mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: R : Type u_1
F : Type u_2
inst✝³ : Semiring R
inst✝² : FunLike F R ℝ
v : F
inst✝¹ : ZeroHomClass F R ℝ
inst✝ : NonnegHomClass F R ℝ
hna : IsNonarchimedean ⇑v
c : ℝ
hc : 0 ≤ c
p q : R[X]
h✝¹ : p ≠ 0
h✝ : q ≠ 0
hpq : p + q ≠ 0
i : ℕ
a✝ : i ∈ (p + q).support
⊢ max (v (p.coeff i) * c ^ i) (v (q.coeff i) * c ^ i) = max (v (p.coeff i) * c ^ i) (v (q.coeff i) * c ^ i)

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: case neg.inr
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E → E
α : ℝ → E
u : Set E
tmin tmax : ℝ
n : ℕ∞
hf : ContDiffOn ℝ (↑n) (uncurry f) (Icc tmin tmax ×ˢ u)
hα : ∀ t ∈ Icc tmin tmax, HasDerivWithinAt α (f t (α t)) (Icc tmin tmax) t
hmem : MapsTo α (Icc tmin tmax) u
h : tmax = tmin
⊢ ContDiffOn ℝ (↑n) α {tmin}

Tactic:
]

Goal state after:
case neg.inr
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E → E
α : ℝ → E
u : Set E
tmin tmax : ℝ
n : ℕ∞
hf : ContDiffOn ℝ (↑n) (uncurry f) (Icc tmin tmax ×ˢ u)
hα : ∀ t ∈ Icc tmin tmax, HasDerivWithinAt α (f t (α t)) (Icc tmin tmax) t
hmem : MapsTo α (Icc tmin tmax) u
h : tmax = tmin
⊢ ContDiffOn ℝ (↑n) α {tmin}
Module: Mathlib.Analysis.Polynomial.Norm

Goal state before:
pp: A : Type u_1
inst✝¹ : SeminormedRing A
inst✝ : NormOneClass A
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: case neg.inr
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E → E
α : ℝ → E
u : Set E
tmin tmax : ℝ
n : ℕ∞
hf : ContDiffOn ℝ (↑n) (uncurry f) (Icc tmin tmax ×ˢ u)
hα : ∀ t ∈ Icc tmin tmax, HasDerivWithinAt α (f t (α t)) (Icc tmin tmax) t
hmem : MapsTo α (Icc tmin tmax) u
h : tmax = tmin
x✝ : ℝ
ht : x✝ ∈ {tmin}
⊢ ContDiffWithinAt ℝ (↑n) α {tmin} tmin

Tactic:
]

Goal state after:
case neg.inr
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E → E
α : ℝ → E
u : Set E
tmin tmax : ℝ
n : ℕ∞
hf : ContDiffOn ℝ (↑n) (uncurry f) (Icc tmin tmax ×ˢ u)
hα : ∀ t ∈ Icc tmin tmax, HasDerivWithinAt α (f t (α t)) (Icc tmin tmax) t
hmem : MapsTo α (Icc tmin tmax) u
h : tmax = tmin
x✝ : ℝ
ht : x✝ ∈ {tmin}
⊢ ContDiffWithinAt ℝ (↑n) α {tmin} tmin
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℝ → E → E
t₀ ε : ℝ
hε : 0 < ε
x₀ : E
a r L K : ℝ≥0
hf : IsPicardLindelof f ⟨t₀, ⋯⟩ x₀ a r L K
a' r' : ℝ≥0
ha : a' ≤ a
hr : r' < a'
ha'r' : 0 < ↑a' - ↑r'
ε' : ℝ := min ε ((↑a' - ↑r') / (↑L + 1))
hε'pos : 0 < ε'
hε'_le : ε' ≤ ε
⊢ ↑L * ((↑a' - ↑r') / (↑L + 1)) = ↑L / (↑L + 1) * (↑a' - ↑r')

Tactic:
by ring

Goal state after:
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: case hbc
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℝ → E → E
t₀ ε : ℝ
hε : 0 < ε
x₀ : E
a r L K : ℝ≥0
hf : IsPicardLindelof f ⟨t₀, ⋯⟩ x₀ a r L K
a' r' : ℝ≥0
ha : a' ≤ a
hr : r' < a'
ha'r' : 0 < ↑a' - ↑r'
ε' : ℝ := min ε ((↑a' - ↑r') / (↑L + 1))
hε'pos : 0 < ε'
hε'_le : ε' ≤ ε
⊢ ↑L ≤ ↑L + 1

Tactic:
]

Goal state after:
case hbc
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℝ → E → E
t₀ ε : ℝ
hε : 0 < ε
x₀ : E
a r L K : ℝ≥0
hf : IsPicardLindelof f ⟨t₀, ⋯⟩ x₀ a r L K
a' r' : ℝ≥0
ha : a' ≤ a
hr : r' < a'
ha'r' : 0 < ↑a' - ↑r'
ε' : ℝ := min ε ((↑a' - ↑r') / (↑L + 1))
hε'pos : 0 < ε'
hε'_le : ε' ≤ ε
⊢ ↑L ≤ ↑L + 1
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: case inr
c : ℝ
R : Type u_3
inst✝ : Ring R
v : AbsoluteValue R ℝ
hna : IsNonarchimedean ⇑v
hc : 0 < c
p q : R[X]
hc0 : 0 ≤ c
i : ℕ
hi_p : gaussNorm v c p = v (p.coeff i) * c ^ i
hlt_p : ∀ j < i, v (p.coeff j) * c ^ j < gaussNorm v c p
j : ℕ
hj_q : gaussNorm v c q = v (q.coeff j) * c ^ j
hlt_q : ∀ j_1 < j, v (q.coeff j_1) * c ^ j_1 < gaussNorm v c q
this :
  ∀ {c : ℝ} {R : Type u_3} [inst : Ring R] {v : AbsoluteValue R ℝ},
    IsNonarchimedean ⇑v →
      0 < c →
        ∀ (p q : R[X]),
          0 ≤ c →
            ∀ (i : ℕ),
              gaussNorm v c p = v (p.coeff i) * c ^ i →
                (∀ j < i, v (p.coeff j) * c ^ j < gaussNorm v c p) →
                  ∀ (j : ℕ),
                    gaussNorm v c q = v (q.coeff j) * c ^ j →
                      (∀ j_1 < j, v (q.coeff j_1) * c ^ j_1 < gaussNorm v c q) →
                        v (p.coeff i) ≠ 0 ∧ v (q.coeff j) ≠ 0 →
                          gaussNorm v c p * gaussNorm v c q ≤ gaussNorm v c (p * q)
hvpq : ¬(v (p.coeff i) ≠ 0 ∧ v (q.coeff j) ≠ 0)
⊢ gaussNorm v c p * gaussNorm v c q ≤ gaussNorm v c (p * q)

Tactic:
grind [mul_mul_mul_comm, gaussNorm_nonneg]

Goal state after:
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → E
x₀ : E
hf : ContDiffAt ℝ 1 f x₀
K : ℝ≥0
s : Set E
hs : s ∈ 𝓝 x₀
hl : LipschitzOnWith K f s
a : ℝ
ha : 0 < a
has : ball x₀ a ⊆ s
L : ℝ := ↑K * a + ‖f x₀‖ + 1
hL : L = ↑K * a + ‖f x₀‖ + 1
hL0 : 0 < L
x : E
hx : x ∈ closedBall x₀ (a / 2)
⊢ ‖f x‖ ≤ ↑K * a + ‖f x₀‖ + 1

Tactic:
]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → E
x₀ : E
hf : ContDiffAt ℝ 1 f x₀
K : ℝ≥0
s : Set E
hs : s ∈ 𝓝 x₀
hl : LipschitzOnWith K f s
a : ℝ
ha : 0 < a
has : ball x₀ a ⊆ s
L : ℝ := ↑K * a + ‖f x₀‖ + 1
hL : L = ↑K * a + ‖f x₀‖ + 1
hL0 : 0 < L
x : E
hx : x ∈ closedBall x₀ (a / 2)
⊢ ‖f x‖ ≤ ↑K * a + ‖f x₀‖ + 1
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: case hbc
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → E
x₀ : E
hf : ContDiffAt ℝ 1 f x₀
K : ℝ≥0
s : Set E
hs : s ∈ 𝓝 x₀
hl : LipschitzOnWith K f s
a : ℝ
ha : 0 < a
has : ball x₀ a ⊆ s
L : ℝ := ↑K * a + ‖f x₀‖ + 1
hL : L = ↑K * a + ‖f x₀‖ + 1
hL0 : 0 < L
x : E
hx : x ∈ closedBall x₀ (a / 2)
⊢ x ∈ closedBall x₀ a

Tactic:
]

Goal state after:
case hbc
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → E
x₀ : E
hf : ContDiffAt ℝ 1 f x₀
K : ℝ≥0
s : Set E
hs : s ∈ 𝓝 x₀
hl : LipschitzOnWith K f s
a : ℝ
ha : 0 < a
has : ball x₀ a ⊆ s
L : ℝ := ↑K * a + ‖f x₀‖ + 1
hL : L = ↑K * a + ‖f x₀‖ + 1
hL0 : 0 < L
x : E
hx : x ∈ closedBall x₀ (a / 2)
⊢ x ∈ closedBall x₀ a
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: c✝ : ℝ
R✝ : Type u_3
inst✝¹ : Ring R✝
v✝ : AbsoluteValue R✝ ℝ
c : ℝ
R : Type u_3
inst✝ : Ring R
v : AbsoluteValue R ℝ
hna : IsNonarchimedean ⇑v
hc : 0 < c
p q : R[X]
hc0 : 0 ≤ c
i : ℕ
hi_p : gaussNorm v c p = v (p.coeff i) * c ^ i
hlt_p : ∀ j < i, v (p.coeff j) * c ^ j < gaussNorm v c p
j : ℕ
hj_q : gaussNorm v c q = v (q.coeff j) * c ^ j
hlt_q : ∀ j_1 < j, v (q.coeff j_1) * c ^ j_1 < gaussNorm v c q
hvpq : v (p.coeff i) ≠ 0 ∧ v (q.coeff j) ≠ 0
this✝ : v (p.coeff i) ≠ 0
this : v (q.coeff j) ≠ 0
⊢ v (p.coeff i) * c ^ i * (v (q.coeff j) * c ^ j) =
    v (p.coeff (i, i + j - i).1 * q.coeff (i, i + j - i).2) * c ^ (i + j)

pp: c✝ : ℝ
R✝ : Type u_3
inst✝¹ : Ring R✝
v✝ : AbsoluteValue R✝ ℝ
c : ℝ
R : Type u_3
inst✝ : Ring R
v : AbsoluteValue R ℝ
hna : IsNonarchimedean ⇑v
hc : 0 < c
p q : R[X]
hc0 : 0 ≤ c
i : ℕ
hi_p : gaussNorm v c p = v (p.coeff i) * c ^ i
hlt_p : ∀ j < i, v (p.coeff j) * c ^ j < gaussNorm v c p
j : ℕ
hj_q : gaussNorm v c q = v (q.coeff j) * c ^ j
hlt_q : ∀ j_1 < j, v (q.coeff j_1) * c ^ j_1 < gaussNorm v c q
hvpq : v (p.coeff i) ≠ 0 ∧ v (q.coeff j) ≠ 0
this✝ : v (p.coeff i) ≠ 0
this : v (q.coeff j) ≠ 0
⊢ ∀ j_1 ∈ range (i + j).succ,
    j_1 ≠ i →
      v (p.coeff (j_1, i + j - j_1).1 * q.coeff (j_1, i + j - j_1).2) <
        v (p.coeff (i, i + j - i).1 * q.coeff (i, i + j - i).2)

Tactic:
]

Goal state after:
c✝ : ℝ
R✝ : Type u_3
inst✝¹ : Ring R✝
v✝ : AbsoluteValue R✝ ℝ
c : ℝ
R : Type u_3
inst✝ : Ring R
v : AbsoluteValue R ℝ
hna : IsNonarchimedean ⇑v
hc : 0 < c
p q : R[X]
hc0 : 0 ≤ c
i : ℕ
hi_p : gaussNorm v c p = v (p.coeff i) * c ^ i
hlt_p : ∀ j < i, v (p.coeff j) * c ^ j < gaussNorm v c p
j : ℕ
hj_q : gaussNorm v c q = v (q.coeff j) * c ^ j
hlt_q : ∀ j_1 < j, v (q.coeff j_1) * c ^ j_1 < gaussNorm v c q
hvpq : v (p.coeff i) ≠ 0 ∧ v (q.coeff j) ≠ 0
this✝ : v (p.coeff i) ≠ 0
this : v (q.coeff j) ≠ 0
⊢ v (p.coeff i) * c ^ i * (v (q.coeff j) * c ^ j) =
    v (p.coeff (i, i + j - i).1 * q.coeff (i, i + j - i).2) * c ^ (i + j)

c✝ : ℝ
R✝ : Type u_3
inst✝¹ : Ring R✝
v✝ : AbsoluteValue R✝ ℝ
c : ℝ
R : Type u_3
inst✝ : Ring R
v : AbsoluteValue R ℝ
hna : IsNonarchimedean ⇑v
hc : 0 < c
p q : R[X]
hc0 : 0 ≤ c
i : ℕ
hi_p : gaussNorm v c p = v (p.coeff i) * c ^ i
hlt_p : ∀ j < i, v (p.coeff j) * c ^ j < gaussNorm v c p
j : ℕ
hj_q : gaussNorm v c q = v (q.coeff j) * c ^ j
hlt_q : ∀ j_1 < j, v (q.coeff j_1) * c ^ j_1 < gaussNorm v c q
hvpq : v (p.coeff i) ≠ 0 ∧ v (q.coeff j) ≠ 0
this✝ : v (p.coeff i) ≠ 0
this : v (q.coeff j) ≠ 0
⊢ ∀ j_1 ∈ range (i + j).succ,
    j_1 ≠ i →
      v (p.coeff (j_1, i + j - j_1).1 * q.coeff (j_1, i + j - j_1).2) <
        v (p.coeff (i, i + j - i).1 * q.coeff (i, i + j - i).2)
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ x : E
a r L K : ℝ≥0
hf : IsPicardLindelof f t₀ x₀ a r L K
hx : x ∈ closedBall x₀ ↑r
α : FunSpace t₀ x₀ r L
hα : IsFixedPt (FunSpace.next hf hx) α
t : ℝ
ht : t ∈ Icc tmin tmax
⊢ picard f (↑t₀) x α.compProj ↑⟨t, ht⟩ = picard f (↑t₀) x α.compProj t

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Polynomial.Order

Goal state before:
pp: case pos
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → y < x
hlc : 0 < P.leadingCoeff
hdeg : P.degree ≤ 0
⊢ 0 < P.coeff P.natDegree

Tactic:
]

Goal state after:
case pos
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → y < x
hlc : 0 < P.leadingCoeff
hdeg : P.degree ≤ 0
⊢ 0 < P.coeff P.natDegree
Module: Mathlib.Analysis.Polynomial.Order

Goal state before:
pp: case pos
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → y ≤ x
hlc : 0 ≤ P.leadingCoeff
y : ℝ
hroot : P.IsRoot y
hle : x ≤ y
⊢ 0 ≤ 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Polynomial.Order

Goal state before:
pp: P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → y < x
hlc : P.leadingCoeff ≤ 0
this : 0 < eval x (-P)
⊢ 0 < eval x (-P)

Tactic:
]

Goal state after:
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → y < x
hlc : P.leadingCoeff ≤ 0
this : 0 < eval x (-P)
⊢ 0 < eval x (-P)
Module: Mathlib.Analysis.Polynomial.Order

Goal state before:
pp: case refine_1
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → y < x
hlc : P.leadingCoeff ≤ 0
y : ℝ
hy : (-P).IsRoot y
⊢ eval y (-P) = 0

Tactic:
]

Goal state after:
case refine_1
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → y < x
hlc : P.leadingCoeff ≤ 0
y : ℝ
hy : (-P).IsRoot y
⊢ eval y (-P) = 0
Module: Mathlib.Analysis.Polynomial.Order

Goal state before:
pp: case refine_2
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → y < x
hlc : P.leadingCoeff ≤ 0
⊢ P.leadingCoeff ≤ 0

Tactic:
]

Goal state after:
case refine_2
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → y < x
hlc : P.leadingCoeff ≤ 0
⊢ P.leadingCoeff ≤ 0
Module: Mathlib.Analysis.Polynomial.Order

Goal state before:
pp: P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → y ≤ x
hlc : P.leadingCoeff ≤ 0
this : 0 ≤ eval x (-P)
⊢ 0 ≤ eval x (-P)

Tactic:
]

Goal state after:
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → y ≤ x
hlc : P.leadingCoeff ≤ 0
this : 0 ≤ eval x (-P)
⊢ 0 ≤ eval x (-P)
Module: Mathlib.Analysis.Polynomial.Order

Goal state before:
pp: case refine_1
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → y ≤ x
hlc : P.leadingCoeff ≤ 0
y : ℝ
hy : (-P).IsRoot y
⊢ eval y (-P) = 0

Tactic:
]

Goal state after:
case refine_1
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → y ≤ x
hlc : P.leadingCoeff ≤ 0
y : ℝ
hy : (-P).IsRoot y
⊢ eval y (-P) = 0
Module: Mathlib.Analysis.Polynomial.Order

Goal state before:
pp: case refine_2
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → y ≤ x
hlc : P.leadingCoeff ≤ 0
⊢ P.leadingCoeff ≤ 0

Tactic:
]

Goal state after:
case refine_2
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → y ≤ x
hlc : P.leadingCoeff ≤ 0
⊢ P.leadingCoeff ≤ 0
Module: Mathlib.Analysis.Polynomial.Order

Goal state before:
pp: P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
y : ℝ
hy : P.IsRoot (-y)
⊢ P.IsRoot (-y)

Tactic:
]

Goal state after:
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
y : ℝ
hy : P.IsRoot (-y)
⊢ P.IsRoot (-y)
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: c✝ : ℝ
R✝ : Type u_3
inst✝¹ : Ring R✝
v✝ : AbsoluteValue R✝ ℝ
c : ℝ
R : Type u_3
inst✝ : Ring R
v : AbsoluteValue R ℝ
hna : IsNonarchimedean ⇑v
hc : 0 < c
p q : R[X]
hc0 : 0 ≤ c
i : ℕ
hi_p : gaussNorm v c p = v (p.coeff i) * c ^ i
hlt_p : ∀ j < i, v (p.coeff j) * c ^ j < gaussNorm v c p
j : ℕ
hj_q : gaussNorm v c q = v (q.coeff j) * c ^ j
hlt_q : ∀ j_1 < j, v (q.coeff j_1) * c ^ j_1 < gaussNorm v c q
hvpq : v (p.coeff i) ≠ 0 ∧ v (q.coeff j) ≠ 0
this✝¹ : v (p.coeff i) ≠ 0
this✝ : v (q.coeff j) ≠ 0
x : ℕ
hx : x ∈ range (i + j).succ
hneq : x ≠ i
this : x + (i + j - x) = i + j
h✝ : x < i
⊢ v (p.coeff x) * c ^ x * (v (q.coeff j) * c ^ j) = v (p.coeff x) * c ^ x * (v (q.coeff j) * c ^ j)

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Polynomial.Order

Goal state before:
pp: P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
hroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x
⊢ 0 ≤ P.leadingCoeff * (↑↑(↑P.natDegree).negOnePow * ↑↑(↑P.natDegree).negOnePow)

Tactic:
]

Goal state after:
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
hroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x
⊢ 0 ≤ P.leadingCoeff * (↑↑(↑P.natDegree).negOnePow * ↑↑(↑P.natDegree).negOnePow)
Module: Mathlib.Analysis.Polynomial.Order

Goal state before:
pp: P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
hroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x
hlc' : 0 ≤ ↑↑1 * (P.comp (-X)).leadingCoeff
h : Even P.natDegree
⊢ 0 < ↑↑1 * eval x P

pp: case hn
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
hroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x
hlc' : 0 ≤ ↑↑1 * (P.comp (-X)).leadingCoeff
h : Even P.natDegree
⊢ Even ↑P.natDegree

pp: case hn
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
hroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x
hlc' : 0 ≤ ↑↑(↑(P.comp (-X)).natDegree).negOnePow * (P.comp (-X)).leadingCoeff
h : Even P.natDegree
⊢ Even ↑(P.comp (-X)).natDegree

Tactic:
]

Goal state after:
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
hroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x
hlc' : 0 ≤ ↑↑1 * (P.comp (-X)).leadingCoeff
h : Even P.natDegree
⊢ 0 < ↑↑1 * eval x P

case hn
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
hroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x
hlc' : 0 ≤ ↑↑1 * (P.comp (-X)).leadingCoeff
h : Even P.natDegree
⊢ Even ↑P.natDegree

case hn
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
hroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x
hlc' : 0 ≤ ↑↑(↑(P.comp (-X)).natDegree).negOnePow * (P.comp (-X)).leadingCoeff
h : Even P.natDegree
⊢ Even ↑(P.comp (-X)).natDegree
Module: Mathlib.Analysis.Polynomial.Order

Goal state before:
pp: P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
hroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x
h : Even P.natDegree
hlc' : 0 ≤ (P.comp (-X)).leadingCoeff
⊢ 0 < eval x P

Tactic:
]

Goal state after:
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
hroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x
h : Even P.natDegree
hlc' : 0 ≤ (P.comp (-X)).leadingCoeff
⊢ 0 < eval x P
Module: Mathlib.Analysis.Polynomial.Order

Goal state before:
pp: P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
hroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x
hlc' : 0 ≤ ↑↑(-1) * (P.comp (-X)).leadingCoeff
h : Odd P.natDegree
⊢ 0 < ↑↑(-1) * eval x P

pp: case hn
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
hroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x
hlc' : 0 ≤ ↑↑(-1) * (P.comp (-X)).leadingCoeff
h : Odd P.natDegree
⊢ Odd ↑P.natDegree

pp: case hn
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
hroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x
hlc' : 0 ≤ ↑↑(↑(P.comp (-X)).natDegree).negOnePow * (P.comp (-X)).leadingCoeff
h : Odd P.natDegree
⊢ Odd ↑(P.comp (-X)).natDegree

Tactic:
]

Goal state after:
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
hroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x
hlc' : 0 ≤ ↑↑(-1) * (P.comp (-X)).leadingCoeff
h : Odd P.natDegree
⊢ 0 < ↑↑(-1) * eval x P

case hn
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
hroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x
hlc' : 0 ≤ ↑↑(-1) * (P.comp (-X)).leadingCoeff
h : Odd P.natDegree
⊢ Odd ↑P.natDegree

case hn
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
hroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x
hlc' : 0 ≤ ↑↑(↑(P.comp (-X)).natDegree).negOnePow * (P.comp (-X)).leadingCoeff
h : Odd P.natDegree
⊢ Odd ↑(P.comp (-X)).natDegree
Module: Mathlib.RingTheory.MvPowerSeries.GaussNorm

Goal state before:
pp: case refine_1
R : Type u_1
σ : Type u_2
v : R → ℝ
c : σ → ℝ
inst✝ : Semiring R
f g : MvPowerSeries σ R
hc : 0 ≤ c
vNonneg : ∀ (a : R), v a ≥ 0
vMul : ∀ (a b : R), v (a * b) ≤ v a * v b
vna : IsNonarchimedean v
vZero : v 0 = 0
hbfd : HasGaussNorm v c f
hbgd : HasGaussNorm v c g
t : σ →₀ ℕ
k : (σ →₀ ℕ) × (σ →₀ ℕ)
hk : (Finset.antidiagonal t).Nonempty → k ∈ Finset.antidiagonal t
hsum : v (∑ a ∈ Finset.antidiagonal t, (coeff a.1) f * (coeff a.2) g) ≤ v ((coeff k.1) f * (coeff k.2) g)
hk' : k.1 + k.2 = t
hprod : (t.prod fun x1 x2 ↦ c x1 ^ x2) = (k.1.prod fun x1 x2 ↦ c x1 ^ x2) * k.2.prod fun x1 x2 ↦ c x1 ^ x2
⊢ v ((coeff t) (f * g)) * ((k.1.prod fun x1 x2 ↦ c x1 ^ x2) * k.2.prod fun x1 x2 ↦ c x1 ^ x2) ≤
    gaussNorm v c f * gaussNorm v c g

Tactic:
]

Goal state after:
case refine_1
R : Type u_1
σ : Type u_2
v : R → ℝ
c : σ → ℝ
inst✝ : Semiring R
f g : MvPowerSeries σ R
hc : 0 ≤ c
vNonneg : ∀ (a : R), v a ≥ 0
vMul : ∀ (a b : R), v (a * b) ≤ v a * v b
vna : IsNonarchimedean v
vZero : v 0 = 0
hbfd : HasGaussNorm v c f
hbgd : HasGaussNorm v c g
t : σ →₀ ℕ
k : (σ →₀ ℕ) × (σ →₀ ℕ)
hk : (Finset.antidiagonal t).Nonempty → k ∈ Finset.antidiagonal t
hsum : v (∑ a ∈ Finset.antidiagonal t, (coeff a.1) f * (coeff a.2) g) ≤ v ((coeff k.1) f * (coeff k.2) g)
hk' : k.1 + k.2 = t
hprod : (t.prod fun x1 x2 ↦ c x1 ^ x2) = (k.1.prod fun x1 x2 ↦ c x1 ^ x2) * k.2.prod fun x1 x2 ↦ c x1 ^ x2
⊢ v ((coeff t) (f * g)) * ((k.1.prod fun x1 x2 ↦ c x1 ^ x2) * k.2.prod fun x1 x2 ↦ c x1 ^ x2) ≤
    gaussNorm v c f * gaussNorm v c g
Module: Mathlib.Analysis.Polynomial.Order

Goal state before:
pp: P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
hroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x
h : Odd P.natDegree
hlc' : 0 ≤ -(P.comp (-X)).leadingCoeff
⊢ 0 < -eval x P

Tactic:
]

Goal state after:
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : 0 ≤ P.leadingCoeff
hroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x
h : Odd P.natDegree
hlc' : 0 ≤ -(P.comp (-X)).leadingCoeff
⊢ 0 < -eval x P
Module: Mathlib.Analysis.Polynomial.MahlerMeasure

Goal state before:
pp: p : ℂ[X]
⊢ p.logMahlerMeasure = log (if p ≠ 0 then rexp p.logMahlerMeasure else 0)

Tactic:
]

Goal state after:
p : ℂ[X]
⊢ p.logMahlerMeasure = log (if p ≠ 0 then rexp p.logMahlerMeasure else 0)
Module: Mathlib.RingTheory.Polynomial.GaussNorm

Goal state before:
pp: c✝ : ℝ
R✝ : Type u_3
inst✝¹ : Ring R✝
v✝ : AbsoluteValue R✝ ℝ
c : ℝ
R : Type u_3
inst✝ : Ring R
v : AbsoluteValue R ℝ
hna : IsNonarchimedean ⇑v
hc : 0 < c
p q : R[X]
hc0 : 0 ≤ c
i : ℕ
hi_p : gaussNorm v c p = v (p.coeff i) * c ^ i
hlt_p : ∀ j < i, v (p.coeff j) * c ^ j < gaussNorm v c p
j : ℕ
hj_q : gaussNorm v c q = v (q.coeff j) * c ^ j
hlt_q : ∀ j_1 < j, v (q.coeff j_1) * c ^ j_1 < gaussNorm v c q
hvpq : v (p.coeff i) ≠ 0 ∧ v (q.coeff j) ≠ 0
this✝¹ : v (p.coeff i) ≠ 0
this✝ : v (q.coeff j) ≠ 0
x : ℕ
hx : x ∈ range (i + j).succ
hneq : x ≠ i
this : x + (i + j - x) = i + j
h✝ : i < x
⊢ v (p.coeff i) * c ^ i * (v (q.coeff (i + j - x)) * c ^ (i + j - x)) =
    v (p.coeff i) * c ^ i * (v (q.coeff (i + j - x)) * c ^ (i + j - x))

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Polynomial.MahlerMeasure

Goal state before:
pp: p : ℂ[X]
⊢ CircleIntegrable (fun z ↦ log ‖eval z p‖) 0 1

Tactic:
]

Goal state after:
p : ℂ[X]
⊢ CircleIntegrable (fun z ↦ log ‖eval z p‖) 0 1
Module: Mathlib.RingTheory.MvPowerSeries.GaussNorm

Goal state before:
pp: R : Type u_1
σ : Type u_2
v : R → ℝ
c : σ → ℝ
inst✝ : Semiring R
f g : MvPowerSeries σ R
hc : 0 ≤ c
vNonneg : ∀ (a : R), v a ≥ 0
vMul : ∀ (a b : R), v (a * b) ≤ v a * v b
vna : IsNonarchimedean v
vZero : v 0 = 0
hbfd : HasGaussNorm v c f
hbgd : HasGaussNorm v c g
t : σ →₀ ℕ
k : (σ →₀ ℕ) × (σ →₀ ℕ)
hk : (Finset.antidiagonal t).Nonempty → k ∈ Finset.antidiagonal t
hsum : v (∑ a ∈ Finset.antidiagonal t, (coeff a.1) f * (coeff a.2) g) ≤ v ((coeff k.1) f * (coeff k.2) g)
hk' : k.1 + k.2 = t
hprod : (t.prod fun x1 x2 ↦ c x1 ^ x2) = (k.1.prod fun x1 x2 ↦ c x1 ^ x2) * k.2.prod fun x1 x2 ↦ c x1 ^ x2
⊢ v ((coeff k.1) f) * v ((coeff k.2) g) * ((k.1.prod fun x1 x2 ↦ c x1 ^ x2) * k.2.prod fun x1 x2 ↦ c x1 ^ x2) =
    (v ((coeff k.1) f) * k.1.prod fun x1 x2 ↦ c x1 ^ x2) * (v ((coeff k.2) g) * k.2.prod fun x1 x2 ↦ c x1 ^ x2)

Tactic:
by ring

Goal state after:
Module: Mathlib.Analysis.Polynomial.MahlerMeasure

Goal state before:
pp: case neg
p q : ℂ[X]
hpq : ¬p = 0 ∧ ¬q = 0
⊢ (p * q).mahlerMeasure = p.mahlerMeasure * q.mahlerMeasure

Tactic:
]

Goal state after:
case neg
p q : ℂ[X]
hpq : ¬p = 0 ∧ ¬q = 0
⊢ (p * q).mahlerMeasure = p.mahlerMeasure * q.mahlerMeasure
Module: Mathlib.Analysis.Polynomial.Order

Goal state before:
pp: case pos
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x ≤ y
hlc : 0 ≤ P.leadingCoeff
y : ℝ
hroot : P.IsRoot y
hle : y ≤ x
⊢ 0 ≤ 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Polynomial.MahlerMeasure

Goal state before:
pp: case neg
p q : ℂ[X]
hpq : ¬p = 0 ∧ ¬q = 0
⊢ rexp (π⁻¹ * 2⁻¹ * ∫ (θ : ℝ) in 0..2 * π, log (‖eval (circleMap 0 1 θ) p‖ * ‖eval (circleMap 0 1 θ) q‖)) =
    rexp
      (π⁻¹ * 2⁻¹ *
        ((∫ (θ : ℝ) in 0..2 * π, log ‖eval (circleMap 0 1 θ) p‖) +
          ∫ (θ : ℝ) in 0..2 * π, log ‖eval (circleMap 0 1 θ) q‖))

Tactic:
]

Goal state after:
case neg
p q : ℂ[X]
hpq : ¬p = 0 ∧ ¬q = 0
⊢ rexp (π⁻¹ * 2⁻¹ * ∫ (θ : ℝ) in 0..2 * π, log (‖eval (circleMap 0 1 θ) p‖ * ‖eval (circleMap 0 1 θ) q‖)) =
    rexp
      (π⁻¹ * 2⁻¹ *
        ((∫ (θ : ℝ) in 0..2 * π, log ‖eval (circleMap 0 1 θ) p‖) +
          ∫ (θ : ℝ) in 0..2 * π, log ‖eval (circleMap 0 1 θ) q‖))
Module: Mathlib.Analysis.Polynomial.MahlerMeasure

Goal state before:
pp: case neg.e_a
p q : ℂ[X]
hpq : ¬p = 0 ∧ ¬q = 0
⊢ ∫ (θ : ℝ) in 0..2 * π, log (‖eval (circleMap 0 1 θ) p‖ * ‖eval (circleMap 0 1 θ) q‖) =
    ∫ (x : ℝ) in 0..2 * π, log ‖eval (circleMap 0 1 x) p‖ + log ‖eval (circleMap 0 1 x) q‖

Tactic:
]

Goal state after:
case neg.e_a
p q : ℂ[X]
hpq : ¬p = 0 ∧ ¬q = 0
⊢ ∫ (θ : ℝ) in 0..2 * π, log (‖eval (circleMap 0 1 θ) p‖ * ‖eval (circleMap 0 1 θ) q‖) =
    ∫ (x : ℝ) in 0..2 * π, log ‖eval (circleMap 0 1 x) p‖ + log ‖eval (circleMap 0 1 x) q‖
Module: Mathlib.Analysis.Polynomial.Order

Goal state before:
pp: case refine_1
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : P.leadingCoeff ≤ 0
y : ℝ
hy : (-P).IsRoot y
⊢ eval y (-P) = 0

Tactic:
]

Goal state after:
case refine_1
P : ℝ[X]
x : ℝ
hroots : ∀ (y : ℝ), P.IsRoot y → x < y
hlc : P.leadingCoeff ≤ 0
y : ℝ
hy : (-P).IsRoot y
⊢ eval y (-P) = 0