mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Order.LiminfLimsup

Goal state before:
pp: case inr
ι : Type u_1
R : Type u_4
inst✝⁷ : ConditionallyCompleteLinearOrder R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : OrderTopology R
F : Filter ι
inst✝⁴ : AddCommSemigroup R
inst✝³ : Sub R
inst✝² : ContinuousSub R
inst✝¹ : OrderedSub R
inst✝ : AddLeftMono R
f : ι → R
c : R
cobdd : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) F f
bdd_below : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) F f
h✝ : F.NeBot
⊢ limsup (fun i ↦ c - f i) F = c - liminf f F

Tactic:
·

Goal state after:
case inr
ι : Type u_1
R : Type u_4
inst✝⁷ : ConditionallyCompleteLinearOrder R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : OrderTopology R
F : Filter ι
inst✝⁴ : AddCommSemigroup R
inst✝³ : Sub R
inst✝² : ContinuousSub R
inst✝¹ : OrderedSub R
inst✝ : AddLeftMono R
f : ι → R
c : R
cobdd : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) F f
bdd_below : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) F f
h✝ : F.NeBot
⊢ limsup (fun i ↦ c - f i) F = c - liminf f F
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
hx : f x = 0
⊢ f x / f x ≤ 1

pp: case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
hx : ¬f x = 0
⊢ f x / f x ≤ 1

Tactic:
·

Goal state after:
case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
hx : f x = 0
⊢ f x / f x ≤ 1
Module: Mathlib.Analysis.Normed.Field.Approximation

Goal state before:
pp: case h.refine_1
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : NormedField L
inst✝ : Algebra K L
hd : DenseRange ⇑(algebraMap K L)
f : L[X]
hf : f.Monic
ε : ℝ
hε : ε > 0
h : ¬f.natDegree = 0
c : ℕ → K
hc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε
hdeg : (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i).natDegree = f.natDegree
⊢ (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i).Monic

pp: case h.refine_2
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : NormedField L
inst✝ : Algebra K L
hd : DenseRange ⇑(algebraMap K L)
f : L[X]
hf : f.Monic
ε : ℝ
hε : ε > 0
h : ¬f.natDegree = 0
c : ℕ → K
hc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε
hdeg : (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i).natDegree = f.natDegree
n : ℕ
⊢ ‖(map (algebraMap K L) (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i)).coeff n - f.coeff n‖ < ε

Tactic:
·

Goal state after:
case h.refine_1
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : NormedField L
inst✝ : Algebra K L
hd : DenseRange ⇑(algebraMap K L)
f : L[X]
hf : f.Monic
ε : ℝ
hε : ε > 0
h : ¬f.natDegree = 0
c : ℕ → K
hc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε
hdeg : (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i).natDegree = f.natDegree
⊢ (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i).Monic
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
hx : ¬f x = 0
⊢ f x / f x ≤ 1

Tactic:
·

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
hx : ¬f x = 0
⊢ f x / f x ≤ 1
Module: Mathlib.Topology.Algebra.Order.LiminfLimsup

Goal state before:
pp: case inl
ι : Type u_1
R : Type u_4
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
inst✝³ : AddCommSemigroup R
inst✝² : Sub R
inst✝¹ : ContinuousSub R
inst✝ : OrderedSub R
f : ι → R
c : R
bdd_above : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) ⊥ f
cobdd : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) ⊥ f
⊢ limsup (fun i ↦ f i - c) ⊥ = limsup f ⊥ - c

pp: case inr
ι : Type u_1
R : Type u_4
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
F : Filter ι
inst✝³ : AddCommSemigroup R
inst✝² : Sub R
inst✝¹ : ContinuousSub R
inst✝ : OrderedSub R
f : ι → R
c : R
bdd_above : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
cobdd : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
h✝ : F.NeBot
⊢ limsup (fun i ↦ f i - c) F = limsup f F - c

Tactic:
·

Goal state after:
case inl
ι : Type u_1
R : Type u_4
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
inst✝³ : AddCommSemigroup R
inst✝² : Sub R
inst✝¹ : ContinuousSub R
inst✝ : OrderedSub R
f : ι → R
c : R
bdd_above : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) ⊥ f
cobdd : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) ⊥ f
⊢ limsup (fun i ↦ f i - c) ⊥ = limsup f ⊥ - c
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case a
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
⊢ 1 ≤ ⨆ y, f y / f y

Tactic:
·

Goal state after:
case a
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
⊢ 1 ≤ ⨆ y, f y / f y
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
y : R
hy : f y = 0
⊢ (fun y ↦ f y / f y) y ≤ 1

pp: case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
y : R
hy : ¬f y = 0
⊢ (fun y ↦ f y / f y) y ≤ 1

Tactic:
·

Goal state after:
case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
y : R
hy : f y = 0
⊢ (fun y ↦ f y / f y) y ≤ 1
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
y : R
hy : ¬f y = 0
⊢ (fun y ↦ f y / f y) y ≤ 1

Tactic:
·

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
y : R
hy : ¬f y = 0
⊢ (fun y ↦ f y / f y) y ≤ 1
Module: Mathlib.Topology.Algebra.Order.LiminfLimsup

Goal state before:
pp: case inl
ι : Type u_1
R : Type u_4
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
inst✝³ : AddCommSemigroup R
inst✝² : Sub R
inst✝¹ : ContinuousSub R
inst✝ : OrderedSub R
f : ι → R
c : R
bdd_above : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) ⊥ f
cobdd : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) ⊥ f
⊢ limsup (fun i ↦ f i - c) ⊥ = limsup f ⊥ - c

Tactic:
have {a : R} : sInf Set.univ ≤ a := by
      apply csInf_le _ (Set.mem_univ a)
      simp only [IsCoboundedUnder, IsCobounded, map_bot, eventually_bot, true_implies] at cobdd
      rcases cobdd with ⟨x, hx⟩
      refine ⟨x, mem_lowerBounds.2 fun y ↦ ?_⟩
      simp only [Set.mem_univ, hx y, implies_true]
    simp only [limsup, limsSup, map_bot, eventually_bot, Set.ofPred_true]
    exact this.antisymm (tsub_le_iff_right.2 this)

Goal state after:
Module: Mathlib.Topology.Algebra.Order.LiminfLimsup

Goal state before:
pp: case inr
ι : Type u_1
R : Type u_4
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
F : Filter ι
inst✝³ : AddCommSemigroup R
inst✝² : Sub R
inst✝¹ : ContinuousSub R
inst✝ : OrderedSub R
f : ι → R
c : R
bdd_above : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
cobdd : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
h✝ : F.NeBot
⊢ limsup (fun i ↦ f i - c) F = limsup f F - c

Tactic:
·

Goal state after:
case inr
ι : Type u_1
R : Type u_4
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
F : Filter ι
inst✝³ : AddCommSemigroup R
inst✝² : Sub R
inst✝¹ : ContinuousSub R
inst✝ : OrderedSub R
f : ι → R
c : R
bdd_above : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
cobdd : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
h✝ : F.NeBot
⊢ limsup (fun i ↦ f i - c) F = limsup f F - c
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
f_ne_zero : f ≠ 0
⊢ seminormFromBounded' f 1 ≤ 1

pp: case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
f_ne_zero : f = 0
⊢ seminormFromBounded' f 1 ≤ 1

Tactic:
·

Goal state after:
case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
f_ne_zero : f ≠ 0
⊢ seminormFromBounded' f 1 ≤ 1
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
f_ne_zero : f = 0
⊢ seminormFromBounded' f 1 ≤ 1

Tactic:
·

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
f_ne_zero : f = 0
⊢ seminormFromBounded' f 1 ≤ 1
Module: Mathlib.Topology.Algebra.Order.LiminfLimsup

Goal state before:
pp: ι : Type u_1
R : Type u_4
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
F : Filter ι
inst✝³ : AddCommSemigroup R
inst✝² : Sub R
inst✝¹ : ContinuousSub R
inst✝ : OrderedSub R
f : ι → R
c : R
bdd_above : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
cobdd : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
h✝ : F.NeBot
⊢ Monotone fun x ↦ x - c

pp: ι : Type u_1
R : Type u_4
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
F : Filter ι
inst✝³ : AddCommSemigroup R
inst✝² : Sub R
inst✝¹ : ContinuousSub R
inst✝ : OrderedSub R
f : ι → R
c : R
bdd_above : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
cobdd : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
h✝ : F.NeBot
⊢ ContinuousAt (fun x ↦ x - c) (map f F).limsSup

Tactic:
·

Goal state after:
ι : Type u_1
R : Type u_4
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
F : Filter ι
inst✝³ : AddCommSemigroup R
inst✝² : Sub R
inst✝¹ : ContinuousSub R
inst✝ : OrderedSub R
f : ι → R
c : R
bdd_above : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
cobdd : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
h✝ : F.NeBot
⊢ Monotone fun x ↦ x - c
Module: Mathlib.Topology.Algebra.Order.LiminfLimsup

Goal state before:
pp: ι : Type u_1
R : Type u_4
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
F : Filter ι
inst✝³ : AddCommSemigroup R
inst✝² : Sub R
inst✝¹ : ContinuousSub R
inst✝ : OrderedSub R
f : ι → R
c : R
bdd_above : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
cobdd : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
h✝ : F.NeBot
⊢ ContinuousAt (fun x ↦ x - c) (map f F).limsSup

Tactic:
·

Goal state after:
ι : Type u_1
R : Type u_4
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
F : Filter ι
inst✝³ : AddCommSemigroup R
inst✝² : Sub R
inst✝¹ : ContinuousSub R
inst✝ : OrderedSub R
f : ι → R
c : R
bdd_above : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
cobdd : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
h✝ : F.NeBot
⊢ ContinuousAt (fun x ↦ x - c) (map f F).limsSup
Module: Mathlib.Analysis.Normed.Field.Approximation

Goal state before:
pp: case h.refine_2
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : NormedField L
inst✝ : Algebra K L
hd : DenseRange ⇑(algebraMap K L)
f : L[X]
hf : f.Monic
ε : ℝ
hε : ε > 0
h : ¬f.natDegree = 0
c : ℕ → K
hc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε
hdeg : (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i).natDegree = f.natDegree
n : ℕ
⊢ ‖(map (algebraMap K L) (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i)).coeff n - f.coeff n‖ < ε

Tactic:
·

Goal state after:
case h.refine_2
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : NormedField L
inst✝ : Algebra K L
hd : DenseRange ⇑(algebraMap K L)
f : L[X]
hf : f.Monic
ε : ℝ
hε : ε > 0
h : ¬f.natDegree = 0
c : ℕ → K
hc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε
hdeg : (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i).natDegree = f.natDegree
n : ℕ
⊢ ‖(map (algebraMap K L) (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i)).coeff n - f.coeff n‖ < ε
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
f_add : ∀ (a b : R), f (a + b) ≤ f a + f b
x y z : R
hz : f z = 0
⊢ f ((x + y) * z) / f z ≤ f (x * z) / f z + f (y * z) / f z

pp: case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
f_add : ∀ (a b : R), f (a + b) ≤ f a + f b
x y z : R
hz : ¬f z = 0
⊢ f ((x + y) * z) / f z ≤ f (x * z) / f z + f (y * z) / f z

Tactic:
·

Goal state after:
case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
f_add : ∀ (a b : R), f (a + b) ≤ f a + f b
x y z : R
hz : f z = 0
⊢ f ((x + y) * z) / f z ≤ f (x * z) / f z + f (y * z) / f z
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
f_add : ∀ (a b : R), f (a + b) ≤ f a + f b
x y z : R
hz : ¬f z = 0
⊢ f ((x + y) * z) / f z ≤ f (x * z) / f z + f (y * z) / f z

Tactic:
·

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
f_add : ∀ (a b : R), f (a + b) ≤ f a + f b
x y z : R
hz : ¬f z = 0
⊢ f ((x + y) * z) / f z ≤ f (x * z) / f z + f (y * z) / f z
Module: Mathlib.Analysis.Normed.Field.Approximation

Goal state before:
pp: case h.refine_2.inl
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : NormedField L
inst✝ : Algebra K L
hd : DenseRange ⇑(algebraMap K L)
f : L[X]
hf : f.Monic
ε : ℝ
hε : ε > 0
h✝ : ¬f.natDegree = 0
c : ℕ → K
hc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε
hdeg : (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i).natDegree = f.natDegree
n : ℕ
h : n < f.natDegree
⊢ ‖(map (algebraMap K L) (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i)).coeff n - f.coeff n‖ < ε

pp: case h.refine_2.inr.inl
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : NormedField L
inst✝ : Algebra K L
hd : DenseRange ⇑(algebraMap K L)
f : L[X]
hf : f.Monic
ε : ℝ
hε : ε > 0
h✝ : ¬f.natDegree = 0
c : ℕ → K
hc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε
hdeg : (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i).natDegree = f.natDegree
n : ℕ
h : n = f.natDegree
⊢ ‖(map (algebraMap K L) (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i)).coeff n - f.coeff n‖ < ε

pp: case h.refine_2.inr.inr
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : NormedField L
inst✝ : Algebra K L
hd : DenseRange ⇑(algebraMap K L)
f : L[X]
hf : f.Monic
ε : ℝ
hε : ε > 0
h✝ : ¬f.natDegree = 0
c : ℕ → K
hc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε
hdeg : (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i).natDegree = f.natDegree
n : ℕ
h : f.natDegree < n
⊢ ‖(map (algebraMap K L) (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i)).coeff n - f.coeff n‖ < ε

Tactic:
·

Goal state after:
case h.refine_2.inl
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : NormedField L
inst✝ : Algebra K L
hd : DenseRange ⇑(algebraMap K L)
f : L[X]
hf : f.Monic
ε : ℝ
hε : ε > 0
h✝ : ¬f.natDegree = 0
c : ℕ → K
hc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε
hdeg : (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i).natDegree = f.natDegree
n : ℕ
h : n < f.natDegree
⊢ ‖(map (algebraMap K L) (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i)).coeff n - f.coeff n‖ < ε
Module: Mathlib.Analysis.Normed.Unbundled.FiniteExtension

Goal state before:
pp: case a
K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
x : K
⊢ (univ.sup' ⋯ fun x_1 ↦ ‖if i = x_1 then x else 0‖) ≤ ‖x‖

pp: case a
K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
x : K
⊢ ‖x‖ ≤ univ.sup' ⋯ fun x_1 ↦ ‖if i = x_1 then x else 0‖

Tactic:
·

Goal state after:
case a
K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
x : K
⊢ (univ.sup' ⋯ fun x_1 ↦ ‖if i = x_1 then x else 0‖) ≤ ‖x‖
Module: Mathlib.Analysis.Normed.Unbundled.FiniteExtension

Goal state before:
pp: case a
K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
x : K
⊢ ‖x‖ ≤ univ.sup' ⋯ fun x_1 ↦ ‖if i = x_1 then x else 0‖

Tactic:
·

Goal state after:
case a
K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
x : K
⊢ ‖x‖ ≤ univ.sup' ⋯ fun x_1 ↦ ‖if i = x_1 then x else 0‖
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case inl
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
hna : IsNonarchimedean f
x y z : R
hfx : f ((x + y) * z) / f z ≤ f (x * z) / f z
⊢ f ((x + y) * z) / f z ≤ seminormFromBounded' f x ∨ f ((x + y) * z) / f z ≤ seminormFromBounded' f y

pp: case inr
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
hna : IsNonarchimedean f
x y z : R
hfy : f ((x + y) * z) / f z ≤ f (y * z) / f z
⊢ f ((x + y) * z) / f z ≤ seminormFromBounded' f x ∨ f ((x + y) * z) / f z ≤ seminormFromBounded' f y

Tactic:
·

Goal state after:
case inl
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
hna : IsNonarchimedean f
x y z : R
hfx : f ((x + y) * z) / f z ≤ f (x * z) / f z
⊢ f ((x + y) * z) / f z ≤ seminormFromBounded' f x ∨ f ((x + y) * z) / f z ≤ seminormFromBounded' f y
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case inr
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
hna : IsNonarchimedean f
x y z : R
hfy : f ((x + y) * z) / f z ≤ f (y * z) / f z
⊢ f ((x + y) * z) / f z ≤ seminormFromBounded' f x ∨ f ((x + y) * z) / f z ≤ seminormFromBounded' f y

Tactic:
·

Goal state after:
case inr
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
hna : IsNonarchimedean f
x y z : R
hfy : f ((x + y) * z) / f z ≤ f (y * z) / f z
⊢ f ((x + y) * z) / f z ≤ seminormFromBounded' f x ∨ f ((x + y) * z) / f z ≤ seminormFromBounded' f y
Module: Mathlib.Analysis.Normed.Field.Approximation

Goal state before:
pp: case h.refine_2.inr.inl
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : NormedField L
inst✝ : Algebra K L
hd : DenseRange ⇑(algebraMap K L)
f : L[X]
hf : f.Monic
ε : ℝ
hε : ε > 0
h✝ : ¬f.natDegree = 0
c : ℕ → K
hc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε
hdeg : (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i).natDegree = f.natDegree
n : ℕ
h : n = f.natDegree
⊢ ‖(map (algebraMap K L) (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i)).coeff n - f.coeff n‖ < ε

pp: case h.refine_2.inr.inr
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : NormedField L
inst✝ : Algebra K L
hd : DenseRange ⇑(algebraMap K L)
f : L[X]
hf : f.Monic
ε : ℝ
hε : ε > 0
h✝ : ¬f.natDegree = 0
c : ℕ → K
hc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε
hdeg : (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i).natDegree = f.natDegree
n : ℕ
h : f.natDegree < n
⊢ ‖(map (algebraMap K L) (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i)).coeff n - f.coeff n‖ < ε

Tactic:
·

Goal state after:
case h.refine_2.inr.inl
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : NormedField L
inst✝ : Algebra K L
hd : DenseRange ⇑(algebraMap K L)
f : L[X]
hf : f.Monic
ε : ℝ
hε : ε > 0
h✝ : ¬f.natDegree = 0
c : ℕ → K
hc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε
hdeg : (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i).natDegree = f.natDegree
n : ℕ
h : n = f.natDegree
⊢ ‖(map (algebraMap K L) (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i)).coeff n - f.coeff n‖ < ε
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
hna : IsNonarchimedean f
x y z : R
hz : f z = 0
⊢ f ((x + y) * z) / f z ≤ f (x * z) / f z ∨ f ((x + y) * z) / f z ≤ f (y * z) / f z

pp: case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
hna : IsNonarchimedean f
x y z : R
hz : ¬f z = 0
⊢ f ((x + y) * z) / f z ≤ f (x * z) / f z ∨ f ((x + y) * z) / f z ≤ f (y * z) / f z

Tactic:
·

Goal state after:
case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
hna : IsNonarchimedean f
x y z : R
hz : f z = 0
⊢ f ((x + y) * z) / f z ≤ f (x * z) / f z ∨ f ((x + y) * z) / f z ≤ f (y * z) / f z
Module: Mathlib.Analysis.Normed.Field.Approximation

Goal state before:
pp: case h.refine_2.inr.inr
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : NormedField L
inst✝ : Algebra K L
hd : DenseRange ⇑(algebraMap K L)
f : L[X]
hf : f.Monic
ε : ℝ
hε : ε > 0
h✝ : ¬f.natDegree = 0
c : ℕ → K
hc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε
hdeg : (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i).natDegree = f.natDegree
n : ℕ
h : f.natDegree < n
⊢ ‖(map (algebraMap K L) (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i)).coeff n - f.coeff n‖ < ε

Tactic:
·

Goal state after:
case h.refine_2.inr.inr
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : NormedField L
inst✝ : Algebra K L
hd : DenseRange ⇑(algebraMap K L)
f : L[X]
hf : f.Monic
ε : ℝ
hε : ε > 0
h✝ : ¬f.natDegree = 0
c : ℕ → K
hc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε
hdeg : (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i).natDegree = f.natDegree
n : ℕ
h : f.natDegree < n
⊢ ‖(map (algebraMap K L) (C 1 * X ^ f.natDegree + ∑ i < f.natDegree, C (c i) * X ^ i)).coeff n - f.coeff n‖ < ε
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
hna : IsNonarchimedean f
x y z : R
hz : ¬f z = 0
⊢ f ((x + y) * z) / f z ≤ f (x * z) / f z ∨ f ((x + y) * z) / f z ≤ f (y * z) / f z

Tactic:
·

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
hna : IsNonarchimedean f
x y z : R
hz : ¬f z = 0
⊢ f ((x + y) * z) / f z ≤ f (x * z) / f z ∨ f ((x + y) * z) / f z ≤ f (y * z) / f z