full_name
stringlengths 3
121
| state
stringlengths 7
9.32k
| tactic
stringlengths 3
5.35k
| target_state
stringlengths 7
19k
| url
stringclasses 1
value | commit
stringclasses 1
value | file_path
stringlengths 21
79
|
|---|---|---|---|---|---|---|
Ordinal.bsup_eq_blsub_iff_succ
|
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}
⊢ sup (o.familyOfBFamily f) = lsub (o.familyOfBFamily f) ↔
∀ a < lsub (o.familyOfBFamily f), succ a < lsub (o.familyOfBFamily f)
|
apply <a>Ordinal.sup_eq_lsub_iff_succ</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
exists_reduced_fraction'
|
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b
|
obtain ⟨⟨a₀, y⟩, H⟩ := <a>IsLocalization.surj</a> (<a>Submonoid.powers</a> x) b
|
case intro.mk
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
H : b * (algebraMap R B) ↑(a₀, y).2 = (algebraMap R B) (a₀, y).1
⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Away/Basic.lean
|
exists_reduced_fraction'
|
case intro.mk
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
H : b * (algebraMap R B) ↑(a₀, y).2 = (algebraMap R B) (a₀, y).1
⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b
|
obtain ⟨d, hy⟩ := (<a>Submonoid.mem_powers_iff</a> y.1 x).<a>Iff.mp</a> y.2
|
case intro.mk.intro
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
H : b * (algebraMap R B) ↑(a₀, y).2 = (algebraMap R B) (a₀, y).1
d : ℕ
hy : x ^ d = ↑y
⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Away/Basic.lean
|
exists_reduced_fraction'
|
case intro.mk.intro
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
H : b * (algebraMap R B) ↑(a₀, y).2 = (algebraMap R B) (a₀, y).1
d : ℕ
hy : x ^ d = ↑y
ha₀ : a₀ ≠ 0
⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b
|
simp only [← hy] at H
|
case intro.mk.intro
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
d : ℕ
hy : x ^ d = ↑y
ha₀ : a₀ ≠ 0
H : b * (algebraMap R B) (x ^ d) = (algebraMap R B) a₀
⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Away/Basic.lean
|
exists_reduced_fraction'
|
case intro.mk.intro
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
d : ℕ
hy : x ^ d = ↑y
ha₀ : a₀ ≠ 0
H : b * (algebraMap R B) (x ^ d) = (algebraMap R B) a₀
⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b
|
obtain ⟨m, a, hyp1, hyp2⟩ := <a>WfDvdMonoid.max_power_factor</a> ha₀ hx
|
case intro.mk.intro.intro.intro.intro
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
d : ℕ
hy : x ^ d = ↑y
ha₀ : a₀ ≠ 0
H : b * (algebraMap R B) (x ^ d) = (algebraMap R B) a₀
m : ℕ
a : R
hyp1 : ¬x ∣ a
hyp2 : a₀ = x ^ m * a
⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Away/Basic.lean
|
exists_reduced_fraction'
|
case intro.mk.intro.intro.intro.intro
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
d : ℕ
hy : x ^ d = ↑y
ha₀ : a₀ ≠ 0
H : b * (algebraMap R B) (x ^ d) = (algebraMap R B) a₀
m : ℕ
a : R
hyp1 : ¬x ∣ a
hyp2 : a₀ = x ^ m * a
⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b
|
refine ⟨a, m - d, ?_⟩
|
case intro.mk.intro.intro.intro.intro
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
d : ℕ
hy : x ^ d = ↑y
ha₀ : a₀ ≠ 0
H : b * (algebraMap R B) (x ^ d) = (algebraMap R B) a₀
m : ℕ
a : R
hyp1 : ¬x ∣ a
hyp2 : a₀ = x ^ m * a
⊢ ¬x ∣ a ∧ selfZPow x B (↑m - ↑d) * (algebraMap R B) a = b
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Away/Basic.lean
|
exists_reduced_fraction'
|
case intro.mk.intro.intro.intro.intro
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
d : ℕ
hy : x ^ d = ↑y
ha₀ : a₀ ≠ 0
H : b * (algebraMap R B) (x ^ d) = (algebraMap R B) a₀
m : ℕ
a : R
hyp1 : ¬x ∣ a
hyp2 : a₀ = x ^ m * a
⊢ ¬x ∣ a ∧ selfZPow x B (↑m - ↑d) * (algebraMap R B) a = b
|
rw [← <a>IsLocalization.mk'_one</a> (M := <a>Submonoid.powers</a> x) B, <a>selfZPow_pow_sub</a>, <a>selfZPow_natCast</a>, <a>selfZPow_natCast</a>, ← <a>map_pow</a> _ _ d, <a>mul_comm</a> _ b, H, hyp2, <a>map_mul</a>, <a>map_pow</a> _ _ m]
|
case intro.mk.intro.intro.intro.intro
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
d : ℕ
hy : x ^ d = ↑y
ha₀ : a₀ ≠ 0
H : b * (algebraMap R B) (x ^ d) = (algebraMap R B) a₀
m : ℕ
a : R
hyp1 : ¬x ∣ a
hyp2 : a₀ = x ^ m * a
⊢ ¬x ∣ a ∧ (algebraMap R B) x ^ m * mk' B a 1 = (algebraMap R B) x ^ m * (algebraMap R B) a
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Away/Basic.lean
|
exists_reduced_fraction'
|
case intro.mk.intro.intro.intro.intro
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
d : ℕ
hy : x ^ d = ↑y
ha₀ : a₀ ≠ 0
H : b * (algebraMap R B) (x ^ d) = (algebraMap R B) a₀
m : ℕ
a : R
hyp1 : ¬x ∣ a
hyp2 : a₀ = x ^ m * a
⊢ ¬x ∣ a ∧ (algebraMap R B) x ^ m * mk' B a 1 = (algebraMap R B) x ^ m * (algebraMap R B) a
|
exact ⟨hyp1, <a>congr_arg</a> _ (<a>IsLocalization.mk'_one</a> _ _)⟩
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Away/Basic.lean
|
exists_reduced_fraction'
|
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
H : b * (algebraMap R B) ↑(a₀, y).2 = (algebraMap R B) (a₀, y).1
d : ℕ
hy : x ^ d = ↑y
⊢ a₀ ≠ 0
|
haveI := @<a>IsLocalization.isDomain_of_le_nonZeroDivisors</a> B _ R _ _ _ (<a>Submonoid.powers</a> x) _ (<a>powers_le_nonZeroDivisors_of_noZeroDivisors</a> hx.ne_zero)
|
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
H : b * (algebraMap R B) ↑(a₀, y).2 = (algebraMap R B) (a₀, y).1
d : ℕ
hy : x ^ d = ↑y
this : IsDomain B
⊢ a₀ ≠ 0
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Away/Basic.lean
|
exists_reduced_fraction'
|
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
H : b * (algebraMap R B) ↑(a₀, y).2 = (algebraMap R B) (a₀, y).1
d : ℕ
hy : x ^ d = ↑y
this : IsDomain B
⊢ a₀ ≠ 0
|
simp only [<a>map_zero</a>, ← hy, <a>map_pow</a>] at H
|
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
d : ℕ
hy : x ^ d = ↑y
this : IsDomain B
H : b * (algebraMap R B) x ^ d = (algebraMap R B) a₀
⊢ a₀ ≠ 0
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Away/Basic.lean
|
exists_reduced_fraction'
|
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
d : ℕ
hy : x ^ d = ↑y
this : IsDomain B
H : b * (algebraMap R B) x ^ d = (algebraMap R B) a₀
⊢ a₀ ≠ 0
|
apply ((<a>injective_iff_map_eq_zero'</a> (<a>algebraMap</a> R B)).<a>Iff.mp</a> _ a₀).mpr.mt
|
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
d : ℕ
hy : x ^ d = ↑y
this : IsDomain B
H : b * (algebraMap R B) x ^ d = (algebraMap R B) a₀
⊢ ¬(algebraMap R B) a₀ = 0
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
d : ℕ
hy : x ^ d = ↑y
this : IsDomain B
H : b * (algebraMap R B) x ^ d = (algebraMap R B) a₀
⊢ Function.Injective ⇑(algebraMap R B)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Away/Basic.lean
|
exists_reduced_fraction'
|
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
d : ℕ
hy : x ^ d = ↑y
this : IsDomain B
H : b * (algebraMap R B) x ^ d = (algebraMap R B) a₀
⊢ ¬(algebraMap R B) a₀ = 0
|
rw [← H]
|
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
d : ℕ
hy : x ^ d = ↑y
this : IsDomain B
H : b * (algebraMap R B) x ^ d = (algebraMap R B) a₀
⊢ ¬b * (algebraMap R B) x ^ d = 0
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Away/Basic.lean
|
exists_reduced_fraction'
|
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
d : ℕ
hy : x ^ d = ↑y
this : IsDomain B
H : b * (algebraMap R B) x ^ d = (algebraMap R B) a₀
⊢ ¬b * (algebraMap R B) x ^ d = 0
|
apply <a>mul_ne_zero</a> hb (<a>pow_ne_zero</a> _ _)
|
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
d : ℕ
hy : x ^ d = ↑y
this : IsDomain B
H : b * (algebraMap R B) x ^ d = (algebraMap R B) a₀
⊢ (algebraMap R B) x ≠ 0
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Away/Basic.lean
|
exists_reduced_fraction'
|
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
d : ℕ
hy : x ^ d = ↑y
this : IsDomain B
H : b * (algebraMap R B) x ^ d = (algebraMap R B) a₀
⊢ (algebraMap R B) x ≠ 0
|
exact <a>IsLocalization.to_map_ne_zero_of_mem_nonZeroDivisors</a> B (<a>powers_le_nonZeroDivisors_of_noZeroDivisors</a> hx.ne_zero) (mem_nonZeroDivisors_iff_ne_zero.mpr hx.ne_zero)
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Away/Basic.lean
|
exists_reduced_fraction'
|
R : Type u_1
inst✝⁵ : CommRing R
x : R
B : Type u_2
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : IsLocalization.Away x B
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
b : B
hb : b ≠ 0
hx : Irreducible x
a₀ : R
y : ↥(Submonoid.powers x)
d : ℕ
hy : x ^ d = ↑y
this : IsDomain B
H : b * (algebraMap R B) x ^ d = (algebraMap R B) a₀
⊢ Function.Injective ⇑(algebraMap R B)
|
exact <a>IsLocalization.injective</a> B (<a>powers_le_nonZeroDivisors_of_noZeroDivisors</a> hx.ne_zero)
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/Localization/Away/Basic.lean
|
Polynomial.hasseDeriv_eq_zero_of_lt_natDegree
|
R : Type u_1
inst✝ : Semiring R
k : ℕ
f p : R[X]
n : ℕ
h : p.natDegree < n
⊢ (hasseDeriv n) p = 0
|
rw [<a>Polynomial.hasseDeriv_apply</a>, <a>Polynomial.sum_def</a>]
|
R : Type u_1
inst✝ : Semiring R
k : ℕ
f p : R[X]
n : ℕ
h : p.natDegree < n
⊢ ∑ n_1 ∈ p.support, (monomial (n_1 - n)) (↑(n_1.choose n) * p.coeff n_1) = 0
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Polynomial/HasseDeriv.lean
|
Polynomial.hasseDeriv_eq_zero_of_lt_natDegree
|
R : Type u_1
inst✝ : Semiring R
k : ℕ
f p : R[X]
n : ℕ
h : p.natDegree < n
⊢ ∑ n_1 ∈ p.support, (monomial (n_1 - n)) (↑(n_1.choose n) * p.coeff n_1) = 0
|
refine <a>Finset.sum_eq_zero</a> fun x hx => ?_
|
R : Type u_1
inst✝ : Semiring R
k : ℕ
f p : R[X]
n : ℕ
h : p.natDegree < n
x : ℕ
hx : x ∈ p.support
⊢ (monomial (x - n)) (↑(x.choose n) * p.coeff x) = 0
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Polynomial/HasseDeriv.lean
|
Polynomial.hasseDeriv_eq_zero_of_lt_natDegree
|
R : Type u_1
inst✝ : Semiring R
k : ℕ
f p : R[X]
n : ℕ
h : p.natDegree < n
x : ℕ
hx : x ∈ p.support
⊢ (monomial (x - n)) (↑(x.choose n) * p.coeff x) = 0
|
simp [<a>Nat.choose_eq_zero_of_lt</a> ((<a>Polynomial.le_natDegree_of_mem_supp</a> _ hx).<a>LE.le.trans_lt</a> h)]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Polynomial/HasseDeriv.lean
|
AddSubgroup.nsmul_mem_zmultiples_iff_exists_sub_div
|
R : Type u_1
inst✝¹ : DivisionRing R
inst✝ : CharZero R
p r : R
n : ℕ
hn : n ≠ 0
⊢ n • r ∈ zmultiples p ↔ ∃ k, r - ↑k • (p / ↑n) ∈ zmultiples p
|
rw [← <a>natCast_zsmul</a> r, <a>AddSubgroup.zsmul_mem_zmultiples_iff_exists_sub_div</a> (Int.natCast_ne_zero.mpr hn), <a>Int.cast_natCast</a>]
|
R : Type u_1
inst✝¹ : DivisionRing R
inst✝ : CharZero R
p r : R
n : ℕ
hn : n ≠ 0
⊢ (∃ k, r - ↑k • (p / ↑n) ∈ zmultiples p) ↔ ∃ k, r - ↑k • (p / ↑n) ∈ zmultiples p
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/CharZero/Quotient.lean
|
AddSubgroup.nsmul_mem_zmultiples_iff_exists_sub_div
|
R : Type u_1
inst✝¹ : DivisionRing R
inst✝ : CharZero R
p r : R
n : ℕ
hn : n ≠ 0
⊢ (∃ k, r - ↑k • (p / ↑n) ∈ zmultiples p) ↔ ∃ k, r - ↑k • (p / ↑n) ∈ zmultiples p
|
rfl
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/CharZero/Quotient.lean
|
IsProperMap.isClosedMap
|
X : Type u_1
Y : Type u_2
Z : Type u_3
W : Type u_4
ι : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace W
f : X → Y
g : Y → Z
h : IsProperMap f
⊢ IsClosedMap f
|
rw [<a>isClosedMap_iff_clusterPt</a>]
|
X : Type u_1
Y : Type u_2
Z : Type u_3
W : Type u_4
ι : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace W
f : X → Y
g : Y → Z
h : IsProperMap f
⊢ ∀ (s : Set X) (y : Y), MapClusterPt y (𝓟 s) f → ∃ x, f x = y ∧ ClusterPt x (𝓟 s)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/ProperMap.lean
|
IsProperMap.isClosedMap
|
X : Type u_1
Y : Type u_2
Z : Type u_3
W : Type u_4
ι : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace W
f : X → Y
g : Y → Z
h : IsProperMap f
⊢ ∀ (s : Set X) (y : Y), MapClusterPt y (𝓟 s) f → ∃ x, f x = y ∧ ClusterPt x (𝓟 s)
|
exact fun s y ↦ h.clusterPt_of_mapClusterPt (ℱ := 𝓟 s) (y := y)
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/ProperMap.lean
|
Finset.map_subtype_embedding_Ico
|
α : Type u_1
β : Type u_2
inst✝² : Preorder α
p : α → Prop
inst✝¹ : DecidablePred p
inst✝ : LocallyFiniteOrder α
a b : Subtype p
hp : ∀ ⦃a b x : α⦄, a ≤ x → x ≤ b → p a → p b → p x
⊢ map (Embedding.subtype p) (Ico a b) = Ico ↑a ↑b
|
rw [<a>Finset.subtype_Ico_eq</a>]
|
α : Type u_1
β : Type u_2
inst✝² : Preorder α
p : α → Prop
inst✝¹ : DecidablePred p
inst✝ : LocallyFiniteOrder α
a b : Subtype p
hp : ∀ ⦃a b x : α⦄, a ≤ x → x ≤ b → p a → p b → p x
⊢ map (Embedding.subtype p) (Finset.subtype p (Ico ↑a ↑b)) = Ico ↑a ↑b
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Interval/Finset/Defs.lean
|
Finset.map_subtype_embedding_Ico
|
α : Type u_1
β : Type u_2
inst✝² : Preorder α
p : α → Prop
inst✝¹ : DecidablePred p
inst✝ : LocallyFiniteOrder α
a b : Subtype p
hp : ∀ ⦃a b x : α⦄, a ≤ x → x ≤ b → p a → p b → p x
⊢ map (Embedding.subtype p) (Finset.subtype p (Ico ↑a ↑b)) = Ico ↑a ↑b
|
refine <a>Finset.subtype_map_of_mem</a> fun x hx => ?_
|
α : Type u_1
β : Type u_2
inst✝² : Preorder α
p : α → Prop
inst✝¹ : DecidablePred p
inst✝ : LocallyFiniteOrder α
a b : Subtype p
hp : ∀ ⦃a b x : α⦄, a ≤ x → x ≤ b → p a → p b → p x
x : α
hx : x ∈ Ico ↑a ↑b
⊢ p x
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Interval/Finset/Defs.lean
|
Finset.map_subtype_embedding_Ico
|
α : Type u_1
β : Type u_2
inst✝² : Preorder α
p : α → Prop
inst✝¹ : DecidablePred p
inst✝ : LocallyFiniteOrder α
a b : Subtype p
hp : ∀ ⦃a b x : α⦄, a ≤ x → x ≤ b → p a → p b → p x
x : α
hx : x ∈ Ico ↑a ↑b
⊢ p x
|
rw [<a>Finset.mem_Ico</a>] at hx
|
α : Type u_1
β : Type u_2
inst✝² : Preorder α
p : α → Prop
inst✝¹ : DecidablePred p
inst✝ : LocallyFiniteOrder α
a b : Subtype p
hp : ∀ ⦃a b x : α⦄, a ≤ x → x ≤ b → p a → p b → p x
x : α
hx : ↑a ≤ x ∧ x < ↑b
⊢ p x
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Interval/Finset/Defs.lean
|
Finset.map_subtype_embedding_Ico
|
α : Type u_1
β : Type u_2
inst✝² : Preorder α
p : α → Prop
inst✝¹ : DecidablePred p
inst✝ : LocallyFiniteOrder α
a b : Subtype p
hp : ∀ ⦃a b x : α⦄, a ≤ x → x ≤ b → p a → p b → p x
x : α
hx : ↑a ≤ x ∧ x < ↑b
⊢ p x
|
exact hp hx.1 hx.2.<a>LT.lt.le</a> a.prop b.prop
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Interval/Finset/Defs.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))
|
letI F : Type _ := E
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))
|
letI : <a>NormedAddCommGroup</a> F := { norm := g dist := fun x y => g (x - y) dist_self := by simp only [<a>sub_self</a>, h1, <a>forall_const</a>] dist_comm := fun _ _ => by dsimp [<a>Dist.dist</a>]; rw [← h2, <a>neg_sub</a>] dist_triangle := fun x y z => by convert h3 (x - y) (y - z) using 1; abel_nf edist := fun x y => .ofReal (g (x - y)) edist_dist := fun _ _ => <a>rfl</a> eq_of_dist_eq_zero := by convert fun _ _ h => <a>eq_of_sub_eq_zero</a> (h4 h) }
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))
|
letI : <a>NormedSpace</a> ℝ F := { norm_smul_le := fun _ _ ↦ h5 _ _ }
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this : NormedSpace ℝ F := NormedSpace.mk ⋯
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this : NormedSpace ℝ F := NormedSpace.mk ⋯
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))
|
letI : <a>TopologicalSpace</a> F := UniformSpace.toTopologicalSpace
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝¹ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝ : NormedSpace ℝ F := NormedSpace.mk ⋯
this : TopologicalSpace F := UniformSpace.toTopologicalSpace
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝¹ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝ : NormedSpace ℝ F := NormedSpace.mk ⋯
this : TopologicalSpace F := UniformSpace.toTopologicalSpace
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))
|
letI : <a>MeasurableSpace</a> F := <a>borel</a> F
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝² : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝¹ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝ : TopologicalSpace F := UniformSpace.toTopologicalSpace
this : MeasurableSpace F := borel F
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝² : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝¹ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝ : TopologicalSpace F := UniformSpace.toTopologicalSpace
this : MeasurableSpace F := borel F
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))
|
have : <a>BorelSpace</a> F := { measurable_eq := <a>rfl</a> }
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝³ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝² : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝¹ : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝ : MeasurableSpace F := borel F
this : BorelSpace F
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝³ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝² : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝¹ : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝ : MeasurableSpace F := borel F
this : BorelSpace F
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))
|
let φ := @<a>LinearEquiv.toContinuousLinearEquiv</a> ℝ _ E _ _ tE _ _ F _ _ _ _ _ _ _ _ _ (<a>LinearEquiv.refl</a> ℝ E : E ≃ₗ[ℝ] F)
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝³ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝² : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝¹ : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝ : MeasurableSpace F := borel F
this : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝³ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝² : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝¹ : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝ : MeasurableSpace F := borel F
this : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))
|
let ν : <a>MeasureTheory.Measure</a> F := @<a>MeasureTheory.Measure.map</a> E F _ mE φ μ
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝³ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝² : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝¹ : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝ : MeasurableSpace F := borel F
this : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝³ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝² : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝¹ : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝ : MeasurableSpace F := borel F
this : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))
|
have : <a>MeasureTheory.Measure.IsAddHaarMeasure</a> ν := @<a>ContinuousLinearEquiv.isAddHaarMeasure_map</a> E F ℝ ℝ _ _ _ _ _ _ tE _ _ _ _ _ _ _ mE _ _ _ φ μ _
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁴ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝³ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝² : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this : ν.IsAddHaarMeasure
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁴ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝³ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝² : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this : ν.IsAddHaarMeasure
⊢ μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))
|
convert (<a>MeasureTheory.measure_unitBall_eq_integral_div_gamma</a> ν hp) using 1
|
case h.e'_2
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁴ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝³ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝² : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this : ν.IsAddHaarMeasure
⊢ μ {x | g x < 1} = ν (Metric.ball 0 1)
case h.e'_3
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁴ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝³ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝² : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this : ν.IsAddHaarMeasure
⊢ ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1)) =
ENNReal.ofReal ((∫ (x : F), Real.exp (-‖x‖ ^ p) ∂ν) / Real.Gamma (↑(finrank ℝ F) / p + 1))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
⊢ ∀ (x : F), dist x x = 0
|
simp only [<a>sub_self</a>, h1, <a>forall_const</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
x✝¹ x✝ : F
⊢ dist x✝¹ x✝ = dist x✝ x✝¹
|
dsimp [<a>Dist.dist</a>]
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
x✝¹ x✝ : F
⊢ g (x✝¹ - x✝) = g (x✝ - x✝¹)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
x✝¹ x✝ : F
⊢ g (x✝¹ - x✝) = g (x✝ - x✝¹)
|
rw [← h2, <a>neg_sub</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
x y z : F
⊢ dist x z ≤ dist x y + dist y z
|
convert h3 (x - y) (y - z) using 1
|
case h.e'_3
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
x y z : F
⊢ dist x z = g (x - y + (y - z))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
case h.e'_3
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
x y z : F
⊢ dist x z = g (x - y + (y - z))
|
abel_nf
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
⊢ ∀ {x y : F}, dist x y = 0 → x = y
|
convert fun _ _ h => <a>eq_of_sub_eq_zero</a> (h4 h)
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
case h.e'_2
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁴ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝³ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝² : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this : ν.IsAddHaarMeasure
⊢ μ {x | g x < 1} = ν (Metric.ball 0 1)
|
rw [@<a>MeasureTheory.Measure.map_apply</a> E F mE _ μ φ _ _ <a>measurableSet_ball</a>]
|
case h.e'_2
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁴ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝³ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝² : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this : ν.IsAddHaarMeasure
⊢ μ {x | g x < 1} = μ (⇑φ ⁻¹' Metric.ball 0 1)
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁴ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝³ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝² : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this : ν.IsAddHaarMeasure
⊢ Measurable ⇑φ
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
case h.e'_2
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁴ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝³ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝² : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this : ν.IsAddHaarMeasure
⊢ μ {x | g x < 1} = μ (⇑φ ⁻¹' Metric.ball 0 1)
|
congr!
|
case h.e'_2.h.e'_6.h.e'_1.h.a
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁴ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝³ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝² : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this : ν.IsAddHaarMeasure
x✝ : E
⊢ g x✝ < 1 ↔ Metric.ball 0 1 x✝
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
case h.e'_2.h.e'_6.h.e'_1.h.a
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁴ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝³ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝² : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this : ν.IsAddHaarMeasure
x✝ : E
⊢ g x✝ < 1 ↔ Metric.ball 0 1 x✝
|
simp_rw [<a>Metric.ball</a>, <a>dist_zero_right</a>]
|
case h.e'_2.h.e'_6.h.e'_1.h.a
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁴ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝³ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝² : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this : ν.IsAddHaarMeasure
x✝ : E
⊢ g x✝ < 1 ↔ {y | ‖y‖ < 1} x✝
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
case h.e'_2.h.e'_6.h.e'_1.h.a
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁴ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝³ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝² : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this : ν.IsAddHaarMeasure
x✝ : E
⊢ g x✝ < 1 ↔ {y | ‖y‖ < 1} x✝
|
rfl
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁴ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝³ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝² : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this : ν.IsAddHaarMeasure
⊢ Measurable ⇑φ
|
refine @<a>Continuous.measurable</a> E F tE mE _ _ _ _ φ ?_
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁴ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝³ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝² : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this : ν.IsAddHaarMeasure
⊢ Continuous ⇑φ
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁴ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝³ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝² : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this : ν.IsAddHaarMeasure
⊢ Continuous ⇑φ
|
exact @<a>ContinuousLinearEquiv.continuous</a> ℝ ℝ _ _ _ _ _ _ E tE _ F _ _ _ _ φ
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
case h.e'_3
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁴ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝³ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝² : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this : ν.IsAddHaarMeasure
⊢ ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1)) =
ENNReal.ofReal ((∫ (x : F), Real.exp (-‖x‖ ^ p) ∂ν) / Real.Gamma (↑(finrank ℝ F) / p + 1))
|
let ψ := @<a>Homeomorph.toMeasurableEquiv</a> E F tE mE _ _ _ _ (@<a>ContinuousLinearEquiv.toHomeomorph</a> ℝ ℝ _ _ _ _ _ _ E tE _ F _ _ _ _ φ)
|
case h.e'_3
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁴ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝³ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝² : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this : ν.IsAddHaarMeasure
ψ : E ≃ᵐ F := φ.toHomeomorph.toMeasurableEquiv
⊢ ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1)) =
ENNReal.ofReal ((∫ (x : F), Real.exp (-‖x‖ ^ p) ∂ν) / Real.Gamma (↑(finrank ℝ F) / p + 1))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
case h.e'_3
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁴ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝³ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝² : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this : ν.IsAddHaarMeasure
ψ : E ≃ᵐ F := φ.toHomeomorph.toMeasurableEquiv
⊢ ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1)) =
ENNReal.ofReal ((∫ (x : F), Real.exp (-‖x‖ ^ p) ∂ν) / Real.Gamma (↑(finrank ℝ F) / p + 1))
|
have : @<a>MeasureTheory.MeasurePreserving</a> E F mE _ ψ μ ν := @<a>Measurable.measurePreserving</a> E F mE _ ψ (@<a>MeasurableEquiv.measurable</a> E F mE _ ψ) _
|
case h.e'_3
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁵ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝⁴ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝³ : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝² : MeasurableSpace F := borel F
this✝¹ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this✝ : ν.IsAddHaarMeasure
ψ : E ≃ᵐ F := φ.toHomeomorph.toMeasurableEquiv
this : MeasurePreserving (⇑ψ) μ ν
⊢ ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1)) =
ENNReal.ofReal ((∫ (x : F), Real.exp (-‖x‖ ^ p) ∂ν) / Real.Gamma (↑(finrank ℝ F) / p + 1))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
case h.e'_3
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁵ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝⁴ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝³ : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝² : MeasurableSpace F := borel F
this✝¹ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this✝ : ν.IsAddHaarMeasure
ψ : E ≃ᵐ F := φ.toHomeomorph.toMeasurableEquiv
this : MeasurePreserving (⇑ψ) μ ν
⊢ ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1)) =
ENNReal.ofReal ((∫ (x : F), Real.exp (-‖x‖ ^ p) ∂ν) / Real.Gamma (↑(finrank ℝ F) / p + 1))
|
erw [← this.integral_comp']
|
case h.e'_3
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁵ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝⁴ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝³ : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝² : MeasurableSpace F := borel F
this✝¹ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this✝ : ν.IsAddHaarMeasure
ψ : E ≃ᵐ F := φ.toHomeomorph.toMeasurableEquiv
this : MeasurePreserving (⇑ψ) μ ν
⊢ ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1)) =
ENNReal.ofReal ((∫ (x : E), Real.exp (-‖ψ x‖ ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ F) / p + 1))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
MeasureTheory.measure_lt_one_eq_integral_div_gamma
|
case h.e'_3
E : Type u_1
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module ℝ E
inst✝⁵ : FiniteDimensional ℝ E
mE : MeasurableSpace E
tE : TopologicalSpace E
inst✝⁴ : TopologicalAddGroup E
inst✝³ : BorelSpace E
inst✝² : T2Space E
inst✝¹ : ContinuousSMul ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
g : E → ℝ
h1 : g 0 = 0
h2 : ∀ (x : E), g (-x) = g x
h3 : ∀ (x y : E), g (x + y) ≤ g x + g y
h4 : ∀ {x : E}, g x = 0 → x = 0
h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x
p : ℝ
hp : 0 < p
F : Type u_1 := E
this✝⁵ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯
this✝⁴ : NormedSpace ℝ F := NormedSpace.mk ⋯
this✝³ : TopologicalSpace F := UniformSpace.toTopologicalSpace
this✝² : MeasurableSpace F := borel F
this✝¹ : BorelSpace F
φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv
ν : Measure F := map (⇑φ) μ
this✝ : ν.IsAddHaarMeasure
ψ : E ≃ᵐ F := φ.toHomeomorph.toMeasurableEquiv
this : MeasurePreserving (⇑ψ) μ ν
⊢ ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1)) =
ENNReal.ofReal ((∫ (x : E), Real.exp (-‖ψ x‖ ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ F) / p + 1))
|
rfl
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
Ordnode.dual_balanceL
|
α : Type u_1
l : Ordnode α
x : α
r : Ordnode α
⊢ (l.balanceL x r).dual = r.dual.balanceR x l.dual
|
unfold <a>Ordnode.balanceL</a> <a>Ordnode.balanceR</a>
|
α : Type u_1
l : Ordnode α
x : α
r : Ordnode α
⊢ (Ordnode.casesOn (motive := fun t => id r = t → Ordnode α) (id r)
(fun h =>
Ordnode.casesOn (motive := fun t => id l = t → Ordnode α) (id l) (fun h => Ordnode.singleton x)
(fun ls ll lx lr h =>
Ordnode.casesOn (motive := fun t => id ll = t → Ordnode α) (id ll)
(fun h =>
Ordnode.casesOn (motive := fun t => lr = t → Ordnode α) lr (fun h => node 2 l x nil)
(fun size l lrx r h => node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x)) ⋯)
(fun lls l x_1 r h =>
Ordnode.casesOn (motive := fun t => id lr = t → Ordnode α) (id lr)
(fun h => node 3 ll lx (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls + 1) ll lx (node (lrs + 1) lr x nil)
else node (ls + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx (node (lrr.size + 1) lrr x nil))
⋯)
⋯)
⋯)
(fun rs l_1 x_1 r_1 h =>
Ordnode.casesOn (motive := fun t => id l = t → Ordnode α) (id l) (fun h => node (rs + 1) nil x r)
(fun ls ll lx lr h =>
if ls > delta * rs then
Ordnode.casesOn (motive := fun t => id ll = t → Ordnode α) (id ll) (fun h => nil)
(fun lls l x_2 r_2 h =>
Ordnode.casesOn (motive := fun t => id lr = t → Ordnode α) (id lr) (fun h => nil)
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls + rs + 1) ll lx (node (rs + lrs + 1) lr x r)
else
node (ls + rs + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx
(node (lrr.size + rs + 1) lrr x r))
⋯)
⋯
else node (ls + rs + 1) l x r)
⋯)
⋯).dual =
Ordnode.casesOn (motive := fun t => id r.dual = t → Ordnode α) (id r.dual)
(fun h =>
Ordnode.casesOn (motive := fun t => id l.dual = t → Ordnode α) (id l.dual) (fun h => Ordnode.singleton x)
(fun rs rl rx rr h =>
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr)
(fun h =>
Ordnode.casesOn (motive := fun t => rl = t → Ordnode α) rl (fun h => node 2 nil x l.dual)
(fun size l rlx r h => node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx)) ⋯)
(fun rrs l x_1 r h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl)
(fun h => node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else node (rs + 1) (node (rll.size + 1) nil x rll) rlx (node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯)
⋯)
(fun ls l_1 x_1 r_1 h =>
Ordnode.casesOn (motive := fun t => id l.dual = t → Ordnode α) (id l.dual) (fun h => node (ls + 1) r.dual x nil)
(fun rs rl rx rr h =>
if rs > delta * ls then
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr) (fun h => nil)
(fun rrs l x_2 r_2 h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl) (fun h => nil)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (ls + rs + 1) (node (ls + rls + 1) r.dual x rl) rx rr
else
node (ls + rs + 1) (node (ls + rll.size + 1) r.dual x rll) rlx
(node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯
else node (ls + rs + 1) r.dual x l.dual)
⋯)
⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.dual_balanceL
|
α : Type u_1
l : Ordnode α
x : α
r : Ordnode α
⊢ (Ordnode.casesOn (motive := fun t => id r = t → Ordnode α) (id r)
(fun h =>
Ordnode.casesOn (motive := fun t => id l = t → Ordnode α) (id l) (fun h => Ordnode.singleton x)
(fun ls ll lx lr h =>
Ordnode.casesOn (motive := fun t => id ll = t → Ordnode α) (id ll)
(fun h =>
Ordnode.casesOn (motive := fun t => lr = t → Ordnode α) lr (fun h => node 2 l x nil)
(fun size l lrx r h => node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x)) ⋯)
(fun lls l x_1 r h =>
Ordnode.casesOn (motive := fun t => id lr = t → Ordnode α) (id lr)
(fun h => node 3 ll lx (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls + 1) ll lx (node (lrs + 1) lr x nil)
else node (ls + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx (node (lrr.size + 1) lrr x nil))
⋯)
⋯)
⋯)
(fun rs l_1 x_1 r_1 h =>
Ordnode.casesOn (motive := fun t => id l = t → Ordnode α) (id l) (fun h => node (rs + 1) nil x r)
(fun ls ll lx lr h =>
if ls > delta * rs then
Ordnode.casesOn (motive := fun t => id ll = t → Ordnode α) (id ll) (fun h => nil)
(fun lls l x_2 r_2 h =>
Ordnode.casesOn (motive := fun t => id lr = t → Ordnode α) (id lr) (fun h => nil)
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls + rs + 1) ll lx (node (rs + lrs + 1) lr x r)
else
node (ls + rs + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx
(node (lrr.size + rs + 1) lrr x r))
⋯)
⋯
else node (ls + rs + 1) l x r)
⋯)
⋯).dual =
Ordnode.casesOn (motive := fun t => id r.dual = t → Ordnode α) (id r.dual)
(fun h =>
Ordnode.casesOn (motive := fun t => id l.dual = t → Ordnode α) (id l.dual) (fun h => Ordnode.singleton x)
(fun rs rl rx rr h =>
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr)
(fun h =>
Ordnode.casesOn (motive := fun t => rl = t → Ordnode α) rl (fun h => node 2 nil x l.dual)
(fun size l rlx r h => node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx)) ⋯)
(fun rrs l x_1 r h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl)
(fun h => node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else node (rs + 1) (node (rll.size + 1) nil x rll) rlx (node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯)
⋯)
(fun ls l_1 x_1 r_1 h =>
Ordnode.casesOn (motive := fun t => id l.dual = t → Ordnode α) (id l.dual) (fun h => node (ls + 1) r.dual x nil)
(fun rs rl rx rr h =>
if rs > delta * ls then
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr) (fun h => nil)
(fun rrs l x_2 r_2 h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl) (fun h => nil)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (ls + rs + 1) (node (ls + rls + 1) r.dual x rl) rx rr
else
node (ls + rs + 1) (node (ls + rll.size + 1) r.dual x rll) rlx
(node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯
else node (ls + rs + 1) r.dual x l.dual)
⋯)
⋯
|
cases' r with rs rl rx rr
|
case nil
α : Type u_1
l : Ordnode α
x : α
⊢ (Ordnode.casesOn (motive := fun t => id nil = t → Ordnode α) (id nil)
(fun h =>
Ordnode.casesOn (motive := fun t => id l = t → Ordnode α) (id l) (fun h => Ordnode.singleton x)
(fun ls ll lx lr h =>
Ordnode.casesOn (motive := fun t => id ll = t → Ordnode α) (id ll)
(fun h =>
Ordnode.casesOn (motive := fun t => lr = t → Ordnode α) lr (fun h => node 2 l x nil)
(fun size l lrx r h => node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x)) ⋯)
(fun lls l x_1 r h =>
Ordnode.casesOn (motive := fun t => id lr = t → Ordnode α) (id lr)
(fun h => node 3 ll lx (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls + 1) ll lx (node (lrs + 1) lr x nil)
else node (ls + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx (node (lrr.size + 1) lrr x nil))
⋯)
⋯)
⋯)
(fun rs l_1 x_1 r h =>
Ordnode.casesOn (motive := fun t => id l = t → Ordnode α) (id l) (fun h => node (rs + 1) nil x nil)
(fun ls ll lx lr h =>
if ls > delta * rs then
Ordnode.casesOn (motive := fun t => id ll = t → Ordnode α) (id ll) (fun h => nil)
(fun lls l x_2 r h =>
Ordnode.casesOn (motive := fun t => id lr = t → Ordnode α) (id lr) (fun h => nil)
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls + rs + 1) ll lx (node (rs + lrs + 1) lr x nil)
else
node (ls + rs + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx
(node (lrr.size + rs + 1) lrr x nil))
⋯)
⋯
else node (ls + rs + 1) l x nil)
⋯)
⋯).dual =
Ordnode.casesOn (motive := fun t => id nil.dual = t → Ordnode α) (id nil.dual)
(fun h =>
Ordnode.casesOn (motive := fun t => id l.dual = t → Ordnode α) (id l.dual) (fun h => Ordnode.singleton x)
(fun rs rl rx rr h =>
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr)
(fun h =>
Ordnode.casesOn (motive := fun t => rl = t → Ordnode α) rl (fun h => node 2 nil x l.dual)
(fun size l rlx r h => node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx)) ⋯)
(fun rrs l x_1 r h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl)
(fun h => node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else node (rs + 1) (node (rll.size + 1) nil x rll) rlx (node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯)
⋯)
(fun ls l_1 x_1 r h =>
Ordnode.casesOn (motive := fun t => id l.dual = t → Ordnode α) (id l.dual)
(fun h => node (ls + 1) nil.dual x nil)
(fun rs rl rx rr h =>
if rs > delta * ls then
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr) (fun h => nil)
(fun rrs l x_2 r h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl) (fun h => nil)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (ls + rs + 1) (node (ls + rls + 1) nil.dual x rl) rx rr
else
node (ls + rs + 1) (node (ls + rll.size + 1) nil.dual x rll) rlx
(node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯
else node (ls + rs + 1) nil.dual x l.dual)
⋯)
⋯
case node
α : Type u_1
l : Ordnode α
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
⊢ (Ordnode.casesOn (motive := fun t => id (node rs rl rx rr) = t → Ordnode α) (id (node rs rl rx rr))
(fun h =>
Ordnode.casesOn (motive := fun t => id l = t → Ordnode α) (id l) (fun h => Ordnode.singleton x)
(fun ls ll lx lr h =>
Ordnode.casesOn (motive := fun t => id ll = t → Ordnode α) (id ll)
(fun h =>
Ordnode.casesOn (motive := fun t => lr = t → Ordnode α) lr (fun h => node 2 l x nil)
(fun size l lrx r h => node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x)) ⋯)
(fun lls l x_1 r h =>
Ordnode.casesOn (motive := fun t => id lr = t → Ordnode α) (id lr)
(fun h => node 3 ll lx (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls + 1) ll lx (node (lrs + 1) lr x nil)
else node (ls + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx (node (lrr.size + 1) lrr x nil))
⋯)
⋯)
⋯)
(fun rs_1 l_1 x_1 r h =>
Ordnode.casesOn (motive := fun t => id l = t → Ordnode α) (id l)
(fun h => node (rs_1 + 1) nil x (node rs rl rx rr))
(fun ls ll lx lr h =>
if ls > delta * rs_1 then
Ordnode.casesOn (motive := fun t => id ll = t → Ordnode α) (id ll) (fun h => nil)
(fun lls l x_2 r h =>
Ordnode.casesOn (motive := fun t => id lr = t → Ordnode α) (id lr) (fun h => nil)
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then
node (ls + rs_1 + 1) ll lx (node (rs_1 + lrs + 1) lr x (node rs rl rx rr))
else
node (ls + rs_1 + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx
(node (lrr.size + rs_1 + 1) lrr x (node rs rl rx rr)))
⋯)
⋯
else node (ls + rs_1 + 1) l x (node rs rl rx rr))
⋯)
⋯).dual =
Ordnode.casesOn (motive := fun t => id (node rs rl rx rr).dual = t → Ordnode α) (id (node rs rl rx rr).dual)
(fun h =>
Ordnode.casesOn (motive := fun t => id l.dual = t → Ordnode α) (id l.dual) (fun h => Ordnode.singleton x)
(fun rs rl rx rr h =>
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr)
(fun h =>
Ordnode.casesOn (motive := fun t => rl = t → Ordnode α) rl (fun h => node 2 nil x l.dual)
(fun size l rlx r h => node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx)) ⋯)
(fun rrs l x_1 r h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl)
(fun h => node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else node (rs + 1) (node (rll.size + 1) nil x rll) rlx (node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯)
⋯)
(fun ls l_1 x_1 r h =>
Ordnode.casesOn (motive := fun t => id l.dual = t → Ordnode α) (id l.dual)
(fun h => node (ls + 1) (node rs rl rx rr).dual x nil)
(fun rs_1 rl_1 rx_1 rr_1 h =>
if rs_1 > delta * ls then
Ordnode.casesOn (motive := fun t => id rr_1 = t → Ordnode α) (id rr_1) (fun h => nil)
(fun rrs l x_2 r h =>
Ordnode.casesOn (motive := fun t => id rl_1 = t → Ordnode α) (id rl_1) (fun h => nil)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then
node (ls + rs_1 + 1) (node (ls + rls + 1) (node rs rl rx rr).dual x rl_1) rx_1 rr_1
else
node (ls + rs_1 + 1) (node (ls + rll.size + 1) (node rs rl rx rr).dual x rll) rlx
(node (rlr.size + rrs + 1) rlr rx_1 rr_1))
⋯)
⋯
else node (ls + rs_1 + 1) (node rs rl rx rr).dual x l.dual)
⋯)
⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.dual_balanceL
|
case nil
α : Type u_1
l : Ordnode α
x : α
⊢ (Ordnode.casesOn (motive := fun t => id nil = t → Ordnode α) (id nil)
(fun h =>
Ordnode.casesOn (motive := fun t => id l = t → Ordnode α) (id l) (fun h => Ordnode.singleton x)
(fun ls ll lx lr h =>
Ordnode.casesOn (motive := fun t => id ll = t → Ordnode α) (id ll)
(fun h =>
Ordnode.casesOn (motive := fun t => lr = t → Ordnode α) lr (fun h => node 2 l x nil)
(fun size l lrx r h => node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x)) ⋯)
(fun lls l x_1 r h =>
Ordnode.casesOn (motive := fun t => id lr = t → Ordnode α) (id lr)
(fun h => node 3 ll lx (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls + 1) ll lx (node (lrs + 1) lr x nil)
else node (ls + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx (node (lrr.size + 1) lrr x nil))
⋯)
⋯)
⋯)
(fun rs l_1 x_1 r h =>
Ordnode.casesOn (motive := fun t => id l = t → Ordnode α) (id l) (fun h => node (rs + 1) nil x nil)
(fun ls ll lx lr h =>
if ls > delta * rs then
Ordnode.casesOn (motive := fun t => id ll = t → Ordnode α) (id ll) (fun h => nil)
(fun lls l x_2 r h =>
Ordnode.casesOn (motive := fun t => id lr = t → Ordnode α) (id lr) (fun h => nil)
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls + rs + 1) ll lx (node (rs + lrs + 1) lr x nil)
else
node (ls + rs + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx
(node (lrr.size + rs + 1) lrr x nil))
⋯)
⋯
else node (ls + rs + 1) l x nil)
⋯)
⋯).dual =
Ordnode.casesOn (motive := fun t => id nil.dual = t → Ordnode α) (id nil.dual)
(fun h =>
Ordnode.casesOn (motive := fun t => id l.dual = t → Ordnode α) (id l.dual) (fun h => Ordnode.singleton x)
(fun rs rl rx rr h =>
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr)
(fun h =>
Ordnode.casesOn (motive := fun t => rl = t → Ordnode α) rl (fun h => node 2 nil x l.dual)
(fun size l rlx r h => node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx)) ⋯)
(fun rrs l x_1 r h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl)
(fun h => node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else node (rs + 1) (node (rll.size + 1) nil x rll) rlx (node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯)
⋯)
(fun ls l_1 x_1 r h =>
Ordnode.casesOn (motive := fun t => id l.dual = t → Ordnode α) (id l.dual)
(fun h => node (ls + 1) nil.dual x nil)
(fun rs rl rx rr h =>
if rs > delta * ls then
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr) (fun h => nil)
(fun rrs l x_2 r h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl) (fun h => nil)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (ls + rs + 1) (node (ls + rls + 1) nil.dual x rl) rx rr
else
node (ls + rs + 1) (node (ls + rll.size + 1) nil.dual x rll) rlx
(node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯
else node (ls + rs + 1) nil.dual x l.dual)
⋯)
⋯
|
cases' l with ls ll lx lr
|
case nil.nil
α : Type u_1
x : α
⊢ (Ordnode.casesOn (motive := fun t => id nil = t → Ordnode α) (id nil)
(fun h =>
Ordnode.casesOn (motive := fun t => id nil = t → Ordnode α) (id nil) (fun h => Ordnode.singleton x)
(fun ls ll lx lr h =>
Ordnode.casesOn (motive := fun t => id ll = t → Ordnode α) (id ll)
(fun h =>
Ordnode.casesOn (motive := fun t => lr = t → Ordnode α) lr (fun h => node 2 nil x nil)
(fun size l lrx r h => node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x)) ⋯)
(fun lls l x_1 r h =>
Ordnode.casesOn (motive := fun t => id lr = t → Ordnode α) (id lr)
(fun h => node 3 ll lx (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls + 1) ll lx (node (lrs + 1) lr x nil)
else node (ls + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx (node (lrr.size + 1) lrr x nil))
⋯)
⋯)
⋯)
(fun rs l x_1 r h =>
Ordnode.casesOn (motive := fun t => id nil = t → Ordnode α) (id nil) (fun h => node (rs + 1) nil x nil)
(fun ls ll lx lr h =>
if ls > delta * rs then
Ordnode.casesOn (motive := fun t => id ll = t → Ordnode α) (id ll) (fun h => nil)
(fun lls l x_2 r h =>
Ordnode.casesOn (motive := fun t => id lr = t → Ordnode α) (id lr) (fun h => nil)
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls + rs + 1) ll lx (node (rs + lrs + 1) lr x nil)
else
node (ls + rs + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx
(node (lrr.size + rs + 1) lrr x nil))
⋯)
⋯
else node (ls + rs + 1) nil x nil)
⋯)
⋯).dual =
Ordnode.casesOn (motive := fun t => id nil.dual = t → Ordnode α) (id nil.dual)
(fun h =>
Ordnode.casesOn (motive := fun t => id nil.dual = t → Ordnode α) (id nil.dual) (fun h => Ordnode.singleton x)
(fun rs rl rx rr h =>
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr)
(fun h =>
Ordnode.casesOn (motive := fun t => rl = t → Ordnode α) rl (fun h => node 2 nil x nil.dual)
(fun size l rlx r h => node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx)) ⋯)
(fun rrs l x_1 r h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl)
(fun h => node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else node (rs + 1) (node (rll.size + 1) nil x rll) rlx (node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯)
⋯)
(fun ls l x_1 r h =>
Ordnode.casesOn (motive := fun t => id nil.dual = t → Ordnode α) (id nil.dual)
(fun h => node (ls + 1) nil.dual x nil)
(fun rs rl rx rr h =>
if rs > delta * ls then
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr) (fun h => nil)
(fun rrs l x_2 r h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl) (fun h => nil)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (ls + rs + 1) (node (ls + rls + 1) nil.dual x rl) rx rr
else
node (ls + rs + 1) (node (ls + rll.size + 1) nil.dual x rll) rlx
(node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯
else node (ls + rs + 1) nil.dual x nil.dual)
⋯)
⋯
case nil.node
α : Type u_1
x : α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
⊢ (Ordnode.casesOn (motive := fun t => id nil = t → Ordnode α) (id nil)
(fun h =>
Ordnode.casesOn (motive := fun t => id (node ls ll lx lr) = t → Ordnode α) (id (node ls ll lx lr))
(fun h => Ordnode.singleton x)
(fun ls_1 ll_1 lx_1 lr_1 h =>
Ordnode.casesOn (motive := fun t => id ll_1 = t → Ordnode α) (id ll_1)
(fun h =>
Ordnode.casesOn (motive := fun t => lr_1 = t → Ordnode α) lr_1
(fun h => node 2 (node ls ll lx lr) x nil)
(fun size l lrx r h => node 3 (Ordnode.singleton lx_1) lrx (Ordnode.singleton x)) ⋯)
(fun lls l x_1 r h =>
Ordnode.casesOn (motive := fun t => id lr_1 = t → Ordnode α) (id lr_1)
(fun h => node 3 ll_1 lx_1 (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls_1 + 1) ll_1 lx_1 (node (lrs + 1) lr_1 x nil)
else
node (ls_1 + 1) (node (lls + lrl.size + 1) ll_1 lx_1 lrl) lrx (node (lrr.size + 1) lrr x nil))
⋯)
⋯)
⋯)
(fun rs l x_1 r h =>
Ordnode.casesOn (motive := fun t => id (node ls ll lx lr) = t → Ordnode α) (id (node ls ll lx lr))
(fun h => node (rs + 1) nil x nil)
(fun ls_1 ll_1 lx_1 lr_1 h =>
if ls_1 > delta * rs then
Ordnode.casesOn (motive := fun t => id ll_1 = t → Ordnode α) (id ll_1) (fun h => nil)
(fun lls l x_2 r h =>
Ordnode.casesOn (motive := fun t => id lr_1 = t → Ordnode α) (id lr_1) (fun h => nil)
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls_1 + rs + 1) ll_1 lx_1 (node (rs + lrs + 1) lr_1 x nil)
else
node (ls_1 + rs + 1) (node (lls + lrl.size + 1) ll_1 lx_1 lrl) lrx
(node (lrr.size + rs + 1) lrr x nil))
⋯)
⋯
else node (ls_1 + rs + 1) (node ls ll lx lr) x nil)
⋯)
⋯).dual =
Ordnode.casesOn (motive := fun t => id nil.dual = t → Ordnode α) (id nil.dual)
(fun h =>
Ordnode.casesOn (motive := fun t => id (node ls ll lx lr).dual = t → Ordnode α) (id (node ls ll lx lr).dual)
(fun h => Ordnode.singleton x)
(fun rs rl rx rr h =>
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr)
(fun h =>
Ordnode.casesOn (motive := fun t => rl = t → Ordnode α) rl
(fun h => node 2 nil x (node ls ll lx lr).dual)
(fun size l rlx r h => node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx)) ⋯)
(fun rrs l x_1 r h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl)
(fun h => node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else node (rs + 1) (node (rll.size + 1) nil x rll) rlx (node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯)
⋯)
(fun ls_1 l x_1 r h =>
Ordnode.casesOn (motive := fun t => id (node ls ll lx lr).dual = t → Ordnode α) (id (node ls ll lx lr).dual)
(fun h => node (ls_1 + 1) nil.dual x nil)
(fun rs rl rx rr h =>
if rs > delta * ls_1 then
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr) (fun h => nil)
(fun rrs l x_2 r h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl) (fun h => nil)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (ls_1 + rs + 1) (node (ls_1 + rls + 1) nil.dual x rl) rx rr
else
node (ls_1 + rs + 1) (node (ls_1 + rll.size + 1) nil.dual x rll) rlx
(node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯
else node (ls_1 + rs + 1) nil.dual x (node ls ll lx lr).dual)
⋯)
⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.dual_balanceL
|
case nil.node
α : Type u_1
x : α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
⊢ (Ordnode.casesOn (motive := fun t => id nil = t → Ordnode α) (id nil)
(fun h =>
Ordnode.casesOn (motive := fun t => id (node ls ll lx lr) = t → Ordnode α) (id (node ls ll lx lr))
(fun h => Ordnode.singleton x)
(fun ls_1 ll_1 lx_1 lr_1 h =>
Ordnode.casesOn (motive := fun t => id ll_1 = t → Ordnode α) (id ll_1)
(fun h =>
Ordnode.casesOn (motive := fun t => lr_1 = t → Ordnode α) lr_1
(fun h => node 2 (node ls ll lx lr) x nil)
(fun size l lrx r h => node 3 (Ordnode.singleton lx_1) lrx (Ordnode.singleton x)) ⋯)
(fun lls l x_1 r h =>
Ordnode.casesOn (motive := fun t => id lr_1 = t → Ordnode α) (id lr_1)
(fun h => node 3 ll_1 lx_1 (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls_1 + 1) ll_1 lx_1 (node (lrs + 1) lr_1 x nil)
else
node (ls_1 + 1) (node (lls + lrl.size + 1) ll_1 lx_1 lrl) lrx (node (lrr.size + 1) lrr x nil))
⋯)
⋯)
⋯)
(fun rs l x_1 r h =>
Ordnode.casesOn (motive := fun t => id (node ls ll lx lr) = t → Ordnode α) (id (node ls ll lx lr))
(fun h => node (rs + 1) nil x nil)
(fun ls_1 ll_1 lx_1 lr_1 h =>
if ls_1 > delta * rs then
Ordnode.casesOn (motive := fun t => id ll_1 = t → Ordnode α) (id ll_1) (fun h => nil)
(fun lls l x_2 r h =>
Ordnode.casesOn (motive := fun t => id lr_1 = t → Ordnode α) (id lr_1) (fun h => nil)
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls_1 + rs + 1) ll_1 lx_1 (node (rs + lrs + 1) lr_1 x nil)
else
node (ls_1 + rs + 1) (node (lls + lrl.size + 1) ll_1 lx_1 lrl) lrx
(node (lrr.size + rs + 1) lrr x nil))
⋯)
⋯
else node (ls_1 + rs + 1) (node ls ll lx lr) x nil)
⋯)
⋯).dual =
Ordnode.casesOn (motive := fun t => id nil.dual = t → Ordnode α) (id nil.dual)
(fun h =>
Ordnode.casesOn (motive := fun t => id (node ls ll lx lr).dual = t → Ordnode α) (id (node ls ll lx lr).dual)
(fun h => Ordnode.singleton x)
(fun rs rl rx rr h =>
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr)
(fun h =>
Ordnode.casesOn (motive := fun t => rl = t → Ordnode α) rl
(fun h => node 2 nil x (node ls ll lx lr).dual)
(fun size l rlx r h => node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx)) ⋯)
(fun rrs l x_1 r h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl)
(fun h => node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else node (rs + 1) (node (rll.size + 1) nil x rll) rlx (node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯)
⋯)
(fun ls_1 l x_1 r h =>
Ordnode.casesOn (motive := fun t => id (node ls ll lx lr).dual = t → Ordnode α) (id (node ls ll lx lr).dual)
(fun h => node (ls_1 + 1) nil.dual x nil)
(fun rs rl rx rr h =>
if rs > delta * ls_1 then
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr) (fun h => nil)
(fun rrs l x_2 r h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl) (fun h => nil)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (ls_1 + rs + 1) (node (ls_1 + rls + 1) nil.dual x rl) rx rr
else
node (ls_1 + rs + 1) (node (ls_1 + rll.size + 1) nil.dual x rll) rlx
(node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯
else node (ls_1 + rs + 1) nil.dual x (node ls ll lx lr).dual)
⋯)
⋯
|
cases' ll with lls lll llx llr <;> cases' lr with lrs lrl lrx lrr <;> dsimp only [<a>Ordnode.dual</a>, <a>id</a>] <;> try rfl
|
case nil.node.node.node
α : Type u_1
x : α
ls : ℕ
lx : α
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
⊢ (if lrs < ratio * lls then node (ls + 1) (node lls lll llx llr) lx (node (lrs + 1) (node lrs lrl lrx lrr) x nil)
else
node (ls + 1) (node (lls + lrl.size + 1) (node lls lll llx llr) lx lrl) lrx
(node (lrr.size + 1) lrr x nil)).dual =
if lrs < ratio * lls then
node (ls + 1) (node (lrs + 1) nil x (node lrs lrr.dual lrx lrl.dual)) lx (node lls llr.dual llx lll.dual)
else
node (ls + 1) (node (lrr.dual.size + 1) nil x lrr.dual) lrx
(node (lrl.dual.size + lls + 1) lrl.dual lx (node lls llr.dual llx lll.dual))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.dual_balanceL
|
case nil.node.node.node
α : Type u_1
x : α
ls : ℕ
lx : α
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
⊢ (if lrs < ratio * lls then node (ls + 1) (node lls lll llx llr) lx (node (lrs + 1) (node lrs lrl lrx lrr) x nil)
else
node (ls + 1) (node (lls + lrl.size + 1) (node lls lll llx llr) lx lrl) lrx
(node (lrr.size + 1) lrr x nil)).dual =
if lrs < ratio * lls then
node (ls + 1) (node (lrs + 1) nil x (node lrs lrr.dual lrx lrl.dual)) lx (node lls llr.dual llx lll.dual)
else
node (ls + 1) (node (lrr.dual.size + 1) nil x lrr.dual) lrx
(node (lrl.dual.size + lls + 1) lrl.dual lx (node lls llr.dual llx lll.dual))
|
split_ifs with h <;> repeat simp [h, <a>add_comm</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.dual_balanceL
|
case nil.nil
α : Type u_1
x : α
⊢ (Ordnode.casesOn (motive := fun t => id nil = t → Ordnode α) (id nil)
(fun h =>
Ordnode.casesOn (motive := fun t => id nil = t → Ordnode α) (id nil) (fun h => Ordnode.singleton x)
(fun ls ll lx lr h =>
Ordnode.casesOn (motive := fun t => id ll = t → Ordnode α) (id ll)
(fun h =>
Ordnode.casesOn (motive := fun t => lr = t → Ordnode α) lr (fun h => node 2 nil x nil)
(fun size l lrx r h => node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x)) ⋯)
(fun lls l x_1 r h =>
Ordnode.casesOn (motive := fun t => id lr = t → Ordnode α) (id lr)
(fun h => node 3 ll lx (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls + 1) ll lx (node (lrs + 1) lr x nil)
else node (ls + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx (node (lrr.size + 1) lrr x nil))
⋯)
⋯)
⋯)
(fun rs l x_1 r h =>
Ordnode.casesOn (motive := fun t => id nil = t → Ordnode α) (id nil) (fun h => node (rs + 1) nil x nil)
(fun ls ll lx lr h =>
if ls > delta * rs then
Ordnode.casesOn (motive := fun t => id ll = t → Ordnode α) (id ll) (fun h => nil)
(fun lls l x_2 r h =>
Ordnode.casesOn (motive := fun t => id lr = t → Ordnode α) (id lr) (fun h => nil)
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls + rs + 1) ll lx (node (rs + lrs + 1) lr x nil)
else
node (ls + rs + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx
(node (lrr.size + rs + 1) lrr x nil))
⋯)
⋯
else node (ls + rs + 1) nil x nil)
⋯)
⋯).dual =
Ordnode.casesOn (motive := fun t => id nil.dual = t → Ordnode α) (id nil.dual)
(fun h =>
Ordnode.casesOn (motive := fun t => id nil.dual = t → Ordnode α) (id nil.dual) (fun h => Ordnode.singleton x)
(fun rs rl rx rr h =>
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr)
(fun h =>
Ordnode.casesOn (motive := fun t => rl = t → Ordnode α) rl (fun h => node 2 nil x nil.dual)
(fun size l rlx r h => node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx)) ⋯)
(fun rrs l x_1 r h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl)
(fun h => node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else node (rs + 1) (node (rll.size + 1) nil x rll) rlx (node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯)
⋯)
(fun ls l x_1 r h =>
Ordnode.casesOn (motive := fun t => id nil.dual = t → Ordnode α) (id nil.dual)
(fun h => node (ls + 1) nil.dual x nil)
(fun rs rl rx rr h =>
if rs > delta * ls then
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr) (fun h => nil)
(fun rrs l x_2 r h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl) (fun h => nil)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (ls + rs + 1) (node (ls + rls + 1) nil.dual x rl) rx rr
else
node (ls + rs + 1) (node (ls + rll.size + 1) nil.dual x rll) rlx
(node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯
else node (ls + rs + 1) nil.dual x nil.dual)
⋯)
⋯
|
rfl
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.dual_balanceL
|
case nil.node.node.nil
α : Type u_1
x : α
ls : ℕ
lx : α
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
⊢ node 3 (node 1 nil x nil) lx (node lls llr.dual llx lll.dual) =
node 3 (Ordnode.singleton x) lx (node lls llr.dual llx lll.dual)
|
rfl
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.dual_balanceL
|
case neg
α : Type u_1
x : α
ls : ℕ
lx : α
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
h : ¬lrs < ratio * lls
⊢ (node (ls + 1) (node (lls + lrl.size + 1) (node lls lll llx llr) lx lrl) lrx (node (lrr.size + 1) lrr x nil)).dual =
node (ls + 1) (node (lrr.dual.size + 1) nil x lrr.dual) lrx
(node (lrl.dual.size + lls + 1) lrl.dual lx (node lls llr.dual llx lll.dual))
|
simp [h, <a>add_comm</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.dual_balanceL
|
case node
α : Type u_1
l : Ordnode α
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
⊢ (Ordnode.casesOn (motive := fun t => id (node rs rl rx rr) = t → Ordnode α) (id (node rs rl rx rr))
(fun h =>
Ordnode.casesOn (motive := fun t => id l = t → Ordnode α) (id l) (fun h => Ordnode.singleton x)
(fun ls ll lx lr h =>
Ordnode.casesOn (motive := fun t => id ll = t → Ordnode α) (id ll)
(fun h =>
Ordnode.casesOn (motive := fun t => lr = t → Ordnode α) lr (fun h => node 2 l x nil)
(fun size l lrx r h => node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x)) ⋯)
(fun lls l x_1 r h =>
Ordnode.casesOn (motive := fun t => id lr = t → Ordnode α) (id lr)
(fun h => node 3 ll lx (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls + 1) ll lx (node (lrs + 1) lr x nil)
else node (ls + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx (node (lrr.size + 1) lrr x nil))
⋯)
⋯)
⋯)
(fun rs_1 l_1 x_1 r h =>
Ordnode.casesOn (motive := fun t => id l = t → Ordnode α) (id l)
(fun h => node (rs_1 + 1) nil x (node rs rl rx rr))
(fun ls ll lx lr h =>
if ls > delta * rs_1 then
Ordnode.casesOn (motive := fun t => id ll = t → Ordnode α) (id ll) (fun h => nil)
(fun lls l x_2 r h =>
Ordnode.casesOn (motive := fun t => id lr = t → Ordnode α) (id lr) (fun h => nil)
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then
node (ls + rs_1 + 1) ll lx (node (rs_1 + lrs + 1) lr x (node rs rl rx rr))
else
node (ls + rs_1 + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx
(node (lrr.size + rs_1 + 1) lrr x (node rs rl rx rr)))
⋯)
⋯
else node (ls + rs_1 + 1) l x (node rs rl rx rr))
⋯)
⋯).dual =
Ordnode.casesOn (motive := fun t => id (node rs rl rx rr).dual = t → Ordnode α) (id (node rs rl rx rr).dual)
(fun h =>
Ordnode.casesOn (motive := fun t => id l.dual = t → Ordnode α) (id l.dual) (fun h => Ordnode.singleton x)
(fun rs rl rx rr h =>
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr)
(fun h =>
Ordnode.casesOn (motive := fun t => rl = t → Ordnode α) rl (fun h => node 2 nil x l.dual)
(fun size l rlx r h => node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx)) ⋯)
(fun rrs l x_1 r h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl)
(fun h => node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else node (rs + 1) (node (rll.size + 1) nil x rll) rlx (node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯)
⋯)
(fun ls l_1 x_1 r h =>
Ordnode.casesOn (motive := fun t => id l.dual = t → Ordnode α) (id l.dual)
(fun h => node (ls + 1) (node rs rl rx rr).dual x nil)
(fun rs_1 rl_1 rx_1 rr_1 h =>
if rs_1 > delta * ls then
Ordnode.casesOn (motive := fun t => id rr_1 = t → Ordnode α) (id rr_1) (fun h => nil)
(fun rrs l x_2 r h =>
Ordnode.casesOn (motive := fun t => id rl_1 = t → Ordnode α) (id rl_1) (fun h => nil)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then
node (ls + rs_1 + 1) (node (ls + rls + 1) (node rs rl rx rr).dual x rl_1) rx_1 rr_1
else
node (ls + rs_1 + 1) (node (ls + rll.size + 1) (node rs rl rx rr).dual x rll) rlx
(node (rlr.size + rrs + 1) rlr rx_1 rr_1))
⋯)
⋯
else node (ls + rs_1 + 1) (node rs rl rx rr).dual x l.dual)
⋯)
⋯
|
cases' l with ls ll lx lr
|
case node.nil
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
⊢ (Ordnode.casesOn (motive := fun t => id (node rs rl rx rr) = t → Ordnode α) (id (node rs rl rx rr))
(fun h =>
Ordnode.casesOn (motive := fun t => id nil = t → Ordnode α) (id nil) (fun h => Ordnode.singleton x)
(fun ls ll lx lr h =>
Ordnode.casesOn (motive := fun t => id ll = t → Ordnode α) (id ll)
(fun h =>
Ordnode.casesOn (motive := fun t => lr = t → Ordnode α) lr (fun h => node 2 nil x nil)
(fun size l lrx r h => node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x)) ⋯)
(fun lls l x_1 r h =>
Ordnode.casesOn (motive := fun t => id lr = t → Ordnode α) (id lr)
(fun h => node 3 ll lx (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls + 1) ll lx (node (lrs + 1) lr x nil)
else node (ls + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx (node (lrr.size + 1) lrr x nil))
⋯)
⋯)
⋯)
(fun rs_1 l x_1 r h =>
Ordnode.casesOn (motive := fun t => id nil = t → Ordnode α) (id nil)
(fun h => node (rs_1 + 1) nil x (node rs rl rx rr))
(fun ls ll lx lr h =>
if ls > delta * rs_1 then
Ordnode.casesOn (motive := fun t => id ll = t → Ordnode α) (id ll) (fun h => nil)
(fun lls l x_2 r h =>
Ordnode.casesOn (motive := fun t => id lr = t → Ordnode α) (id lr) (fun h => nil)
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then
node (ls + rs_1 + 1) ll lx (node (rs_1 + lrs + 1) lr x (node rs rl rx rr))
else
node (ls + rs_1 + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx
(node (lrr.size + rs_1 + 1) lrr x (node rs rl rx rr)))
⋯)
⋯
else node (ls + rs_1 + 1) nil x (node rs rl rx rr))
⋯)
⋯).dual =
Ordnode.casesOn (motive := fun t => id (node rs rl rx rr).dual = t → Ordnode α) (id (node rs rl rx rr).dual)
(fun h =>
Ordnode.casesOn (motive := fun t => id nil.dual = t → Ordnode α) (id nil.dual) (fun h => Ordnode.singleton x)
(fun rs rl rx rr h =>
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr)
(fun h =>
Ordnode.casesOn (motive := fun t => rl = t → Ordnode α) rl (fun h => node 2 nil x nil.dual)
(fun size l rlx r h => node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx)) ⋯)
(fun rrs l x_1 r h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl)
(fun h => node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else node (rs + 1) (node (rll.size + 1) nil x rll) rlx (node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯)
⋯)
(fun ls l x_1 r h =>
Ordnode.casesOn (motive := fun t => id nil.dual = t → Ordnode α) (id nil.dual)
(fun h => node (ls + 1) (node rs rl rx rr).dual x nil)
(fun rs_1 rl_1 rx_1 rr_1 h =>
if rs_1 > delta * ls then
Ordnode.casesOn (motive := fun t => id rr_1 = t → Ordnode α) (id rr_1) (fun h => nil)
(fun rrs l x_2 r h =>
Ordnode.casesOn (motive := fun t => id rl_1 = t → Ordnode α) (id rl_1) (fun h => nil)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then
node (ls + rs_1 + 1) (node (ls + rls + 1) (node rs rl rx rr).dual x rl_1) rx_1 rr_1
else
node (ls + rs_1 + 1) (node (ls + rll.size + 1) (node rs rl rx rr).dual x rll) rlx
(node (rlr.size + rrs + 1) rlr rx_1 rr_1))
⋯)
⋯
else node (ls + rs_1 + 1) (node rs rl rx rr).dual x nil.dual)
⋯)
⋯
case node.node
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
⊢ (Ordnode.casesOn (motive := fun t => id (node rs rl rx rr) = t → Ordnode α) (id (node rs rl rx rr))
(fun h =>
Ordnode.casesOn (motive := fun t => id (node ls ll lx lr) = t → Ordnode α) (id (node ls ll lx lr))
(fun h => Ordnode.singleton x)
(fun ls_1 ll_1 lx_1 lr_1 h =>
Ordnode.casesOn (motive := fun t => id ll_1 = t → Ordnode α) (id ll_1)
(fun h =>
Ordnode.casesOn (motive := fun t => lr_1 = t → Ordnode α) lr_1
(fun h => node 2 (node ls ll lx lr) x nil)
(fun size l lrx r h => node 3 (Ordnode.singleton lx_1) lrx (Ordnode.singleton x)) ⋯)
(fun lls l x_1 r h =>
Ordnode.casesOn (motive := fun t => id lr_1 = t → Ordnode α) (id lr_1)
(fun h => node 3 ll_1 lx_1 (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls_1 + 1) ll_1 lx_1 (node (lrs + 1) lr_1 x nil)
else
node (ls_1 + 1) (node (lls + lrl.size + 1) ll_1 lx_1 lrl) lrx (node (lrr.size + 1) lrr x nil))
⋯)
⋯)
⋯)
(fun rs_1 l x_1 r h =>
Ordnode.casesOn (motive := fun t => id (node ls ll lx lr) = t → Ordnode α) (id (node ls ll lx lr))
(fun h => node (rs_1 + 1) nil x (node rs rl rx rr))
(fun ls_1 ll_1 lx_1 lr_1 h =>
if ls_1 > delta * rs_1 then
Ordnode.casesOn (motive := fun t => id ll_1 = t → Ordnode α) (id ll_1) (fun h => nil)
(fun lls l x_2 r h =>
Ordnode.casesOn (motive := fun t => id lr_1 = t → Ordnode α) (id lr_1) (fun h => nil)
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then
node (ls_1 + rs_1 + 1) ll_1 lx_1 (node (rs_1 + lrs + 1) lr_1 x (node rs rl rx rr))
else
node (ls_1 + rs_1 + 1) (node (lls + lrl.size + 1) ll_1 lx_1 lrl) lrx
(node (lrr.size + rs_1 + 1) lrr x (node rs rl rx rr)))
⋯)
⋯
else node (ls_1 + rs_1 + 1) (node ls ll lx lr) x (node rs rl rx rr))
⋯)
⋯).dual =
Ordnode.casesOn (motive := fun t => id (node rs rl rx rr).dual = t → Ordnode α) (id (node rs rl rx rr).dual)
(fun h =>
Ordnode.casesOn (motive := fun t => id (node ls ll lx lr).dual = t → Ordnode α) (id (node ls ll lx lr).dual)
(fun h => Ordnode.singleton x)
(fun rs rl rx rr h =>
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr)
(fun h =>
Ordnode.casesOn (motive := fun t => rl = t → Ordnode α) rl
(fun h => node 2 nil x (node ls ll lx lr).dual)
(fun size l rlx r h => node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx)) ⋯)
(fun rrs l x_1 r h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl)
(fun h => node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else node (rs + 1) (node (rll.size + 1) nil x rll) rlx (node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯)
⋯)
(fun ls_1 l x_1 r h =>
Ordnode.casesOn (motive := fun t => id (node ls ll lx lr).dual = t → Ordnode α) (id (node ls ll lx lr).dual)
(fun h => node (ls_1 + 1) (node rs rl rx rr).dual x nil)
(fun rs_1 rl_1 rx_1 rr_1 h =>
if rs_1 > delta * ls_1 then
Ordnode.casesOn (motive := fun t => id rr_1 = t → Ordnode α) (id rr_1) (fun h => nil)
(fun rrs l x_2 r h =>
Ordnode.casesOn (motive := fun t => id rl_1 = t → Ordnode α) (id rl_1) (fun h => nil)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then
node (ls_1 + rs_1 + 1) (node (ls_1 + rls + 1) (node rs rl rx rr).dual x rl_1) rx_1 rr_1
else
node (ls_1 + rs_1 + 1) (node (ls_1 + rll.size + 1) (node rs rl rx rr).dual x rll) rlx
(node (rlr.size + rrs + 1) rlr rx_1 rr_1))
⋯)
⋯
else node (ls_1 + rs_1 + 1) (node rs rl rx rr).dual x (node ls ll lx lr).dual)
⋯)
⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.dual_balanceL
|
case node.node
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
⊢ (Ordnode.casesOn (motive := fun t => id (node rs rl rx rr) = t → Ordnode α) (id (node rs rl rx rr))
(fun h =>
Ordnode.casesOn (motive := fun t => id (node ls ll lx lr) = t → Ordnode α) (id (node ls ll lx lr))
(fun h => Ordnode.singleton x)
(fun ls_1 ll_1 lx_1 lr_1 h =>
Ordnode.casesOn (motive := fun t => id ll_1 = t → Ordnode α) (id ll_1)
(fun h =>
Ordnode.casesOn (motive := fun t => lr_1 = t → Ordnode α) lr_1
(fun h => node 2 (node ls ll lx lr) x nil)
(fun size l lrx r h => node 3 (Ordnode.singleton lx_1) lrx (Ordnode.singleton x)) ⋯)
(fun lls l x_1 r h =>
Ordnode.casesOn (motive := fun t => id lr_1 = t → Ordnode α) (id lr_1)
(fun h => node 3 ll_1 lx_1 (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls_1 + 1) ll_1 lx_1 (node (lrs + 1) lr_1 x nil)
else
node (ls_1 + 1) (node (lls + lrl.size + 1) ll_1 lx_1 lrl) lrx (node (lrr.size + 1) lrr x nil))
⋯)
⋯)
⋯)
(fun rs_1 l x_1 r h =>
Ordnode.casesOn (motive := fun t => id (node ls ll lx lr) = t → Ordnode α) (id (node ls ll lx lr))
(fun h => node (rs_1 + 1) nil x (node rs rl rx rr))
(fun ls_1 ll_1 lx_1 lr_1 h =>
if ls_1 > delta * rs_1 then
Ordnode.casesOn (motive := fun t => id ll_1 = t → Ordnode α) (id ll_1) (fun h => nil)
(fun lls l x_2 r h =>
Ordnode.casesOn (motive := fun t => id lr_1 = t → Ordnode α) (id lr_1) (fun h => nil)
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then
node (ls_1 + rs_1 + 1) ll_1 lx_1 (node (rs_1 + lrs + 1) lr_1 x (node rs rl rx rr))
else
node (ls_1 + rs_1 + 1) (node (lls + lrl.size + 1) ll_1 lx_1 lrl) lrx
(node (lrr.size + rs_1 + 1) lrr x (node rs rl rx rr)))
⋯)
⋯
else node (ls_1 + rs_1 + 1) (node ls ll lx lr) x (node rs rl rx rr))
⋯)
⋯).dual =
Ordnode.casesOn (motive := fun t => id (node rs rl rx rr).dual = t → Ordnode α) (id (node rs rl rx rr).dual)
(fun h =>
Ordnode.casesOn (motive := fun t => id (node ls ll lx lr).dual = t → Ordnode α) (id (node ls ll lx lr).dual)
(fun h => Ordnode.singleton x)
(fun rs rl rx rr h =>
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr)
(fun h =>
Ordnode.casesOn (motive := fun t => rl = t → Ordnode α) rl
(fun h => node 2 nil x (node ls ll lx lr).dual)
(fun size l rlx r h => node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx)) ⋯)
(fun rrs l x_1 r h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl)
(fun h => node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else node (rs + 1) (node (rll.size + 1) nil x rll) rlx (node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯)
⋯)
(fun ls_1 l x_1 r h =>
Ordnode.casesOn (motive := fun t => id (node ls ll lx lr).dual = t → Ordnode α) (id (node ls ll lx lr).dual)
(fun h => node (ls_1 + 1) (node rs rl rx rr).dual x nil)
(fun rs_1 rl_1 rx_1 rr_1 h =>
if rs_1 > delta * ls_1 then
Ordnode.casesOn (motive := fun t => id rr_1 = t → Ordnode α) (id rr_1) (fun h => nil)
(fun rrs l x_2 r h =>
Ordnode.casesOn (motive := fun t => id rl_1 = t → Ordnode α) (id rl_1) (fun h => nil)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then
node (ls_1 + rs_1 + 1) (node (ls_1 + rls + 1) (node rs rl rx rr).dual x rl_1) rx_1 rr_1
else
node (ls_1 + rs_1 + 1) (node (ls_1 + rll.size + 1) (node rs rl rx rr).dual x rll) rlx
(node (rlr.size + rrs + 1) rlr rx_1 rr_1))
⋯)
⋯
else node (ls_1 + rs_1 + 1) (node rs rl rx rr).dual x (node ls ll lx lr).dual)
⋯)
⋯
|
dsimp only [<a>Ordnode.dual</a>, <a>id</a>]
|
case node.node
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
⊢ (if ls > delta * rs then
rec (motive := fun t => ll = t → Ordnode α) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr = t → Ordnode α) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then node (ls + rs + 1) ll lx (node (rs + size_1 + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + l.size + 1) ll lx l) x_2
(node (r.size + rs + 1) r x (node rs rl rx rr)))
lr ⋯)
ll ⋯
else node (ls + rs + 1) (node ls ll lx lr) x (node rs rl rx rr)).dual =
if ls > delta * rs then
rec (motive := fun t => ll.dual = t → Ordnode α) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr.dual = t → Ordnode α) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then
node (rs + ls + 1) (node (rs + size_1 + 1) (node rs rr.dual rx rl.dual) x lr.dual) lx ll.dual
else
node (rs + ls + 1) (node (rs + l.size + 1) (node rs rr.dual rx rl.dual) x l) x_2
(node (r.size + size + 1) r lx ll.dual))
lr.dual ⋯)
ll.dual ⋯
else node (rs + ls + 1) (node rs rr.dual rx rl.dual) x (node ls lr.dual lx ll.dual)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.dual_balanceL
|
case node.node
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
⊢ (if ls > delta * rs then
rec (motive := fun t => ll = t → Ordnode α) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr = t → Ordnode α) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then node (ls + rs + 1) ll lx (node (rs + size_1 + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + l.size + 1) ll lx l) x_2
(node (r.size + rs + 1) r x (node rs rl rx rr)))
lr ⋯)
ll ⋯
else node (ls + rs + 1) (node ls ll lx lr) x (node rs rl rx rr)).dual =
if ls > delta * rs then
rec (motive := fun t => ll.dual = t → Ordnode α) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr.dual = t → Ordnode α) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then
node (rs + ls + 1) (node (rs + size_1 + 1) (node rs rr.dual rx rl.dual) x lr.dual) lx ll.dual
else
node (rs + ls + 1) (node (rs + l.size + 1) (node rs rr.dual rx rl.dual) x l) x_2
(node (r.size + size + 1) r lx ll.dual))
lr.dual ⋯)
ll.dual ⋯
else node (rs + ls + 1) (node rs rr.dual rx rl.dual) x (node ls lr.dual lx ll.dual)
|
split_ifs
|
case pos
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
h✝ : ls > delta * rs
⊢ (rec (motive := fun t => ll = t → Ordnode α) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr = t → Ordnode α) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then node (ls + rs + 1) ll lx (node (rs + size_1 + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + l.size + 1) ll lx l) x_2
(node (r.size + rs + 1) r x (node rs rl rx rr)))
lr ⋯)
ll ⋯).dual =
rec (motive := fun t => ll.dual = t → Ordnode α) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr.dual = t → Ordnode α) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then
node (rs + ls + 1) (node (rs + size_1 + 1) (node rs rr.dual rx rl.dual) x lr.dual) lx ll.dual
else
node (rs + ls + 1) (node (rs + l.size + 1) (node rs rr.dual rx rl.dual) x l) x_2
(node (r.size + size + 1) r lx ll.dual))
lr.dual ⋯)
ll.dual ⋯
case neg
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
h✝ : ¬ls > delta * rs
⊢ (node (ls + rs + 1) (node ls ll lx lr) x (node rs rl rx rr)).dual =
node (rs + ls + 1) (node rs rr.dual rx rl.dual) x (node ls lr.dual lx ll.dual)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.dual_balanceL
|
case pos
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
h✝ : ls > delta * rs
⊢ (rec (motive := fun t => ll = t → Ordnode α) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr = t → Ordnode α) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then node (ls + rs + 1) ll lx (node (rs + size_1 + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + l.size + 1) ll lx l) x_2
(node (r.size + rs + 1) r x (node rs rl rx rr)))
lr ⋯)
ll ⋯).dual =
rec (motive := fun t => ll.dual = t → Ordnode α) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr.dual = t → Ordnode α) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then
node (rs + ls + 1) (node (rs + size_1 + 1) (node rs rr.dual rx rl.dual) x lr.dual) lx ll.dual
else
node (rs + ls + 1) (node (rs + l.size + 1) (node rs rr.dual rx rl.dual) x l) x_2
(node (r.size + size + 1) r lx ll.dual))
lr.dual ⋯)
ll.dual ⋯
case neg
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
h✝ : ¬ls > delta * rs
⊢ (node (ls + rs + 1) (node ls ll lx lr) x (node rs rl rx rr)).dual =
node (rs + ls + 1) (node rs rr.dual rx rl.dual) x (node ls lr.dual lx ll.dual)
|
swap
|
case neg
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
h✝ : ¬ls > delta * rs
⊢ (node (ls + rs + 1) (node ls ll lx lr) x (node rs rl rx rr)).dual =
node (rs + ls + 1) (node rs rr.dual rx rl.dual) x (node ls lr.dual lx ll.dual)
case pos
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
h✝ : ls > delta * rs
⊢ (rec (motive := fun t => ll = t → Ordnode α) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr = t → Ordnode α) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then node (ls + rs + 1) ll lx (node (rs + size_1 + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + l.size + 1) ll lx l) x_2
(node (r.size + rs + 1) r x (node rs rl rx rr)))
lr ⋯)
ll ⋯).dual =
rec (motive := fun t => ll.dual = t → Ordnode α) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr.dual = t → Ordnode α) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then
node (rs + ls + 1) (node (rs + size_1 + 1) (node rs rr.dual rx rl.dual) x lr.dual) lx ll.dual
else
node (rs + ls + 1) (node (rs + l.size + 1) (node rs rr.dual rx rl.dual) x l) x_2
(node (r.size + size + 1) r lx ll.dual))
lr.dual ⋯)
ll.dual ⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.dual_balanceL
|
case pos
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
h✝ : ls > delta * rs
⊢ (rec (motive := fun t => ll = t → Ordnode α) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr = t → Ordnode α) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then node (ls + rs + 1) ll lx (node (rs + size_1 + 1) lr x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + l.size + 1) ll lx l) x_2
(node (r.size + rs + 1) r x (node rs rl rx rr)))
lr ⋯)
ll ⋯).dual =
rec (motive := fun t => ll.dual = t → Ordnode α) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => lr.dual = t → Ordnode α) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then
node (rs + ls + 1) (node (rs + size_1 + 1) (node rs rr.dual rx rl.dual) x lr.dual) lx ll.dual
else
node (rs + ls + 1) (node (rs + l.size + 1) (node rs rr.dual rx rl.dual) x l) x_2
(node (r.size + size + 1) r lx ll.dual))
lr.dual ⋯)
ll.dual ⋯
|
cases' ll with lls lll llx llr <;> cases' lr with lrs lrl lrx lrr <;> try rfl
|
case pos.node.node
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
ls : ℕ
lx : α
h✝ : ls > delta * rs
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
⊢ (rec (motive := fun t => node lls lll llx llr = t → Ordnode α) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => node lrs lrl lrx lrr = t → Ordnode α) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then
node (ls + rs + 1) (node lls lll llx llr) lx
(node (rs + size_1 + 1) (node lrs lrl lrx lrr) x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + l.size + 1) (node lls lll llx llr) lx l) x_2
(node (r.size + rs + 1) r x (node rs rl rx rr)))
(node lrs lrl lrx lrr) ⋯)
(node lls lll llx llr) ⋯).dual =
rec (motive := fun t => (node lls lll llx llr).dual = t → Ordnode α) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => (node lrs lrl lrx lrr).dual = t → Ordnode α) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then
node (rs + ls + 1) (node (rs + size_1 + 1) (node rs rr.dual rx rl.dual) x (node lrs lrl lrx lrr).dual) lx
(node lls lll llx llr).dual
else
node (rs + ls + 1) (node (rs + l.size + 1) (node rs rr.dual rx rl.dual) x l) x_2
(node (r.size + size + 1) r lx (node lls lll llx llr).dual))
(node lrs lrl lrx lrr).dual ⋯)
(node lls lll llx llr).dual ⋯
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.dual_balanceL
|
case pos.node.node
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
ls : ℕ
lx : α
h✝ : ls > delta * rs
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
⊢ (rec (motive := fun t => node lls lll llx llr = t → Ordnode α) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => node lrs lrl lrx lrr = t → Ordnode α) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then
node (ls + rs + 1) (node lls lll llx llr) lx
(node (rs + size_1 + 1) (node lrs lrl lrx lrr) x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + l.size + 1) (node lls lll llx llr) lx l) x_2
(node (r.size + rs + 1) r x (node rs rl rx rr)))
(node lrs lrl lrx lrr) ⋯)
(node lls lll llx llr) ⋯).dual =
rec (motive := fun t => (node lls lll llx llr).dual = t → Ordnode α) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => (node lrs lrl lrx lrr).dual = t → Ordnode α) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then
node (rs + ls + 1) (node (rs + size_1 + 1) (node rs rr.dual rx rl.dual) x (node lrs lrl lrx lrr).dual) lx
(node lls lll llx llr).dual
else
node (rs + ls + 1) (node (rs + l.size + 1) (node rs rr.dual rx rl.dual) x l) x_2
(node (r.size + size + 1) r lx (node lls lll llx llr).dual))
(node lrs lrl lrx lrr).dual ⋯)
(node lls lll llx llr).dual ⋯
|
dsimp only [<a>Ordnode.dual</a>, <a>id</a>]
|
case pos.node.node
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
ls : ℕ
lx : α
h✝ : ls > delta * rs
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
⊢ (if lrs < ratio * lls then
node (ls + rs + 1) (node lls lll llx llr) lx (node (rs + lrs + 1) (node lrs lrl lrx lrr) x (node rs rl rx rr))
else
node (ls + rs + 1) (node (lls + lrl.size + 1) (node lls lll llx llr) lx lrl) lrx
(node (lrr.size + rs + 1) lrr x (node rs rl rx rr))).dual =
if lrs < ratio * lls then
node (rs + ls + 1) (node (rs + lrs + 1) (node rs rr.dual rx rl.dual) x (node lrs lrr.dual lrx lrl.dual)) lx
(node lls llr.dual llx lll.dual)
else
node (rs + ls + 1) (node (rs + lrr.dual.size + 1) (node rs rr.dual rx rl.dual) x lrr.dual) lrx
(node (lrl.dual.size + lls + 1) lrl.dual lx (node lls llr.dual llx lll.dual))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.dual_balanceL
|
case pos.node.node
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
ls : ℕ
lx : α
h✝ : ls > delta * rs
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
⊢ (if lrs < ratio * lls then
node (ls + rs + 1) (node lls lll llx llr) lx (node (rs + lrs + 1) (node lrs lrl lrx lrr) x (node rs rl rx rr))
else
node (ls + rs + 1) (node (lls + lrl.size + 1) (node lls lll llx llr) lx lrl) lrx
(node (lrr.size + rs + 1) lrr x (node rs rl rx rr))).dual =
if lrs < ratio * lls then
node (rs + ls + 1) (node (rs + lrs + 1) (node rs rr.dual rx rl.dual) x (node lrs lrr.dual lrx lrl.dual)) lx
(node lls llr.dual llx lll.dual)
else
node (rs + ls + 1) (node (rs + lrr.dual.size + 1) (node rs rr.dual rx rl.dual) x lrr.dual) lrx
(node (lrl.dual.size + lls + 1) lrl.dual lx (node lls llr.dual llx lll.dual))
|
split_ifs with h <;> simp [h, <a>add_comm</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.dual_balanceL
|
case node.nil
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
⊢ (Ordnode.casesOn (motive := fun t => id (node rs rl rx rr) = t → Ordnode α) (id (node rs rl rx rr))
(fun h =>
Ordnode.casesOn (motive := fun t => id nil = t → Ordnode α) (id nil) (fun h => Ordnode.singleton x)
(fun ls ll lx lr h =>
Ordnode.casesOn (motive := fun t => id ll = t → Ordnode α) (id ll)
(fun h =>
Ordnode.casesOn (motive := fun t => lr = t → Ordnode α) lr (fun h => node 2 nil x nil)
(fun size l lrx r h => node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x)) ⋯)
(fun lls l x_1 r h =>
Ordnode.casesOn (motive := fun t => id lr = t → Ordnode α) (id lr)
(fun h => node 3 ll lx (Ordnode.singleton x))
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then node (ls + 1) ll lx (node (lrs + 1) lr x nil)
else node (ls + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx (node (lrr.size + 1) lrr x nil))
⋯)
⋯)
⋯)
(fun rs_1 l x_1 r h =>
Ordnode.casesOn (motive := fun t => id nil = t → Ordnode α) (id nil)
(fun h => node (rs_1 + 1) nil x (node rs rl rx rr))
(fun ls ll lx lr h =>
if ls > delta * rs_1 then
Ordnode.casesOn (motive := fun t => id ll = t → Ordnode α) (id ll) (fun h => nil)
(fun lls l x_2 r h =>
Ordnode.casesOn (motive := fun t => id lr = t → Ordnode α) (id lr) (fun h => nil)
(fun lrs lrl lrx lrr h =>
if lrs < ratio * lls then
node (ls + rs_1 + 1) ll lx (node (rs_1 + lrs + 1) lr x (node rs rl rx rr))
else
node (ls + rs_1 + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx
(node (lrr.size + rs_1 + 1) lrr x (node rs rl rx rr)))
⋯)
⋯
else node (ls + rs_1 + 1) nil x (node rs rl rx rr))
⋯)
⋯).dual =
Ordnode.casesOn (motive := fun t => id (node rs rl rx rr).dual = t → Ordnode α) (id (node rs rl rx rr).dual)
(fun h =>
Ordnode.casesOn (motive := fun t => id nil.dual = t → Ordnode α) (id nil.dual) (fun h => Ordnode.singleton x)
(fun rs rl rx rr h =>
Ordnode.casesOn (motive := fun t => id rr = t → Ordnode α) (id rr)
(fun h =>
Ordnode.casesOn (motive := fun t => rl = t → Ordnode α) rl (fun h => node 2 nil x nil.dual)
(fun size l rlx r h => node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx)) ⋯)
(fun rrs l x_1 r h =>
Ordnode.casesOn (motive := fun t => id rl = t → Ordnode α) (id rl)
(fun h => node 3 (Ordnode.singleton x) rx rr)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
else node (rs + 1) (node (rll.size + 1) nil x rll) rlx (node (rlr.size + rrs + 1) rlr rx rr))
⋯)
⋯)
⋯)
(fun ls l x_1 r h =>
Ordnode.casesOn (motive := fun t => id nil.dual = t → Ordnode α) (id nil.dual)
(fun h => node (ls + 1) (node rs rl rx rr).dual x nil)
(fun rs_1 rl_1 rx_1 rr_1 h =>
if rs_1 > delta * ls then
Ordnode.casesOn (motive := fun t => id rr_1 = t → Ordnode α) (id rr_1) (fun h => nil)
(fun rrs l x_2 r h =>
Ordnode.casesOn (motive := fun t => id rl_1 = t → Ordnode α) (id rl_1) (fun h => nil)
(fun rls rll rlx rlr h =>
if rls < ratio * rrs then
node (ls + rs_1 + 1) (node (ls + rls + 1) (node rs rl rx rr).dual x rl_1) rx_1 rr_1
else
node (ls + rs_1 + 1) (node (ls + rll.size + 1) (node rs rl rx rr).dual x rll) rlx
(node (rlr.size + rrs + 1) rlr rx_1 rr_1))
⋯)
⋯
else node (ls + rs_1 + 1) (node rs rl rx rr).dual x nil.dual)
⋯)
⋯
|
rfl
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.dual_balanceL
|
case neg
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
h✝ : ¬ls > delta * rs
⊢ (node (ls + rs + 1) (node ls ll lx lr) x (node rs rl rx rr)).dual =
node (rs + ls + 1) (node rs rr.dual rx rl.dual) x (node ls lr.dual lx ll.dual)
|
simp [<a>add_comm</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.dual_balanceL
|
case pos.node.nil
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
ls : ℕ
lx : α
h✝ : ls > delta * rs
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
⊢ (rec (motive := fun t => node lls lll llx llr = t → Ordnode α) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => nil = t → Ordnode α) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then
node (ls + rs + 1) (node lls lll llx llr) lx (node (rs + size_1 + 1) nil x (node rs rl rx rr))
else
node (ls + rs + 1) (node (size + l.size + 1) (node lls lll llx llr) lx l) x_2
(node (r.size + rs + 1) r x (node rs rl rx rr)))
nil ⋯)
(node lls lll llx llr) ⋯).dual =
rec (motive := fun t => (node lls lll llx llr).dual = t → Ordnode α) (fun h => nil)
(fun size l x_1 r l_ih r_ih h =>
rec (motive := fun t => nil.dual = t → Ordnode α) (fun h => nil)
(fun size_1 l x_2 r l_ih r_ih h =>
if size_1 < ratio * size then
node (rs + ls + 1) (node (rs + size_1 + 1) (node rs rr.dual rx rl.dual) x nil.dual) lx
(node lls lll llx llr).dual
else
node (rs + ls + 1) (node (rs + l.size + 1) (node rs rr.dual rx rl.dual) x l) x_2
(node (r.size + size + 1) r lx (node lls lll llx llr).dual))
nil.dual ⋯)
(node lls lll llx llr).dual ⋯
|
rfl
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Ordmap/Ordset.lean
|
pow_sub₀
|
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a✝ b c d : G₀
m n : ℕ
a : G₀
ha : a ≠ 0
h : n ≤ m
⊢ a ^ (m - n) = a ^ m * (a ^ n)⁻¹
|
have h1 : m - n + n = m := <a>Nat.sub_add_cancel</a> h
|
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a✝ b c d : G₀
m n : ℕ
a : G₀
ha : a ≠ 0
h : n ≤ m
h1 : m - n + n = m
⊢ a ^ (m - n) = a ^ m * (a ^ n)⁻¹
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/GroupWithZero/Units/Basic.lean
|
pow_sub₀
|
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a✝ b c d : G₀
m n : ℕ
a : G₀
ha : a ≠ 0
h : n ≤ m
h1 : m - n + n = m
⊢ a ^ (m - n) = a ^ m * (a ^ n)⁻¹
|
have h2 : a ^ (m - n) * a ^ n = a ^ m := by rw [← <a>pow_add</a>, h1]
|
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a✝ b c d : G₀
m n : ℕ
a : G₀
ha : a ≠ 0
h : n ≤ m
h1 : m - n + n = m
h2 : a ^ (m - n) * a ^ n = a ^ m
⊢ a ^ (m - n) = a ^ m * (a ^ n)⁻¹
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/GroupWithZero/Units/Basic.lean
|
pow_sub₀
|
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a✝ b c d : G₀
m n : ℕ
a : G₀
ha : a ≠ 0
h : n ≤ m
h1 : m - n + n = m
h2 : a ^ (m - n) * a ^ n = a ^ m
⊢ a ^ (m - n) = a ^ m * (a ^ n)⁻¹
|
simpa only [<a>div_eq_mul_inv</a>] using <a>eq_div_of_mul_eq</a> (<a>pow_ne_zero</a> _ ha) h2
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/GroupWithZero/Units/Basic.lean
|
pow_sub₀
|
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a✝ b c d : G₀
m n : ℕ
a : G₀
ha : a ≠ 0
h : n ≤ m
h1 : m - n + n = m
⊢ a ^ (m - n) * a ^ n = a ^ m
|
rw [← <a>pow_add</a>, h1]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/GroupWithZero/Units/Basic.lean
|
contMDiffOn_iff_source_of_mem_maximalAtlas
|
𝕜 : Type u_1
inst✝³⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³⁶ : NormedAddCommGroup E
inst✝³⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝³⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³³ : TopologicalSpace M
inst✝³² : ChartedSpace H M
inst✝³¹ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝³⁰ : NormedAddCommGroup E'
inst✝²⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝²⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝²⁷ : TopologicalSpace M'
inst✝²⁶ : ChartedSpace H' M'
inst✝²⁵ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝²² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝²¹ : TopologicalSpace M''
inst✝²⁰ : ChartedSpace H'' M''
F : Type u_11
inst✝¹⁹ : NormedAddCommGroup F
inst✝¹⁸ : NormedSpace 𝕜 F
G : Type u_12
inst✝¹⁷ : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_13
inst✝¹⁶ : TopologicalSpace N
inst✝¹⁵ : ChartedSpace G N
inst✝¹⁴ : SmoothManifoldWithCorners J N
F' : Type u_14
inst✝¹³ : NormedAddCommGroup F'
inst✝¹² : NormedSpace 𝕜 F'
G' : Type u_15
inst✝¹¹ : TopologicalSpace G'
J' : ModelWithCorners 𝕜 F' G'
N' : Type u_16
inst✝¹⁰ : TopologicalSpace N'
inst✝⁹ : ChartedSpace G' N'
inst✝⁸ : SmoothManifoldWithCorners J' N'
F₁ : Type u_17
inst✝⁷ : NormedAddCommGroup F₁
inst✝⁶ : NormedSpace 𝕜 F₁
F₂ : Type u_18
inst✝⁵ : NormedAddCommGroup F₂
inst✝⁴ : NormedSpace 𝕜 F₂
F₃ : Type u_19
inst✝³ : NormedAddCommGroup F₃
inst✝² : NormedSpace 𝕜 F₃
F₄ : Type u_20
inst✝¹ : NormedAddCommGroup F₄
inst✝ : NormedSpace 𝕜 F₄
e : PartialHomeomorph M H
e' : PartialHomeomorph M' H'
f f₁ : M → M'
s s₁ t : Set M
x : M
m n : ℕ∞
he : e ∈ maximalAtlas I M
hs : s ⊆ e.source
⊢ ContMDiffOn I I' n f s ↔ ContMDiffOn 𝓘(𝕜, E) I' n (f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s)
|
simp_rw [<a>ContMDiffOn</a>, <a>Set.forall_mem_image</a>]
|
𝕜 : Type u_1
inst✝³⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³⁶ : NormedAddCommGroup E
inst✝³⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝³⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³³ : TopologicalSpace M
inst✝³² : ChartedSpace H M
inst✝³¹ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝³⁰ : NormedAddCommGroup E'
inst✝²⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝²⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝²⁷ : TopologicalSpace M'
inst✝²⁶ : ChartedSpace H' M'
inst✝²⁵ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝²² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝²¹ : TopologicalSpace M''
inst✝²⁰ : ChartedSpace H'' M''
F : Type u_11
inst✝¹⁹ : NormedAddCommGroup F
inst✝¹⁸ : NormedSpace 𝕜 F
G : Type u_12
inst✝¹⁷ : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_13
inst✝¹⁶ : TopologicalSpace N
inst✝¹⁵ : ChartedSpace G N
inst✝¹⁴ : SmoothManifoldWithCorners J N
F' : Type u_14
inst✝¹³ : NormedAddCommGroup F'
inst✝¹² : NormedSpace 𝕜 F'
G' : Type u_15
inst✝¹¹ : TopologicalSpace G'
J' : ModelWithCorners 𝕜 F' G'
N' : Type u_16
inst✝¹⁰ : TopologicalSpace N'
inst✝⁹ : ChartedSpace G' N'
inst✝⁸ : SmoothManifoldWithCorners J' N'
F₁ : Type u_17
inst✝⁷ : NormedAddCommGroup F₁
inst✝⁶ : NormedSpace 𝕜 F₁
F₂ : Type u_18
inst✝⁵ : NormedAddCommGroup F₂
inst✝⁴ : NormedSpace 𝕜 F₂
F₃ : Type u_19
inst✝³ : NormedAddCommGroup F₃
inst✝² : NormedSpace 𝕜 F₃
F₄ : Type u_20
inst✝¹ : NormedAddCommGroup F₄
inst✝ : NormedSpace 𝕜 F₄
e : PartialHomeomorph M H
e' : PartialHomeomorph M' H'
f f₁ : M → M'
s s₁ t : Set M
x : M
m n : ℕ∞
he : e ∈ maximalAtlas I M
hs : s ⊆ e.source
⊢ (∀ x ∈ s, ContMDiffWithinAt I I' n f s x) ↔
∀ ⦃x : M⦄, x ∈ s → ContMDiffWithinAt 𝓘(𝕜, E) I' n (f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s) (↑(e.extend I) x)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Geometry/Manifold/ContMDiff/Defs.lean
|
contMDiffOn_iff_source_of_mem_maximalAtlas
|
𝕜 : Type u_1
inst✝³⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³⁶ : NormedAddCommGroup E
inst✝³⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝³⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³³ : TopologicalSpace M
inst✝³² : ChartedSpace H M
inst✝³¹ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝³⁰ : NormedAddCommGroup E'
inst✝²⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝²⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝²⁷ : TopologicalSpace M'
inst✝²⁶ : ChartedSpace H' M'
inst✝²⁵ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝²² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝²¹ : TopologicalSpace M''
inst✝²⁰ : ChartedSpace H'' M''
F : Type u_11
inst✝¹⁹ : NormedAddCommGroup F
inst✝¹⁸ : NormedSpace 𝕜 F
G : Type u_12
inst✝¹⁷ : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_13
inst✝¹⁶ : TopologicalSpace N
inst✝¹⁵ : ChartedSpace G N
inst✝¹⁴ : SmoothManifoldWithCorners J N
F' : Type u_14
inst✝¹³ : NormedAddCommGroup F'
inst✝¹² : NormedSpace 𝕜 F'
G' : Type u_15
inst✝¹¹ : TopologicalSpace G'
J' : ModelWithCorners 𝕜 F' G'
N' : Type u_16
inst✝¹⁰ : TopologicalSpace N'
inst✝⁹ : ChartedSpace G' N'
inst✝⁸ : SmoothManifoldWithCorners J' N'
F₁ : Type u_17
inst✝⁷ : NormedAddCommGroup F₁
inst✝⁶ : NormedSpace 𝕜 F₁
F₂ : Type u_18
inst✝⁵ : NormedAddCommGroup F₂
inst✝⁴ : NormedSpace 𝕜 F₂
F₃ : Type u_19
inst✝³ : NormedAddCommGroup F₃
inst✝² : NormedSpace 𝕜 F₃
F₄ : Type u_20
inst✝¹ : NormedAddCommGroup F₄
inst✝ : NormedSpace 𝕜 F₄
e : PartialHomeomorph M H
e' : PartialHomeomorph M' H'
f f₁ : M → M'
s s₁ t : Set M
x : M
m n : ℕ∞
he : e ∈ maximalAtlas I M
hs : s ⊆ e.source
⊢ (∀ x ∈ s, ContMDiffWithinAt I I' n f s x) ↔
∀ ⦃x : M⦄, x ∈ s → ContMDiffWithinAt 𝓘(𝕜, E) I' n (f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s) (↑(e.extend I) x)
|
refine <a>forall₂_congr</a> fun x hx => ?_
|
𝕜 : Type u_1
inst✝³⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³⁶ : NormedAddCommGroup E
inst✝³⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝³⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³³ : TopologicalSpace M
inst✝³² : ChartedSpace H M
inst✝³¹ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝³⁰ : NormedAddCommGroup E'
inst✝²⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝²⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝²⁷ : TopologicalSpace M'
inst✝²⁶ : ChartedSpace H' M'
inst✝²⁵ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝²² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝²¹ : TopologicalSpace M''
inst✝²⁰ : ChartedSpace H'' M''
F : Type u_11
inst✝¹⁹ : NormedAddCommGroup F
inst✝¹⁸ : NormedSpace 𝕜 F
G : Type u_12
inst✝¹⁷ : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_13
inst✝¹⁶ : TopologicalSpace N
inst✝¹⁵ : ChartedSpace G N
inst✝¹⁴ : SmoothManifoldWithCorners J N
F' : Type u_14
inst✝¹³ : NormedAddCommGroup F'
inst✝¹² : NormedSpace 𝕜 F'
G' : Type u_15
inst✝¹¹ : TopologicalSpace G'
J' : ModelWithCorners 𝕜 F' G'
N' : Type u_16
inst✝¹⁰ : TopologicalSpace N'
inst✝⁹ : ChartedSpace G' N'
inst✝⁸ : SmoothManifoldWithCorners J' N'
F₁ : Type u_17
inst✝⁷ : NormedAddCommGroup F₁
inst✝⁶ : NormedSpace 𝕜 F₁
F₂ : Type u_18
inst✝⁵ : NormedAddCommGroup F₂
inst✝⁴ : NormedSpace 𝕜 F₂
F₃ : Type u_19
inst✝³ : NormedAddCommGroup F₃
inst✝² : NormedSpace 𝕜 F₃
F₄ : Type u_20
inst✝¹ : NormedAddCommGroup F₄
inst✝ : NormedSpace 𝕜 F₄
e : PartialHomeomorph M H
e' : PartialHomeomorph M' H'
f f₁ : M → M'
s s₁ t : Set M
x✝ : M
m n : ℕ∞
he : e ∈ maximalAtlas I M
hs : s ⊆ e.source
x : M
hx : x ∈ s
⊢ ContMDiffWithinAt I I' n f s x ↔
ContMDiffWithinAt 𝓘(𝕜, E) I' n (f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s) (↑(e.extend I) x)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Geometry/Manifold/ContMDiff/Defs.lean
|
contMDiffOn_iff_source_of_mem_maximalAtlas
|
𝕜 : Type u_1
inst✝³⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³⁶ : NormedAddCommGroup E
inst✝³⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝³⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³³ : TopologicalSpace M
inst✝³² : ChartedSpace H M
inst✝³¹ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝³⁰ : NormedAddCommGroup E'
inst✝²⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝²⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝²⁷ : TopologicalSpace M'
inst✝²⁶ : ChartedSpace H' M'
inst✝²⁵ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝²² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝²¹ : TopologicalSpace M''
inst✝²⁰ : ChartedSpace H'' M''
F : Type u_11
inst✝¹⁹ : NormedAddCommGroup F
inst✝¹⁸ : NormedSpace 𝕜 F
G : Type u_12
inst✝¹⁷ : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_13
inst✝¹⁶ : TopologicalSpace N
inst✝¹⁵ : ChartedSpace G N
inst✝¹⁴ : SmoothManifoldWithCorners J N
F' : Type u_14
inst✝¹³ : NormedAddCommGroup F'
inst✝¹² : NormedSpace 𝕜 F'
G' : Type u_15
inst✝¹¹ : TopologicalSpace G'
J' : ModelWithCorners 𝕜 F' G'
N' : Type u_16
inst✝¹⁰ : TopologicalSpace N'
inst✝⁹ : ChartedSpace G' N'
inst✝⁸ : SmoothManifoldWithCorners J' N'
F₁ : Type u_17
inst✝⁷ : NormedAddCommGroup F₁
inst✝⁶ : NormedSpace 𝕜 F₁
F₂ : Type u_18
inst✝⁵ : NormedAddCommGroup F₂
inst✝⁴ : NormedSpace 𝕜 F₂
F₃ : Type u_19
inst✝³ : NormedAddCommGroup F₃
inst✝² : NormedSpace 𝕜 F₃
F₄ : Type u_20
inst✝¹ : NormedAddCommGroup F₄
inst✝ : NormedSpace 𝕜 F₄
e : PartialHomeomorph M H
e' : PartialHomeomorph M' H'
f f₁ : M → M'
s s₁ t : Set M
x✝ : M
m n : ℕ∞
he : e ∈ maximalAtlas I M
hs : s ⊆ e.source
x : M
hx : x ∈ s
⊢ ContMDiffWithinAt I I' n f s x ↔
ContMDiffWithinAt 𝓘(𝕜, E) I' n (f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s) (↑(e.extend I) x)
|
rw [<a>contMDiffWithinAt_iff_source_of_mem_maximalAtlas</a> he (hs hx)]
|
𝕜 : Type u_1
inst✝³⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³⁶ : NormedAddCommGroup E
inst✝³⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝³⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³³ : TopologicalSpace M
inst✝³² : ChartedSpace H M
inst✝³¹ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝³⁰ : NormedAddCommGroup E'
inst✝²⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝²⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝²⁷ : TopologicalSpace M'
inst✝²⁶ : ChartedSpace H' M'
inst✝²⁵ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝²² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝²¹ : TopologicalSpace M''
inst✝²⁰ : ChartedSpace H'' M''
F : Type u_11
inst✝¹⁹ : NormedAddCommGroup F
inst✝¹⁸ : NormedSpace 𝕜 F
G : Type u_12
inst✝¹⁷ : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_13
inst✝¹⁶ : TopologicalSpace N
inst✝¹⁵ : ChartedSpace G N
inst✝¹⁴ : SmoothManifoldWithCorners J N
F' : Type u_14
inst✝¹³ : NormedAddCommGroup F'
inst✝¹² : NormedSpace 𝕜 F'
G' : Type u_15
inst✝¹¹ : TopologicalSpace G'
J' : ModelWithCorners 𝕜 F' G'
N' : Type u_16
inst✝¹⁰ : TopologicalSpace N'
inst✝⁹ : ChartedSpace G' N'
inst✝⁸ : SmoothManifoldWithCorners J' N'
F₁ : Type u_17
inst✝⁷ : NormedAddCommGroup F₁
inst✝⁶ : NormedSpace 𝕜 F₁
F₂ : Type u_18
inst✝⁵ : NormedAddCommGroup F₂
inst✝⁴ : NormedSpace 𝕜 F₂
F₃ : Type u_19
inst✝³ : NormedAddCommGroup F₃
inst✝² : NormedSpace 𝕜 F₃
F₄ : Type u_20
inst✝¹ : NormedAddCommGroup F₄
inst✝ : NormedSpace 𝕜 F₄
e : PartialHomeomorph M H
e' : PartialHomeomorph M' H'
f f₁ : M → M'
s s₁ t : Set M
x✝ : M
m n : ℕ∞
he : e ∈ maximalAtlas I M
hs : s ⊆ e.source
x : M
hx : x ∈ s
⊢ ContMDiffWithinAt 𝓘(𝕜, E) I' n (f ∘ ↑(e.extend I).symm) (↑(e.extend I).symm ⁻¹' s ∩ range ↑I) (↑(e.extend I) x) ↔
ContMDiffWithinAt 𝓘(𝕜, E) I' n (f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s) (↑(e.extend I) x)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Geometry/Manifold/ContMDiff/Defs.lean
|
contMDiffOn_iff_source_of_mem_maximalAtlas
|
𝕜 : Type u_1
inst✝³⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³⁶ : NormedAddCommGroup E
inst✝³⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝³⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³³ : TopologicalSpace M
inst✝³² : ChartedSpace H M
inst✝³¹ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝³⁰ : NormedAddCommGroup E'
inst✝²⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝²⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝²⁷ : TopologicalSpace M'
inst✝²⁶ : ChartedSpace H' M'
inst✝²⁵ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝²² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝²¹ : TopologicalSpace M''
inst✝²⁰ : ChartedSpace H'' M''
F : Type u_11
inst✝¹⁹ : NormedAddCommGroup F
inst✝¹⁸ : NormedSpace 𝕜 F
G : Type u_12
inst✝¹⁷ : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_13
inst✝¹⁶ : TopologicalSpace N
inst✝¹⁵ : ChartedSpace G N
inst✝¹⁴ : SmoothManifoldWithCorners J N
F' : Type u_14
inst✝¹³ : NormedAddCommGroup F'
inst✝¹² : NormedSpace 𝕜 F'
G' : Type u_15
inst✝¹¹ : TopologicalSpace G'
J' : ModelWithCorners 𝕜 F' G'
N' : Type u_16
inst✝¹⁰ : TopologicalSpace N'
inst✝⁹ : ChartedSpace G' N'
inst✝⁸ : SmoothManifoldWithCorners J' N'
F₁ : Type u_17
inst✝⁷ : NormedAddCommGroup F₁
inst✝⁶ : NormedSpace 𝕜 F₁
F₂ : Type u_18
inst✝⁵ : NormedAddCommGroup F₂
inst✝⁴ : NormedSpace 𝕜 F₂
F₃ : Type u_19
inst✝³ : NormedAddCommGroup F₃
inst✝² : NormedSpace 𝕜 F₃
F₄ : Type u_20
inst✝¹ : NormedAddCommGroup F₄
inst✝ : NormedSpace 𝕜 F₄
e : PartialHomeomorph M H
e' : PartialHomeomorph M' H'
f f₁ : M → M'
s s₁ t : Set M
x✝ : M
m n : ℕ∞
he : e ∈ maximalAtlas I M
hs : s ⊆ e.source
x : M
hx : x ∈ s
⊢ ContMDiffWithinAt 𝓘(𝕜, E) I' n (f ∘ ↑(e.extend I).symm) (↑(e.extend I).symm ⁻¹' s ∩ range ↑I) (↑(e.extend I) x) ↔
ContMDiffWithinAt 𝓘(𝕜, E) I' n (f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s) (↑(e.extend I) x)
|
apply <a>contMDiffWithinAt_congr_nhds</a>
|
case hst
𝕜 : Type u_1
inst✝³⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³⁶ : NormedAddCommGroup E
inst✝³⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝³⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³³ : TopologicalSpace M
inst✝³² : ChartedSpace H M
inst✝³¹ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝³⁰ : NormedAddCommGroup E'
inst✝²⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝²⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝²⁷ : TopologicalSpace M'
inst✝²⁶ : ChartedSpace H' M'
inst✝²⁵ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝²² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝²¹ : TopologicalSpace M''
inst✝²⁰ : ChartedSpace H'' M''
F : Type u_11
inst✝¹⁹ : NormedAddCommGroup F
inst✝¹⁸ : NormedSpace 𝕜 F
G : Type u_12
inst✝¹⁷ : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_13
inst✝¹⁶ : TopologicalSpace N
inst✝¹⁵ : ChartedSpace G N
inst✝¹⁴ : SmoothManifoldWithCorners J N
F' : Type u_14
inst✝¹³ : NormedAddCommGroup F'
inst✝¹² : NormedSpace 𝕜 F'
G' : Type u_15
inst✝¹¹ : TopologicalSpace G'
J' : ModelWithCorners 𝕜 F' G'
N' : Type u_16
inst✝¹⁰ : TopologicalSpace N'
inst✝⁹ : ChartedSpace G' N'
inst✝⁸ : SmoothManifoldWithCorners J' N'
F₁ : Type u_17
inst✝⁷ : NormedAddCommGroup F₁
inst✝⁶ : NormedSpace 𝕜 F₁
F₂ : Type u_18
inst✝⁵ : NormedAddCommGroup F₂
inst✝⁴ : NormedSpace 𝕜 F₂
F₃ : Type u_19
inst✝³ : NormedAddCommGroup F₃
inst✝² : NormedSpace 𝕜 F₃
F₄ : Type u_20
inst✝¹ : NormedAddCommGroup F₄
inst✝ : NormedSpace 𝕜 F₄
e : PartialHomeomorph M H
e' : PartialHomeomorph M' H'
f f₁ : M → M'
s s₁ t : Set M
x✝ : M
m n : ℕ∞
he : e ∈ maximalAtlas I M
hs : s ⊆ e.source
x : M
hx : x ∈ s
⊢ 𝓝[↑(e.extend I).symm ⁻¹' s ∩ range ↑I] ↑(e.extend I) x = 𝓝[↑(e.extend I) '' s] ↑(e.extend I) x
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Geometry/Manifold/ContMDiff/Defs.lean
|
contMDiffOn_iff_source_of_mem_maximalAtlas
|
case hst
𝕜 : Type u_1
inst✝³⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³⁶ : NormedAddCommGroup E
inst✝³⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝³⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³³ : TopologicalSpace M
inst✝³² : ChartedSpace H M
inst✝³¹ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝³⁰ : NormedAddCommGroup E'
inst✝²⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝²⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝²⁷ : TopologicalSpace M'
inst✝²⁶ : ChartedSpace H' M'
inst✝²⁵ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝²² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝²¹ : TopologicalSpace M''
inst✝²⁰ : ChartedSpace H'' M''
F : Type u_11
inst✝¹⁹ : NormedAddCommGroup F
inst✝¹⁸ : NormedSpace 𝕜 F
G : Type u_12
inst✝¹⁷ : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_13
inst✝¹⁶ : TopologicalSpace N
inst✝¹⁵ : ChartedSpace G N
inst✝¹⁴ : SmoothManifoldWithCorners J N
F' : Type u_14
inst✝¹³ : NormedAddCommGroup F'
inst✝¹² : NormedSpace 𝕜 F'
G' : Type u_15
inst✝¹¹ : TopologicalSpace G'
J' : ModelWithCorners 𝕜 F' G'
N' : Type u_16
inst✝¹⁰ : TopologicalSpace N'
inst✝⁹ : ChartedSpace G' N'
inst✝⁸ : SmoothManifoldWithCorners J' N'
F₁ : Type u_17
inst✝⁷ : NormedAddCommGroup F₁
inst✝⁶ : NormedSpace 𝕜 F₁
F₂ : Type u_18
inst✝⁵ : NormedAddCommGroup F₂
inst✝⁴ : NormedSpace 𝕜 F₂
F₃ : Type u_19
inst✝³ : NormedAddCommGroup F₃
inst✝² : NormedSpace 𝕜 F₃
F₄ : Type u_20
inst✝¹ : NormedAddCommGroup F₄
inst✝ : NormedSpace 𝕜 F₄
e : PartialHomeomorph M H
e' : PartialHomeomorph M' H'
f f₁ : M → M'
s s₁ t : Set M
x✝ : M
m n : ℕ∞
he : e ∈ maximalAtlas I M
hs : s ⊆ e.source
x : M
hx : x ∈ s
⊢ 𝓝[↑(e.extend I).symm ⁻¹' s ∩ range ↑I] ↑(e.extend I) x = 𝓝[↑(e.extend I) '' s] ↑(e.extend I) x
|
simp_rw [<a>nhdsWithin_eq_iff_eventuallyEq</a>, e.extend_symm_preimage_inter_range_eventuallyEq I hs (hs hx)]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Geometry/Manifold/ContMDiff/Defs.lean
|
UnitAddCircle.norm_eq
|
x : ℝ
⊢ ‖↑x‖ = |x - ↑(round x)|
|
simp [<a>AddCircle.norm_eq</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/Normed/Group/AddCircle.lean
|
InnerProductSpace.Core.inner_zero_right
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : _root_.RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : Core 𝕜 F
x : F
⊢ ⟪x, 0⟫_𝕜 = 0
|
rw [← <a>InnerProductSpace.Core.inner_conj_symm</a>, <a>InnerProductSpace.Core.inner_zero_left</a>]
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : _root_.RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : Core 𝕜 F
x : F
⊢ (starRingEnd 𝕜) 0 = 0
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
InnerProductSpace.Core.inner_zero_right
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : _root_.RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : Core 𝕜 F
x : F
⊢ (starRingEnd 𝕜) 0 = 0
|
simp only [<a>RingHom.map_zero</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
NNReal.sqrt_one
|
x y : ℝ≥0
⊢ sqrt 1 = 1
|
simp
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Real/Sqrt.lean
|
Ordering.compares_swap
|
α : Type u_1
β : Type u_2
inst✝ : LT α
a b : α
o : Ordering
⊢ o.swap.Compares a b ↔ o.Compares b a
|
cases o
|
case lt
α : Type u_1
β : Type u_2
inst✝ : LT α
a b : α
⊢ lt.swap.Compares a b ↔ lt.Compares b a
case eq
α : Type u_1
β : Type u_2
inst✝ : LT α
a b : α
⊢ eq.swap.Compares a b ↔ eq.Compares b a
case gt
α : Type u_1
β : Type u_2
inst✝ : LT α
a b : α
⊢ gt.swap.Compares a b ↔ gt.Compares b a
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Compare.lean
|
Ordering.compares_swap
|
case lt
α : Type u_1
β : Type u_2
inst✝ : LT α
a b : α
⊢ lt.swap.Compares a b ↔ lt.Compares b a
|
exact <a>Iff.rfl</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Compare.lean
|
Ordering.compares_swap
|
case eq
α : Type u_1
β : Type u_2
inst✝ : LT α
a b : α
⊢ eq.swap.Compares a b ↔ eq.Compares b a
|
exact <a>eq_comm</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Compare.lean
|
Ordering.compares_swap
|
case gt
α : Type u_1
β : Type u_2
inst✝ : LT α
a b : α
⊢ gt.swap.Compares a b ↔ gt.Compares b a
|
exact <a>Iff.rfl</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Compare.lean
|
Set.iInter₂_union
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : (i : ι) → κ i → Set α
t : Set α
⊢ (⋂ i, ⋂ j, s i j) ∪ t = ⋂ i, ⋂ j, s i j ∪ t
|
simp_rw [<a>Set.iInter_union</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Set/Lattice.lean
|
Finset.image_diag_union_image_offDiag
|
α : Type u_1
inst✝ : DecidableEq α
s t : Finset α
a b : α
m : Sym2 α
⊢ image Sym2.mk s.diag ∪ image Sym2.mk s.offDiag = s.sym2
|
rw [← <a>Finset.image_union</a>, <a>Finset.diag_union_offDiag</a>, <a>Finset.sym2_eq_image</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Finset/Sym.lean
|
AdjoinRoot.algHomOfDvd_apply_root
|
K : Type u_1
L : Type ?u.589078
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
p q : K[X]
hpq : q ∣ p
⊢ (algHomOfDvd hpq) (root p) = root q
|
rw [<a>AdjoinRoot.algHomOfDvd</a>, <a>AdjoinRoot.liftHom_root</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/FieldTheory/Adjoin.lean
|
Real.deriv_Gamma_nat
|
n : ℕ
⊢ deriv Gamma (↑n + 1) = ↑n ! * (-γ + ↑(harmonic n))
|
let f := <a>Real.log</a> ∘ <a>Real.Gamma</a>
|
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
⊢ deriv Gamma (↑n + 1) = ↑n ! * (-γ + ↑(harmonic n))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/Harmonic/GammaDeriv.lean
|
Real.deriv_Gamma_nat
|
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
⊢ deriv Gamma (↑n + 1) = ↑n ! * (-γ + ↑(harmonic n))
|
suffices <a>deriv</a> (<a>Real.log</a> ∘ <a>Real.Gamma</a>) (n + 1) = -γ + <a>harmonic</a> n by rwa [<a>Function.comp_def</a>, <a>deriv.log</a> (<a>Real.differentiableAt_Gamma</a> (fun m ↦ by linarith)) (by positivity), <a>Real.Gamma_nat_eq_factorial</a>, <a>div_eq_iff_mul_eq</a> (by positivity), <a>mul_comm</a>, <a>Eq.comm</a>] at this
|
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
⊢ deriv (log ∘ Gamma) (↑n + 1) = -γ + ↑(harmonic n)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/Harmonic/GammaDeriv.lean
|
Real.deriv_Gamma_nat
|
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
⊢ deriv (log ∘ Gamma) (↑n + 1) = -γ + ↑(harmonic n)
|
have hc : <a>ConvexOn</a> ℝ (<a>Set.Ioi</a> 0) f := <a>Real.convexOn_log_Gamma</a>
|
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
⊢ deriv (log ∘ Gamma) (↑n + 1) = -γ + ↑(harmonic n)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/Harmonic/GammaDeriv.lean
|
Real.deriv_Gamma_nat
|
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
⊢ deriv (log ∘ Gamma) (↑n + 1) = -γ + ↑(harmonic n)
|
have h_rec (x : ℝ) (hx : 0 < x) : f (x + 1) = f x + <a>Real.log</a> x := by simp only [f, <a>Function.comp_apply</a>, <a>Real.Gamma_add_one</a> hx.ne', <a>Real.log_mul</a> hx.ne' (<a>Real.Gamma_pos_of_pos</a> hx).<a>LT.lt.ne'</a>, <a>add_comm</a>]
|
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
⊢ deriv (log ∘ Gamma) (↑n + 1) = -γ + ↑(harmonic n)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/Harmonic/GammaDeriv.lean
|
Real.deriv_Gamma_nat
|
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
⊢ deriv (log ∘ Gamma) (↑n + 1) = -γ + ↑(harmonic n)
|
have hder {x : ℝ} (hx : 0 < x) : <a>DifferentiableAt</a> ℝ f x := by refine ((<a>Real.differentiableAt_Gamma</a> ?_).<a>DifferentiableAt.log</a> (<a>Real.Gamma_ne_zero</a> ?_)) <;> exact fun m ↦ <a>ne_of_gt</a> (by linarith)
|
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
⊢ deriv (log ∘ Gamma) (↑n + 1) = -γ + ↑(harmonic n)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/Harmonic/GammaDeriv.lean
|
Real.deriv_Gamma_nat
|
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
⊢ deriv (log ∘ Gamma) (↑n + 1) = -γ + ↑(harmonic n)
|
have hder_rec (x : ℝ) (hx : 0 < x) : <a>deriv</a> f (x + 1) = <a>deriv</a> f x + 1 / x := by rw [← <a>deriv_comp_add_const</a> _ _ (hder <| by positivity), <a>one_div</a>, ← <a>Real.deriv_log</a>, ← <a>deriv_add</a> (hder <| by positivity) (<a>Real.differentiableAt_log</a> hx.ne')] apply <a>Filter.EventuallyEq.deriv_eq</a> filter_upwards [<a>eventually_gt_nhds</a> hx] using h_rec
|
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
⊢ deriv (log ∘ Gamma) (↑n + 1) = -γ + ↑(harmonic n)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/Harmonic/GammaDeriv.lean
|
Real.deriv_Gamma_nat
|
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
⊢ deriv (log ∘ Gamma) (↑n + 1) = -γ + ↑(harmonic n)
|
suffices -<a>deriv</a> f 1 = γ by rw [hder_nat n, ← this, <a>neg_neg</a>]
|
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
⊢ -deriv f 1 = γ
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/Harmonic/GammaDeriv.lean
|
Real.deriv_Gamma_nat
|
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
⊢ -deriv f 1 = γ
|
have derivLB (n : ℕ) (hn : 0 < n) : <a>Real.log</a> n ≤ <a>deriv</a> f (n + 1) := by refine (<a>le_of_eq</a> ?_).<a>LE.le.trans</a> <| hc.slope_le_deriv (mem_Ioi.mpr <| Nat.cast_pos.mpr hn) (by positivity : _ < (_ : ℝ)) (by linarith) (hder <| by positivity) rw [<a>slope_def_field</a>, show n + 1 - n = (1 : ℝ) by ring, <a>div_one</a>, h_rec n (by positivity), <a>add_sub_cancel_left</a>]
|
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
derivLB : ∀ (n : ℕ), 0 < n → log ↑n ≤ deriv f (↑n + 1)
⊢ -deriv f 1 = γ
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/Harmonic/GammaDeriv.lean
|
Real.deriv_Gamma_nat
|
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
derivLB : ∀ (n : ℕ), 0 < n → log ↑n ≤ deriv f (↑n + 1)
⊢ -deriv f 1 = γ
|
have derivUB (n : ℕ) : <a>deriv</a> f (n + 1) ≤ <a>Real.log</a> (n + 1) := by refine (hc.deriv_le_slope (by positivity : (0 : ℝ) < n + 1) (by positivity : (0 : ℝ) < n + 2) (by linarith) (hder <| by positivity)).<a>LE.le.trans</a> (<a>le_of_eq</a> ?_) rw [<a>slope_def_field</a>, show n + 2 - (n + 1) = (1 : ℝ) by ring, <a>div_one</a>, show n + 2 = (n + 1) + (1 : ℝ) by ring, h_rec (n + 1) (by positivity), <a>add_sub_cancel_left</a>]
|
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
derivLB : ∀ (n : ℕ), 0 < n → log ↑n ≤ deriv f (↑n + 1)
derivUB : ∀ (n : ℕ), deriv f (↑n + 1) ≤ log (↑n + 1)
⊢ -deriv f 1 = γ
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/Harmonic/GammaDeriv.lean
|
Real.deriv_Gamma_nat
|
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
derivLB : ∀ (n : ℕ), 0 < n → log ↑n ≤ deriv f (↑n + 1)
derivUB : ∀ (n : ℕ), deriv f (↑n + 1) ≤ log (↑n + 1)
⊢ -deriv f 1 = γ
|
apply <a>le_antisymm</a>
|
case a
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
derivLB : ∀ (n : ℕ), 0 < n → log ↑n ≤ deriv f (↑n + 1)
derivUB : ∀ (n : ℕ), deriv f (↑n + 1) ≤ log (↑n + 1)
⊢ -deriv f 1 ≤ γ
case a
n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
derivLB : ∀ (n : ℕ), 0 < n → log ↑n ≤ deriv f (↑n + 1)
derivUB : ∀ (n : ℕ), deriv f (↑n + 1) ≤ log (↑n + 1)
⊢ γ ≤ -deriv f 1
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/Harmonic/GammaDeriv.lean
|
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