dslope c)^[0] f
case succ
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
⊢ ((Function.swap dslope c)^[n + 1] fun z => a • f z) = a • (Function.swap dslope c)^[n + 1] f | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
⊢ ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Analytic.lean | leadingCoeff_const_smul | [354, 1] | [368, 31] | funext | case zero
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
⊢ ((Function.swap dslope c)^[0] fun z => a • f z) = a • (Function.swap dslope c)^[0] f
case succ
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
⊢ ((Function.swap dslope c)^[n + 1] fun z => a • f z) = a • (Function.swap dslope c)^[n + 1] f | case zero.h
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
x✝ : 𝕜
⊢ (Function.swap dslope c)^[0] (fun z => a • f z) x✝ = (a • (Function.swap dslope c)^[0] f) x✝
case succ
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
⊢ ((Function.swap dslope c)^[n + 1] fun z => a • f z) = a • (Function.swap dslope c)^[n + 1] f | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
⊢ ((Function.swap dslope c)^[0] fun z => a • f z) = a • (Function.swap dslope c)^[0] f
case succ
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
⊢ ((Function.swap dslope c)^[n + 1] fun z => a • f z) = a • (Function.swap dslope c)^[n + 1] f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Analytic.lean | leadingCoeff_const_smul | [354, 1] | [368, 31] | simp only [Function.iterate_zero_apply, Pi.smul_apply] | case zero.h
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
x✝ : 𝕜
⊢ (Function.swap dslope c)^[0] (fun z => a • f z) x✝ = (a • (Function.swap dslope c)^[0] f) x✝
case succ
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
⊢ ((Function.swap dslope c)^[n + 1] fun z => a • f z) = a • (Function.swap dslope c)^[n + 1] f | case succ
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
⊢ ((Function.swap dslope c)^[n + 1] fun z => a • f z) = a • (Function.swap dslope c)^[n + 1] f | Please generate a tactic in lean4 to solve the state.
STATE:
case zero.h
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
x✝ : 𝕜
⊢ (Function.swap dslope c)^[0] (fun z => a • f z) x✝ = (a • (Function.swap dslope c)^[0] f) x✝
case succ
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
⊢ ((Function.swap dslope c)^[n + 1] fun z => a • f z) = a • (Function.swap dslope c)^[n + 1] f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Analytic.lean | leadingCoeff_const_smul | [354, 1] | [368, 31] | generalize hg : (Function.swap dslope c)^[n] f = g | case succ
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
⊢ ((Function.swap dslope c)^[n + 1] fun z => a • f z) = a • (Function.swap dslope c)^[n + 1] f | case succ
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
g : 𝕜 → E
hg : (Function.swap dslope c)^[n] f = g
⊢ ((Function.swap dslope c)^[n + 1] fun z => a • f z) = a • (Function.swap dslope c)^[n + 1] f | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
⊢ ((Function.swap dslope c)^[n + 1] fun z => a • f z) = a • (Function.swap dslope c)^[n + 1] f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Analytic.lean | leadingCoeff_const_smul | [354, 1] | [368, 31] | simp only [Function.iterate_succ_apply', h, hg] | case succ
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
g : 𝕜 → E
hg : (Function.swap dslope c)^[n] f = g
⊢ ((Function.swap dslope c)^[n + 1] fun z => a • f z) = a • (Function.swap dslope c)^[n + 1] f | case succ
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
g : 𝕜 → E
hg : (Function.swap dslope c)^[n] f = g
⊢ Function.swap dslope c (a • g) = a • Function.swap dslope c g | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
g : 𝕜 → E
hg : (Function.swap dslope c)^[n] f = g
⊢ ((Function.swap dslope c)^[n + 1] fun z => a • f z) = a • (Function.swap dslope c)^[n + 1] f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Analytic.lean | leadingCoeff_const_smul | [354, 1] | [368, 31] | funext x | case succ
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
g : 𝕜 → E
hg : (Function.swap dslope c)^[n] f = g
⊢ Function.swap dslope c (a • g) = a • Function.swap dslope c g | case succ.h
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
g : 𝕜 → E
hg : (Function.swap dslope c)^[n] f = g
x : 𝕜
⊢ Function.swap dslope c (a • g) x = (a • Function.swap dslope c g) x | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
g : 𝕜 → E
hg : (Function.swap dslope c)^[n] f = g
⊢ Function.swap dslope c (a • g) = a • Function.swap dslope c g
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Analytic.lean | leadingCoeff_const_smul | [354, 1] | [368, 31] | simp only [Function.swap] | case succ.h
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
g : 𝕜 → E
hg : (Function.swap dslope c)^[n] f = g
x : 𝕜
⊢ Function.swap dslope c (a • g) x = (a • Function.swap dslope c g) x | case succ.h
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
g : 𝕜 → E
hg : (Function.swap dslope c)^[n] f = g
x : 𝕜
⊢ dslope (a • g) c x = (a • dslope g c) x | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.h
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
g : 𝕜 → E
hg : (Function.swap dslope c)^[n] f = g
x : 𝕜
⊢ Function.swap dslope c (a • g) x = (a • Function.swap dslope c g) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Analytic.lean | leadingCoeff_const_smul | [354, 1] | [368, 31] | by_cases cx : x = c | case succ.h
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
g : 𝕜 → E
hg : (Function.swap dslope c)^[n] f = g
x : 𝕜
⊢ dslope (a • g) c x = (a • dslope g c) x | case pos
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
g : 𝕜 → E
hg : (Function.swap dslope c)^[n] f = g
x : 𝕜
cx : x = c
⊢ dslope (a • g) c x = (a • dslope g c) x
case neg
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
g : 𝕜 → E
hg : (Function.swap dslope c)^[n] f = g
x : 𝕜
cx : ¬x = c
⊢ dslope (a • g) c x = (a • dslope g c) x | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.h
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
g : 𝕜 → E
hg : (Function.swap dslope c)^[n] f = g
x : 𝕜
⊢ dslope (a • g) c x = (a • dslope g c) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Analytic.lean | leadingCoeff_const_smul | [354, 1] | [368, 31] | simp only [cx, dslope_same, Pi.smul_apply, Pi.smul_def, deriv_const_smul'] | case pos
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
g : 𝕜 → E
hg : (Function.swap dslope c)^[n] f = g
x : 𝕜
cx : x = c
⊢ dslope (a • g) c x = (a • dslope g c) x
case neg
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
g : 𝕜 → E
hg : (Function.swap dslope c)^[n] f = g
x : 𝕜
cx : ¬x = c
⊢ dslope (a • g) c x = (a • dslope g c) x | case neg
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
g : 𝕜 → E
hg : (Function.swap dslope c)^[n] f = g
x : 𝕜
cx : ¬x = c
⊢ dslope (a • g) c x = (a • dslope g c) x | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
g : 𝕜 → E
hg : (Function.swap dslope c)^[n] f = g
x : 𝕜
cx : x = c
⊢ dslope (a • g) c x = (a • dslope g c) x
case neg
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
g : 𝕜 → E
hg : (Function.swap dslope c)^[n] f = g
x : 𝕜
cx : ¬x = c
⊢ dslope (a • g) c x = (a • dslope g c) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Analytic.lean | leadingCoeff_const_smul | [354, 1] | [368, 31] | simp only [dslope_of_ne _ cx, Pi.smul_apply, slope, vsub_eq_sub, ← smul_sub, smul_comm _ a] | case neg
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
g : 𝕜 → E
hg : (Function.swap dslope c)^[n] f = g
x : 𝕜
cx : ¬x = c
⊢ dslope (a • g) c x = (a • dslope g c) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c a : 𝕜
a0 : ¬a = 0
n : ℕ
h : ((Function.swap dslope c)^[n] fun z => a • f z) = a • (Function.swap dslope c)^[n] f
g : 𝕜 → E
hg : (Function.swap dslope c)^[n] f = g
x : 𝕜
cx : ¬x = c
⊢ dslope (a • g) c x = (a • dslope g c) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Analytic.lean | leadingCoeff_ne_zero | [371, 1] | [375, 95] | rcases fa with ⟨p, fp⟩ | 𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c : 𝕜
fa : AnalyticAt 𝕜 f c
o0 : orderAt f c ≠ 0
⊢ leadingCoeff f c ≠ 0 | case intro
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c : 𝕜
o0 : orderAt f c ≠ 0
p : FormalMultilinearSeries 𝕜 𝕜 E
fp : HasFPowerSeriesAt f p c
⊢ leadingCoeff f c ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c : 𝕜
fa : AnalyticAt 𝕜 f c
o0 : orderAt f c ≠ 0
⊢ leadingCoeff f c ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Analytic.lean | leadingCoeff_ne_zero | [371, 1] | [375, 95] | simp only [fp.orderAt_unique, leadingCoeff] at o0 ⊢ | case intro
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c : 𝕜
o0 : orderAt f c ≠ 0
p : FormalMultilinearSeries 𝕜 𝕜 E
fp : HasFPowerSeriesAt f p c
⊢ leadingCoeff f c ≠ 0 | case intro
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c : 𝕜
p : FormalMultilinearSeries 𝕜 𝕜 E
fp : HasFPowerSeriesAt f p c
o0 : p.order ≠ 0
⊢ (Function.swap dslope c)^[p.order] f c ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c : 𝕜
o0 : orderAt f c ≠ 0
p : FormalMultilinearSeries 𝕜 𝕜 E
fp : HasFPowerSeriesAt f p c
⊢ leadingCoeff f c ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Analytic.lean | leadingCoeff_ne_zero | [371, 1] | [375, 95] | exact fp.iterate_dslope_fslope_ne_zero (FormalMultilinearSeries.ne_zero_of_order_ne_zero o0) | case intro
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c : 𝕜
p : FormalMultilinearSeries 𝕜 𝕜 E
fp : HasFPowerSeriesAt f p c
o0 : p.order ≠ 0
⊢ (Function.swap dslope c)^[p.order] f c ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
𝕜 : Type
inst✝¹² : NontriviallyNormedField 𝕜
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : CompleteSpace E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
G : Type
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : CompleteSpace G
H : Type
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
inst✝ : CompleteSpace H
f : 𝕜 → E
c : 𝕜
p : FormalMultilinearSeries 𝕜 𝕜 E
fp : HasFPowerSeriesAt f p c
o0 : p.order ≠ 0
⊢ (Function.swap dslope c)^[p.order] f c ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_pos_lt_sq | [15, 1] | [15, 55] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
h : 0 < x
⊢ x ^ 2 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
h : 0 < x
⊢ x ^ 2 > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_pos_gt_sq | [16, 1] | [16, 55] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
h : x > 0
⊢ x ^ 2 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
h : x > 0
⊢ x ^ 2 > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_pos_mul | [17, 1] | [17, 67] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
p : x > 0
q : y > 0
⊢ x * y > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
p : x > 0
q : y > 0
⊢ x * y > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_pos_div | [18, 1] | [18, 67] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
p : x > 0
q : y > 0
⊢ x / y > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
p : x > 0
q : y > 0
⊢ x / y > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_pos_4 | [19, 1] | [19, 37] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 < 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 < 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_pos_7 | [20, 1] | [20, 37] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 < 7 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 < 7
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_pos_4_real | [21, 1] | [21, 48] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 < 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 < 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_pos_7_real | [22, 1] | [22, 48] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 < 7 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 < 7
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_pos_zero_one | [23, 1] | [23, 50] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_pos_nnreal | [24, 1] | [24, 60] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
h : u > 0
⊢ 0 < ↑u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
h : u > 0
⊢ 0 < ↑u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_pos_pow | [25, 1] | [25, 41] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 < 2 ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 < 2 ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_pos_inv | [26, 1] | [26, 47] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 < 1⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 < 1⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_abs | [34, 1] | [34, 39] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 ≤ Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_abs_ge | [35, 1] | [35, 42] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ Complex.abs z ≥ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ Complex.abs z ≥ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_nn_sq | [36, 1] | [36, 39] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ x ^ 2 ≥ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ x ^ 2 ≥ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_mul | [37, 1] | [37, 63] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
p : x ≥ 0
q : y ≥ 0
⊢ x * y ≥ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
p : x ≥ 0
q : y ≥ 0
⊢ x * y ≥ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_div | [38, 1] | [38, 63] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
p : x ≥ 0
q : y ≥ 0
⊢ x / y ≥ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
p : x ≥ 0
q : y ≥ 0
⊢ x / y ≥ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_add | [39, 1] | [39, 63] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
p : x ≥ 0
q : y ≥ 0
⊢ x + y ≥ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
p : x ≥ 0
q : y ≥ 0
⊢ x + y ≥ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_nat | [40, 1] | [40, 41] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ ↑n ≥ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ ↑n ≥ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_num | [41, 1] | [41, 35] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 ≤ 7 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 ≤ 7
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_num_real | [42, 1] | [42, 46] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 ≤ 7 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 ≤ 7
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_zero_one | [43, 1] | [43, 46] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_nnreal | [44, 1] | [44, 44] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 ≤ ↑u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 ≤ ↑u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_zero | [45, 1] | [45, 42] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 ≤ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 ≤ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_pow | [46, 1] | [46, 37] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 ≤ 2 ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 ≤ 2 ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_inv | [47, 1] | [47, 43] | bound | n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 ≤ 0⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
x y : ℝ
u : ℝ≥0
z : ℂ
⊢ 0 ≤ 0⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_sq | [54, 1] | [54, 62] | bound | a b c x y : ℝ
z : ℂ
n✝ : ℕ
n : x ≥ 0
h : x ≤ y
⊢ x ^ 2 ≤ y ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : ℝ
z : ℂ
n✝ : ℕ
n : x ≥ 0
h : x ≤ y
⊢ x ^ 2 ≤ y ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_sq_ge | [55, 1] | [55, 65] | bound | a b c x y : ℝ
z : ℂ
n✝ : ℕ
n : x ≥ 0
h : x ≤ y
⊢ y ^ 2 ≥ x ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : ℝ
z : ℂ
n✝ : ℕ
n : x ≥ 0
h : x ≤ y
⊢ y ^ 2 ≥ x ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_mul_left | [56, 1] | [56, 72] | bound | a b c x y : ℝ
z : ℂ
n✝ : ℕ
n : a ≥ 0
h : x ≤ y
⊢ a * x ≤ a * y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : ℝ
z : ℂ
n✝ : ℕ
n : a ≥ 0
h : x ≤ y
⊢ a * x ≤ a * y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_mul_right | [57, 1] | [57, 73] | bound | a b c x y : ℝ
z : ℂ
n✝ : ℕ
n : a ≥ 0
h : x ≤ y
⊢ x * a ≤ y * a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : ℝ
z : ℂ
n✝ : ℕ
n : a ≥ 0
h : x ≤ y
⊢ x * a ≤ y * a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_mul_both | [58, 1] | [58, 100] | bound | a b c x y : ℝ
z : ℂ
n : ℕ
bp : b ≥ 0
xp : x ≥ 0
ab : a ≤ b
xy : x ≤ y
⊢ a * x ≤ b * y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : ℝ
z : ℂ
n : ℕ
bp : b ≥ 0
xp : x ≥ 0
ab : a ≤ b
xy : x ≤ y
⊢ a * x ≤ b * y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_abs_mul | [59, 1] | [59, 67] | bound | a b c x y : ℝ
z : ℂ
n : ℕ
h : x ≤ y
⊢ Complex.abs z * x ≤ Complex.abs z * y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : ℝ
z : ℂ
n : ℕ
h : x ≤ y
⊢ Complex.abs z * x ≤ Complex.abs z * y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_add_left | [60, 1] | [60, 60] | bound | a b c x y : ℝ
z : ℂ
n : ℕ
h : x ≤ y
⊢ a + x ≤ a + y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : ℝ
z : ℂ
n : ℕ
h : x ≤ y
⊢ a + x ≤ a + y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_add_right | [61, 1] | [61, 61] | bound | a b c x y : ℝ
z : ℂ
n : ℕ
h : x ≤ y
⊢ x + a ≤ y + a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : ℝ
z : ℂ
n : ℕ
h : x ≤ y
⊢ x + a ≤ y + a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_add_both | [62, 1] | [62, 74] | bound | a b c x y : ℝ
z : ℂ
n : ℕ
ab : a ≤ b
xy : x ≤ y
⊢ a + x ≤ b + y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : ℝ
z : ℂ
n : ℕ
ab : a ≤ b
xy : x ≤ y
⊢ a + x ≤ b + y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_sub_left | [63, 1] | [63, 60] | bound | a b c x y : ℝ
z : ℂ
n : ℕ
h : x ≥ y
⊢ a - x ≤ a - y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : ℝ
z : ℂ
n : ℕ
h : x ≥ y
⊢ a - x ≤ a - y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_sub_right | [64, 1] | [64, 61] | bound | a b c x y : ℝ
z : ℂ
n : ℕ
h : x ≤ y
⊢ x - a ≤ y - a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : ℝ
z : ℂ
n : ℕ
h : x ≤ y
⊢ x - a ≤ y - a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_sub_both | [65, 1] | [65, 74] | bound | a b c x y : ℝ
z : ℂ
n : ℕ
ab : a ≤ b
xy : x ≥ y
⊢ a - x ≤ b - y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : ℝ
z : ℂ
n : ℕ
ab : a ≤ b
xy : x ≥ y
⊢ a - x ≤ b - y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_sub_pos | [66, 1] | [66, 55] | bound | a b c x y : ℝ
z : ℂ
n : ℕ
h : x < y
⊢ y - x > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : ℝ
z : ℂ
n : ℕ
h : x < y
⊢ y - x > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_le_of_lt | [67, 1] | [67, 52] | bound | a b c x y : ℝ
z : ℂ
n : ℕ
h : x > 0
⊢ x ≥ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : ℝ
z : ℂ
n : ℕ
h : x > 0
⊢ x ≥ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_extra | [68, 1] | [68, 76] | bound [h n] | a b c x y : ℝ
z : ℂ
n : ℕ
f : ℕ → ℝ
h : ∀ (n : ℕ), f n ≥ 0
⊢ f n ≥ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : ℝ
z : ℂ
n : ℕ
f : ℕ → ℝ
h : ∀ (n : ℕ), f n ≥ 0
⊢ f n ≥ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_1_4 | [69, 1] | [69, 41] | bound | a b c x y : ℝ
z : ℂ
n : ℕ
⊢ 1 < 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : ℝ
z : ℂ
n : ℕ
⊢ 1 < 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_2_4 | [70, 1] | [70, 41] | bound | a b c x y : ℝ
z : ℂ
n : ℕ
⊢ 2 < 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : ℝ
z : ℂ
n : ℕ
⊢ 2 < 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_div_left | [71, 1] | [71, 73] | bound | a b c x y : ℝ
z : ℂ
n : ℕ
hc : c ≥ 0
h : a ≤ b
⊢ a / c ≤ b / c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : ℝ
z : ℂ
n : ℕ
hc : c ≥ 0
h : a ≤ b
⊢ a / c ≤ b / c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_div_right | [72, 1] | [72, 87] | bound | a b c x y : ℝ
z : ℂ
n : ℕ
ha : a ≥ 0
hc : c > 0
h : b ≥ c
⊢ a / b ≤ a / c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : ℝ
z : ℂ
n : ℕ
ha : a ≥ 0
hc : c > 0
h : b ≥ c
⊢ a / b ≤ a / c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_coe | [73, 1] | [73, 65] | bound | a b c x✝ y✝ : ℝ
z : ℂ
n : ℕ
x y : ℝ≥0
h : x < y
⊢ ↑x < ↑y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x✝ y✝ : ℝ
z : ℂ
n : ℕ
x y : ℝ≥0
h : x < y
⊢ ↑x < ↑y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_dist | [74, 1] | [74, 61] | bound | a b c x y : ℝ
z : ℂ
n : ℕ
⊢ dist a c ≤ dist a b + dist b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x y : ℝ
z : ℂ
n : ℕ
⊢ dist a c ≤ dist a b + dist b c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_log | [75, 1] | [75, 78] | bound | a b c x✝ y✝ : ℝ
z : ℂ
n : ℕ
x y : ℝ
x0 : 0 < x
h : x ≤ y
⊢ x.log ≤ y.log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c x✝ y✝ : ℝ
z : ℂ
n : ℕ
x y : ℝ
x0 : 0 < x
h : x ≤ y
⊢ x.log ≤ y.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_le_max_of_le_left | [81, 1] | [81, 67] | bound | a b c : ℝ
n m : ℕ
h : a ≤ b
⊢ a ≤ max b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : ℝ
n m : ℕ
h : a ≤ b
⊢ a ≤ max b c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_le_max_of_le_right | [82, 1] | [82, 68] | bound | a b c : ℝ
n m : ℕ
h : a ≤ c
⊢ a ≤ max b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : ℝ
n m : ℕ
h : a ≤ c
⊢ a ≤ max b c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_lt_max_of_lt_left | [83, 1] | [83, 67] | bound | a b c : ℝ
n m : ℕ
h : a < b
⊢ a < max b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : ℝ
n m : ℕ
h : a < b
⊢ a < max b c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_lt_max_of_lt_right | [84, 1] | [84, 68] | bound | a b c : ℝ
n m : ℕ
h : a < c
⊢ a < max b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : ℝ
n m : ℕ
h : a < c
⊢ a < max b c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_min_le_of_left_le | [85, 1] | [85, 67] | bound | a b c : ℝ
n m : ℕ
h : a ≤ c
⊢ min a b ≤ c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : ℝ
n m : ℕ
h : a ≤ c
⊢ min a b ≤ c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_min_le_of_right_le | [86, 1] | [86, 68] | bound | a b c : ℝ
n m : ℕ
h : b ≤ c
⊢ min a b ≤ c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : ℝ
n m : ℕ
h : b ≤ c
⊢ min a b ≤ c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_min_lt_of_left_lt | [87, 1] | [87, 67] | bound | a b c : ℝ
n m : ℕ
h : a < c
⊢ min a b < c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : ℝ
n m : ℕ
h : a < c
⊢ min a b < c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_min_lt_of_right_lt | [88, 1] | [88, 68] | bound | a b c : ℝ
n m : ℕ
h : b < c
⊢ min a b < c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : ℝ
n m : ℕ
h : b < c
⊢ min a b < c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_pow_le_pow_right | [89, 1] | [89, 77] | bound | a b c : ℝ
n m : ℕ
a1 : 1 ≤ a
h : m ≤ n
⊢ a ^ m ≤ a ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : ℝ
n m : ℕ
a1 : 1 ≤ a
h : m ≤ n
⊢ a ^ m ≤ a ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_pow_le_pow_of_le_one | [90, 1] | [90, 94] | bound | a b c : ℝ
n m : ℕ
a0 : 0 ≤ a
a1 : a ≤ 1
h : n ≤ m
⊢ a ^ m ≤ a ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : ℝ
n m : ℕ
a0 : 0 ≤ a
a1 : a ≤ 1
h : n ≤ m
⊢ a ^ m ≤ a ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_rpow_le_rpow_of_exponent_le | [91, 1] | [91, 88] | bound | a b c : ℝ
n m : ℕ
a1 : 1 ≤ a
h : b ≤ c
⊢ a ^ b ≤ a ^ c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : ℝ
n m : ℕ
a1 : 1 ≤ a
h : b ≤ c
⊢ a ^ b ≤ a ^ c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_rpow_le_rpow_of_exponent_ge | [92, 1] | [92, 101] | bound | a b c : ℝ
n m : ℕ
a0 : 0 < a
a1 : a ≤ 1
h : c ≤ b
⊢ a ^ b ≤ a ^ c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : ℝ
n m : ℕ
a0 : 0 < a
a1 : a ≤ 1
h : c ≤ b
⊢ a ^ b ≤ a ^ c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_with_context | [96, 1] | [101, 8] | rw [Metric.isOpen_iff] at o | s : Set ℂ
o : IsOpen s
z : ℂ
h : z ∈ s
⊢ ∃ r, r > 0 | s : Set ℂ
o : ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s
z : ℂ
h : z ∈ s
⊢ ∃ r, r > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
s : Set ℂ
o : IsOpen s
z : ℂ
h : z ∈ s
⊢ ∃ r, r > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_with_context | [96, 1] | [101, 8] | rcases o z h with ⟨t, tp, bs⟩ | s : Set ℂ
o : ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s
z : ℂ
h : z ∈ s
⊢ ∃ r, r > 0 | case intro.intro
s : Set ℂ
o : ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s
z : ℂ
h : z ∈ s
t : ℝ
tp : t > 0
bs : Metric.ball z t ⊆ s
⊢ ∃ r, r > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
s : Set ℂ
o : ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s
z : ℂ
h : z ∈ s
⊢ ∃ r, r > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_with_context | [96, 1] | [101, 8] | exists t/2 | case intro.intro
s : Set ℂ
o : ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s
z : ℂ
h : z ∈ s
t : ℝ
tp : t > 0
bs : Metric.ball z t ⊆ s
⊢ ∃ r, r > 0 | case intro.intro
s : Set ℂ
o : ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s
z : ℂ
h : z ∈ s
t : ℝ
tp : t > 0
bs : Metric.ball z t ⊆ s
⊢ t / 2 > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s : Set ℂ
o : ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s
z : ℂ
h : z ∈ s
t : ℝ
tp : t > 0
bs : Metric.ball z t ⊆ s
⊢ ∃ r, r > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_with_context | [96, 1] | [101, 8] | clear o h bs z s | case intro.intro
s : Set ℂ
o : ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s
z : ℂ
h : z ∈ s
t : ℝ
tp : t > 0
bs : Metric.ball z t ⊆ s
⊢ t / 2 > 0 | case intro.intro
t : ℝ
tp : t > 0
⊢ t / 2 > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s : Set ℂ
o : ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s
z : ℂ
h : z ∈ s
t : ℝ
tp : t > 0
bs : Metric.ball z t ⊆ s
⊢ t / 2 > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_with_context | [96, 1] | [101, 8] | bound | case intro.intro
t : ℝ
tp : t > 0
⊢ t / 2 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
t : ℝ
tp : t > 0
⊢ t / 2 > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_try_elab | [104, 1] | [110, 50] | calc abs (f z) = abs (f w - (f w - f z)) := by ring_nf
_ ≤ abs (f w) + abs (f w - f z) := by bound
_ ≤ c * abs w + e := by bound [h w wr, sc wz] | f : ℂ → ℂ
z w : ℂ
s r c e : ℝ
sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e
wz : Complex.abs (w - z) < s
wr : Complex.abs w < r
h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z
⊢ Complex.abs (f z) ≤ c * Complex.abs w + e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
z w : ℂ
s r c e : ℝ
sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e
wz : Complex.abs (w - z) < s
wr : Complex.abs w < r
h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z
⊢ Complex.abs (f z) ≤ c * Complex.abs w + e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_try_elab | [104, 1] | [110, 50] | ring_nf | f : ℂ → ℂ
z w : ℂ
s r c e : ℝ
sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e
wz : Complex.abs (w - z) < s
wr : Complex.abs w < r
h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z
⊢ Complex.abs (f z) = Complex.abs (f w - (f w - f z)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
z w : ℂ
s r c e : ℝ
sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e
wz : Complex.abs (w - z) < s
wr : Complex.abs w < r
h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z
⊢ Complex.abs (f z) = Complex.abs (f w - (f w - f z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_try_elab | [104, 1] | [110, 50] | bound | f : ℂ → ℂ
z w : ℂ
s r c e : ℝ
sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e
wz : Complex.abs (w - z) < s
wr : Complex.abs w < r
h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z
⊢ Complex.abs (f w - (f w - f z)) ≤ Complex.abs (f w) + Complex.abs (f w - f z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
z w : ℂ
s r c e : ℝ
sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e
wz : Complex.abs (w - z) < s
wr : Complex.abs w < r
h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z
⊢ Complex.abs (f w - (f w - f z)) ≤ Complex.abs (f w) + Complex.abs (f w - f z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_try_elab | [104, 1] | [110, 50] | bound [h w wr, sc wz] | f : ℂ → ℂ
z w : ℂ
s r c e : ℝ
sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e
wz : Complex.abs (w - z) < s
wr : Complex.abs w < r
h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z
⊢ Complex.abs (f w) + Complex.abs (f w - f z) ≤ c * Complex.abs w + e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
z w : ℂ
s r c e : ℝ
sc : ∀ {w : ℂ}, Complex.abs (w - z) < s → Complex.abs (f w - f z) < e
wz : Complex.abs (w - z) < s
wr : Complex.abs w < r
h : ∀ (z : ℂ), Complex.abs z < r → Complex.abs (f z) ≤ c * Complex.abs z
⊢ Complex.abs (f w) + Complex.abs (f w - f z) ≤ c * Complex.abs w + e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_fun_inference | [113, 1] | [115, 8] | bound | α : Type
s : Finset α
f g : α → ℂ
⊢ ‖s.sum fun x => f x + g x‖ ≤ s.sum fun x => ‖f x + g x‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
s : Finset α
f g : α → ℂ
⊢ ‖s.sum fun x => f x + g x‖ ≤ s.sum fun x => ‖f x + g x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_whnf | [118, 1] | [119, 14] | bound [h.1] | x y : ℝ
h : x < y ∧ True
⊢ x ≤ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
h : x < y ∧ True
⊢ x ≤ y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_unknown_identifier | [122, 1] | [123, 24] | intro n | f : ℕ → ℝ
le : ∀ (n : ℕ), f n ≤ ↑n
⊢ ∀ (n : ℕ), f n ≤ ↑n | f : ℕ → ℝ
le : ∀ (n : ℕ), f n ≤ ↑n
n : ℕ
⊢ f n ≤ ↑n | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℝ
le : ∀ (n : ℕ), f n ≤ ↑n
⊢ ∀ (n : ℕ), f n ≤ ↑n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | test_unknown_identifier | [122, 1] | [123, 24] | bound [le n] | f : ℕ → ℝ
le : ∀ (n : ℕ), f n ≤ ↑n
n : ℕ
⊢ f n ≤ ↑n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℝ
le : ∀ (n : ℕ), f n ≤ ↑n
n : ℕ
⊢ f n ≤ ↑n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | le_sqr_add | [126, 1] | [132, 30] | calc Complex.abs (z^2 + c)
_ ≥ Complex.abs (z^2) - abs c := by bound
_ ≥ Complex.abs (z^2) - abs z := by bound
_ ≥ (abs z - 1) * abs z := by rw [mul_comm, mul_sub_one, ←pow_two, ←Complex.abs.map_pow]
_ ≥ 2 * abs z := by bound | c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ 2 * Complex.abs z ≤ Complex.abs (z ^ 2 + c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ 2 * Complex.abs z ≤ Complex.abs (z ^ 2 + c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | le_sqr_add | [126, 1] | [132, 30] | bound | c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ Complex.abs (z ^ 2 + c) ≥ Complex.abs (z ^ 2) - Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ Complex.abs (z ^ 2 + c) ≥ Complex.abs (z ^ 2) - Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | le_sqr_add | [126, 1] | [132, 30] | bound | c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ Complex.abs (z ^ 2) - Complex.abs c ≥ Complex.abs (z ^ 2) - Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ Complex.abs (z ^ 2) - Complex.abs c ≥ Complex.abs (z ^ 2) - Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | le_sqr_add | [126, 1] | [132, 30] | rw [mul_comm, mul_sub_one, ←pow_two, ←Complex.abs.map_pow] | c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ Complex.abs (z ^ 2) - Complex.abs z ≥ (Complex.abs z - 1) * Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ Complex.abs (z ^ 2) - Complex.abs z ≥ (Complex.abs z - 1) * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Tactic/BoundTests.lean | le_sqr_add | [126, 1] | [132, 30] | bound | c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ (Complex.abs z - 1) * Complex.abs z ≥ 2 * Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ (Complex.abs z - 1) * Complex.abs z ≥ 2 * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.ray_inv | [44, 1] | [49, 35] | rw [← s.ray_bij.image_eq] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ ∃ b, HolomorphicOn (I.prod I) I (uncurry b) s.post ∧ ∀ y ∈ s.ext, b y.1 (s.ray y.1 y.2) = y.2 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ ∃ b,
HolomorphicOn (I.prod I) I (uncurry b) ((fun y => (y.1, s.ray y.1 y.2)) '' s.ext) ∧
∀ y ∈ s.ext, b y.1 (s.ray y.1 y.2) = y.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ ∃ b, HolomorphicOn (I.prod I) I (uncurry b) s.post ∧ ∀ y ∈ s.ext, b y.1 (s.ray y.1 y.2) = y.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.ray_inv | [44, 1] | [49, 35] | exact global_complex_inverse_fun_open s.ray_holomorphicOn (fun _ m ↦ s.ray_noncritical m)
s.ray_bij.injOn s.isOpen_ext | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ ∃ b,
HolomorphicOn (I.prod I) I (uncurry b) ((fun y => (y.1, s.ray y.1 y.2)) '' s.ext) ∧
∀ y ∈ s.ext, b y.1 (s.ray y.1 y.2) = y.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ ∃ b,
HolomorphicOn (I.prod I) I (uncurry b) ((fun y => (y.1, s.ray y.1 y.2)) '' s.ext) ∧
∀ y ∈ s.ext, b y.1 (s.ray y.1 y.2) = y.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eq_bottcherPost | [67, 1] | [72, 22] | have h : ∃ n, (c, (f c)^[n] z) ∈ s.post := ⟨0, by simpa only [Function.iterate_zero_apply]⟩ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
⊢ s.bottcher c z = s.bottcherPost c z | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
h : ∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ s.bottcher c z = s.bottcherPost c z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
⊢ s.bottcher c z = s.bottcherPost c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eq_bottcherPost | [67, 1] | [72, 22] | have h0 := (Nat.find_eq_zero h).mpr m | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
h : ∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ s.bottcher c z = s.bottcherPost c z | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
h : ∃ n, (c, (f c)^[n] z) ∈ s.post
h0 : Nat.find h = 0
⊢ s.bottcher c z = s.bottcherPost c z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
h : ∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ s.bottcher c z = s.bottcherPost c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eq_bottcherPost | [67, 1] | [72, 22] | simp only [Super.bottcher, h, dif_pos, h0, Function.iterate_zero_apply, pow_zero, inv_one,
Complex.cpow_one] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
h : ∃ n, (c, (f c)^[n] z) ∈ s.post
h0 : Nat.find h = 0
⊢ s.bottcher c z = s.bottcherPost c z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
h : ∃ n, (c, (f c)^[n] z) ∈ s.post
h0 : Nat.find h = 0
⊢ s.bottcher c z = s.bottcherPost c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eq_bottcherPost | [67, 1] | [72, 22] | simpa only [Function.iterate_zero_apply] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
⊢ (c, (f c)^[0] z) ∈ s.post | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
⊢ (c, (f c)^[0] z) ∈ s.post
TACTIC:
|
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