state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
α : Type u_2
β✝ : Type ?u.24150
inst✝⁴ : LinearOrderedField α
inst✝³ : Ring β✝
abv✝ : β✝ → α
inst✝² : IsAbsoluteValue abv✝
β : Type u_1
inst✝¹ : DivisionRing β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε K : α
ε0 : 0 < ε
K0 : 0 < K
a b : β
ha : K ≤ abv a
hb : K ≤ abv b
h : abv (a - b) < K * ε * K
a0 : 0 < abv a
b0 : 0 < ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | refine' h.trans_le _ | theorem rat_inv_continuous_lemma {β : Type*} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv]
{ε K : α} (ε0 : 0 < ε) (K0 : 0 < K) :
∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε := by
refine' ⟨K * ε * K, mul_pos (mul_pos K0 ε0) K0, fun {a b} ha hb h => _⟩
have a0 :=... | Mathlib.Data.Real.CauSeq.76_0.b72JnnMmhSc5wVP | theorem rat_inv_continuous_lemma {β : Type*} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv]
{ε K : α} (ε0 : 0 < ε) (K0 : 0 < K) :
∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_2
β✝ : Type ?u.24150
inst✝⁴ : LinearOrderedField α
inst✝³ : Ring β✝
abv✝ : β✝ → α
inst✝² : IsAbsoluteValue abv✝
β : Type u_1
inst✝¹ : DivisionRing β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε K : α
ε0 : 0 < ε
K0 : 0 < K
a b : β
ha : K ≤ abv a
hb : K ≤ abv b
h : abv (a - b) < K * ε * K
a0 : 0 < abv a
b0 : 0 < ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | gcongr | theorem rat_inv_continuous_lemma {β : Type*} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv]
{ε K : α} (ε0 : 0 < ε) (K0 : 0 < K) :
∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε := by
refine' ⟨K * ε * K, mul_pos (mul_pos K0 ε0) K0, fun {a b} ha hb h => _⟩
have a0 :=... | Mathlib.Data.Real.CauSeq.76_0.b72JnnMmhSc5wVP | theorem rat_inv_continuous_lemma {β : Type*} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv]
{ε K : α} (ε0 : 0 < ε) (K0 : 0 < K) :
∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : ℕ → β
hf : IsCauSeq abv f
ε : α
ε0 : 0 < ε
⊢ ∃ i, ∀ j ≥ i, ∀ k ≥ i, abv (f j - f k) < ε | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | refine' (hf _ (half_pos ε0)).imp fun i hi j ij k ik => _ | theorem cauchy₂ (hf : IsCauSeq abv f) {ε : α} (ε0 : 0 < ε) :
∃ i, ∀ j ≥ i, ∀ k ≥ i, abv (f j - f k) < ε := by
| Mathlib.Data.Real.CauSeq.104_0.b72JnnMmhSc5wVP | theorem cauchy₂ (hf : IsCauSeq abv f) {ε : α} (ε0 : 0 < ε) :
∃ i, ∀ j ≥ i, ∀ k ≥ i, abv (f j - f k) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : ℕ → β
hf : IsCauSeq abv f
ε : α
ε0 : 0 < ε
i : ℕ
hi : ∀ j ≥ i, abv (f j - f i) < ε / 2
j : ℕ
ij : j ≥ i
k : ℕ
ik : k ≥ i
⊢ abv (f j - f k) < ε | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rw [← add_halves ε] | theorem cauchy₂ (hf : IsCauSeq abv f) {ε : α} (ε0 : 0 < ε) :
∃ i, ∀ j ≥ i, ∀ k ≥ i, abv (f j - f k) < ε := by
refine' (hf _ (half_pos ε0)).imp fun i hi j ij k ik => _
| Mathlib.Data.Real.CauSeq.104_0.b72JnnMmhSc5wVP | theorem cauchy₂ (hf : IsCauSeq abv f) {ε : α} (ε0 : 0 < ε) :
∃ i, ∀ j ≥ i, ∀ k ≥ i, abv (f j - f k) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : ℕ → β
hf : IsCauSeq abv f
ε : α
ε0 : 0 < ε
i : ℕ
hi : ∀ j ≥ i, abv (f j - f i) < ε / 2
j : ℕ
ij : j ≥ i
k : ℕ
ik : k ≥ i
⊢ abv (f j - f k) < ε / 2 + ε / 2 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | refine' lt_of_le_of_lt (abv_sub_le abv _ _ _) (add_lt_add (hi _ ij) _) | theorem cauchy₂ (hf : IsCauSeq abv f) {ε : α} (ε0 : 0 < ε) :
∃ i, ∀ j ≥ i, ∀ k ≥ i, abv (f j - f k) < ε := by
refine' (hf _ (half_pos ε0)).imp fun i hi j ij k ik => _
rw [← add_halves ε]
| Mathlib.Data.Real.CauSeq.104_0.b72JnnMmhSc5wVP | theorem cauchy₂ (hf : IsCauSeq abv f) {ε : α} (ε0 : 0 < ε) :
∃ i, ∀ j ≥ i, ∀ k ≥ i, abv (f j - f k) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : ℕ → β
hf : IsCauSeq abv f
ε : α
ε0 : 0 < ε
i : ℕ
hi : ∀ j ≥ i, abv (f j - f i) < ε / 2
j : ℕ
ij : j ≥ i
k : ℕ
ik : k ≥ i
⊢ abv (f i - f k) < ε / 2 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rw [abv_sub abv] | theorem cauchy₂ (hf : IsCauSeq abv f) {ε : α} (ε0 : 0 < ε) :
∃ i, ∀ j ≥ i, ∀ k ≥ i, abv (f j - f k) < ε := by
refine' (hf _ (half_pos ε0)).imp fun i hi j ij k ik => _
rw [← add_halves ε]
refine' lt_of_le_of_lt (abv_sub_le abv _ _ _) (add_lt_add (hi _ ij) _)
| Mathlib.Data.Real.CauSeq.104_0.b72JnnMmhSc5wVP | theorem cauchy₂ (hf : IsCauSeq abv f) {ε : α} (ε0 : 0 < ε) :
∃ i, ∀ j ≥ i, ∀ k ≥ i, abv (f j - f k) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : ℕ → β
hf : IsCauSeq abv f
ε : α
ε0 : 0 < ε
i : ℕ
hi : ∀ j ≥ i, abv (f j - f i) < ε / 2
j : ℕ
ij : j ≥ i
k : ℕ
ik : k ≥ i
⊢ abv (f k - f i) < ε / 2 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | exact hi _ ik | theorem cauchy₂ (hf : IsCauSeq abv f) {ε : α} (ε0 : 0 < ε) :
∃ i, ∀ j ≥ i, ∀ k ≥ i, abv (f j - f k) < ε := by
refine' (hf _ (half_pos ε0)).imp fun i hi j ij k ik => _
rw [← add_halves ε]
refine' lt_of_le_of_lt (abv_sub_le abv _ _ _) (add_lt_add (hi _ ij) _)
rw [abv_sub abv]; | Mathlib.Data.Real.CauSeq.104_0.b72JnnMmhSc5wVP | theorem cauchy₂ (hf : IsCauSeq abv f) {ε : α} (ε0 : 0 < ε) :
∃ i, ∀ j ≥ i, ∀ k ≥ i, abv (f j - f k) < ε | Mathlib_Data_Real_CauSeq |
α : Type ?u.42637
β : Type ?u.42640
inst✝¹ : LinearOrderedField α
inst✝ : Ring β
abv : β → α
f : CauSeq β abv
g : ℕ → β
e : ∀ (i : ℕ), ↑f i = g i
ε : α
⊢ ε > 0 → ∃ i, ∀ j ≥ i, abv (g j - g i) < ε | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rw [show g = f from (funext e).symm] | /-- Given a Cauchy sequence `f`, create a Cauchy sequence from a sequence `g` with
the same values as `f`. -/
def ofEq (f : CauSeq β abv) (g : ℕ → β) (e : ∀ i, f i = g i) : CauSeq β abv :=
⟨g, fun ε => by | Mathlib.Data.Real.CauSeq.162_0.b72JnnMmhSc5wVP | /-- Given a Cauchy sequence `f`, create a Cauchy sequence from a sequence `g` with
the same values as `f`. -/
def ofEq (f : CauSeq β abv) (g : ℕ → β) (e : ∀ i, f i = g i) : CauSeq β abv | Mathlib_Data_Real_CauSeq |
α : Type ?u.42637
β : Type ?u.42640
inst✝¹ : LinearOrderedField α
inst✝ : Ring β
abv : β → α
f : CauSeq β abv
g : ℕ → β
e : ∀ (i : ℕ), ↑f i = g i
ε : α
⊢ ε > 0 → ∃ i, ∀ j ≥ i, abv (↑f j - ↑f i) < ε | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | exact f.cauchy | /-- Given a Cauchy sequence `f`, create a Cauchy sequence from a sequence `g` with
the same values as `f`. -/
def ofEq (f : CauSeq β abv) (g : ℕ → β) (e : ∀ i, f i = g i) : CauSeq β abv :=
⟨g, fun ε => by rw [show g = f from (funext e).symm]; | Mathlib.Data.Real.CauSeq.162_0.b72JnnMmhSc5wVP | /-- Given a Cauchy sequence `f`, create a Cauchy sequence from a sequence `g` with
the same values as `f`. -/
def ofEq (f : CauSeq β abv) (g : ℕ → β) (e : ∀ i, f i = g i) : CauSeq β abv | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
⊢ ∃ r, ∀ (i : ℕ), abv (↑f i) < r | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | cases' f.cauchy zero_lt_one with i h | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r := by
| Mathlib.Data.Real.CauSeq.181_0.b72JnnMmhSc5wVP | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r | Mathlib_Data_Real_CauSeq |
case intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
i : ℕ
h : ∀ j ≥ i, abv (↑f j - ↑f i) < 1
⊢ ∃ r, ∀ (i : ℕ), abv (↑f i) < r | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | set R : ℕ → α := @Nat.rec (fun _ => α) (abv (f 0)) fun i c => max c (abv (f i.succ)) with hR | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r := by
cases' f.cauchy zero_lt_one with i h
| Mathlib.Data.Real.CauSeq.181_0.b72JnnMmhSc5wVP | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r | Mathlib_Data_Real_CauSeq |
case intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
i : ℕ
h : ∀ j ≥ i, abv (↑f j - ↑f i) < 1
R : ℕ → α := Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat.succ i)))
hR : R = Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat.suc... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | have : ∀ i, ∀ j ≤ i, abv (f j) ≤ R i := by
refine' Nat.rec (by simp [hR]) _
rintro i hi j (rfl | hj)
· simp
exact (hi j hj).trans (le_max_left _ _) | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r := by
cases' f.cauchy zero_lt_one with i h
set R : ℕ → α := @Nat.rec (fun _ => α) (abv (f 0)) fun i c => max c (abv (f i.succ)) with hR
| Mathlib.Data.Real.CauSeq.181_0.b72JnnMmhSc5wVP | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
i : ℕ
h : ∀ j ≥ i, abv (↑f j - ↑f i) < 1
R : ℕ → α := Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat.succ i)))
hR : R = Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat.succ i)))
⊢ ∀ ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | refine' Nat.rec (by simp [hR]) _ | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r := by
cases' f.cauchy zero_lt_one with i h
set R : ℕ → α := @Nat.rec (fun _ => α) (abv (f 0)) fun i c => max c (abv (f i.succ)) with hR
have : ∀ i, ∀ j ≤ i, abv (f j) ≤ R i := by
| Mathlib.Data.Real.CauSeq.181_0.b72JnnMmhSc5wVP | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
i : ℕ
h : ∀ j ≥ i, abv (↑f j - ↑f i) < 1
R : ℕ → α := Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat.succ i)))
hR : R = Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat.succ i)))
⊢ ∀ ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simp [hR] | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r := by
cases' f.cauchy zero_lt_one with i h
set R : ℕ → α := @Nat.rec (fun _ => α) (abv (f 0)) fun i c => max c (abv (f i.succ)) with hR
have : ∀ i, ∀ j ≤ i, abv (f j) ≤ R i := by
refine' Nat.rec (by | Mathlib.Data.Real.CauSeq.181_0.b72JnnMmhSc5wVP | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
i : ℕ
h : ∀ j ≥ i, abv (↑f j - ↑f i) < 1
R : ℕ → α := Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat.succ i)))
hR : R = Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat.succ i)))
⊢ ∀ ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rintro i hi j (rfl | hj) | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r := by
cases' f.cauchy zero_lt_one with i h
set R : ℕ → α := @Nat.rec (fun _ => α) (abv (f 0)) fun i c => max c (abv (f i.succ)) with hR
have : ∀ i, ∀ j ≤ i, abv (f j) ≤ R i := by
refine' Nat.rec (by simp [hR]) _
| Mathlib.Data.Real.CauSeq.181_0.b72JnnMmhSc5wVP | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r | Mathlib_Data_Real_CauSeq |
case refl
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
i✝ : ℕ
h : ∀ j ≥ i✝, abv (↑f j - ↑f i✝) < 1
R : ℕ → α := Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat.succ i)))
hR : R = Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat.s... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simp | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r := by
cases' f.cauchy zero_lt_one with i h
set R : ℕ → α := @Nat.rec (fun _ => α) (abv (f 0)) fun i c => max c (abv (f i.succ)) with hR
have : ∀ i, ∀ j ≤ i, abv (f j) ≤ R i := by
refine' Nat.rec (by simp [hR]) _
rintro i hi j (rfl | hj)
· | Mathlib.Data.Real.CauSeq.181_0.b72JnnMmhSc5wVP | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r | Mathlib_Data_Real_CauSeq |
case step
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
i✝ : ℕ
h : ∀ j ≥ i✝, abv (↑f j - ↑f i✝) < 1
R : ℕ → α := Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat.succ i)))
hR : R = Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat.s... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | exact (hi j hj).trans (le_max_left _ _) | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r := by
cases' f.cauchy zero_lt_one with i h
set R : ℕ → α := @Nat.rec (fun _ => α) (abv (f 0)) fun i c => max c (abv (f i.succ)) with hR
have : ∀ i, ∀ j ≤ i, abv (f j) ≤ R i := by
refine' Nat.rec (by simp [hR]) _
rintro i hi j (rfl | hj)
· si... | Mathlib.Data.Real.CauSeq.181_0.b72JnnMmhSc5wVP | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r | Mathlib_Data_Real_CauSeq |
case intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
i : ℕ
h : ∀ j ≥ i, abv (↑f j - ↑f i) < 1
R : ℕ → α := Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat.succ i)))
hR : R = Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat.suc... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | refine' ⟨R i + 1, fun j => _⟩ | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r := by
cases' f.cauchy zero_lt_one with i h
set R : ℕ → α := @Nat.rec (fun _ => α) (abv (f 0)) fun i c => max c (abv (f i.succ)) with hR
have : ∀ i, ∀ j ≤ i, abv (f j) ≤ R i := by
refine' Nat.rec (by simp [hR]) _
rintro i hi j (rfl | hj)
· si... | Mathlib.Data.Real.CauSeq.181_0.b72JnnMmhSc5wVP | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r | Mathlib_Data_Real_CauSeq |
case intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
i : ℕ
h : ∀ j ≥ i, abv (↑f j - ↑f i) < 1
R : ℕ → α := Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat.succ i)))
hR : R = Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat.suc... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | cases' lt_or_le j i with ij ij | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r := by
cases' f.cauchy zero_lt_one with i h
set R : ℕ → α := @Nat.rec (fun _ => α) (abv (f 0)) fun i c => max c (abv (f i.succ)) with hR
have : ∀ i, ∀ j ≤ i, abv (f j) ≤ R i := by
refine' Nat.rec (by simp [hR]) _
rintro i hi j (rfl | hj)
· si... | Mathlib.Data.Real.CauSeq.181_0.b72JnnMmhSc5wVP | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r | Mathlib_Data_Real_CauSeq |
case intro.inl
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
i : ℕ
h : ∀ j ≥ i, abv (↑f j - ↑f i) < 1
R : ℕ → α := Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat.succ i)))
hR : R = Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | exact lt_of_le_of_lt (this i _ (le_of_lt ij)) (lt_add_one _) | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r := by
cases' f.cauchy zero_lt_one with i h
set R : ℕ → α := @Nat.rec (fun _ => α) (abv (f 0)) fun i c => max c (abv (f i.succ)) with hR
have : ∀ i, ∀ j ≤ i, abv (f j) ≤ R i := by
refine' Nat.rec (by simp [hR]) _
rintro i hi j (rfl | hj)
· si... | Mathlib.Data.Real.CauSeq.181_0.b72JnnMmhSc5wVP | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r | Mathlib_Data_Real_CauSeq |
case intro.inr
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
i : ℕ
h : ∀ j ≥ i, abv (↑f j - ↑f i) < 1
R : ℕ → α := Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat.succ i)))
hR : R = Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | have := lt_of_le_of_lt (abv_add abv _ _) (add_lt_add_of_le_of_lt (this i _ le_rfl) (h _ ij)) | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r := by
cases' f.cauchy zero_lt_one with i h
set R : ℕ → α := @Nat.rec (fun _ => α) (abv (f 0)) fun i c => max c (abv (f i.succ)) with hR
have : ∀ i, ∀ j ≤ i, abv (f j) ≤ R i := by
refine' Nat.rec (by simp [hR]) _
rintro i hi j (rfl | hj)
· si... | Mathlib.Data.Real.CauSeq.181_0.b72JnnMmhSc5wVP | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r | Mathlib_Data_Real_CauSeq |
case intro.inr
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
i : ℕ
h : ∀ j ≥ i, abv (↑f j - ↑f i) < 1
R : ℕ → α := Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat.succ i)))
hR : R = Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rw [add_sub, add_comm] at this | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r := by
cases' f.cauchy zero_lt_one with i h
set R : ℕ → α := @Nat.rec (fun _ => α) (abv (f 0)) fun i c => max c (abv (f i.succ)) with hR
have : ∀ i, ∀ j ≤ i, abv (f j) ≤ R i := by
refine' Nat.rec (by simp [hR]) _
rintro i hi j (rfl | hj)
· si... | Mathlib.Data.Real.CauSeq.181_0.b72JnnMmhSc5wVP | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r | Mathlib_Data_Real_CauSeq |
case intro.inr
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
i : ℕ
h : ∀ j ≥ i, abv (↑f j - ↑f i) < 1
R : ℕ → α := Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat.succ i)))
hR : R = Nat.rec (abv (↑f 0)) fun i c => max c (abv (↑f (Nat... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa using this | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r := by
cases' f.cauchy zero_lt_one with i h
set R : ℕ → α := @Nat.rec (fun _ => α) (abv (f 0)) fun i c => max c (abv (f i.succ)) with hR
have : ∀ i, ∀ j ≤ i, abv (f j) ≤ R i := by
refine' Nat.rec (by simp [hR]) _
rintro i hi j (rfl | hj)
· si... | Mathlib.Data.Real.CauSeq.181_0.b72JnnMmhSc5wVP | theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r | Mathlib_Data_Real_CauSeq |
α : Type ?u.63580
β : Type ?u.63583
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
x : β
ε : α
ε0 : ε > 0
j : ℕ
x✝ : j ≥ 0
⊢ abv ((fun x_1 => x) j - (fun x_1 => x) 0) < ε | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa [abv_zero] using ε0 | /-- The constant Cauchy sequence. -/
def const (x : β) : CauSeq β abv :=
⟨fun _ => x, fun ε ε0 => ⟨0, fun j _ => by | Mathlib.Data.Real.CauSeq.218_0.b72JnnMmhSc5wVP | /-- The constant Cauchy sequence. -/
def const (x : β) : CauSeq β abv | Mathlib_Data_Real_CauSeq |
α : Type ?u.90076
β : Type ?u.90079
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
i : ℕ
⊢ ↑(const (-1) * f) i = (fun x => -↑f x) i | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simp | instance : Neg (CauSeq β abv) :=
⟨fun f => ofEq (const (-1) * f) (fun x => -f x) fun i => by | Mathlib.Data.Real.CauSeq.310_0.b72JnnMmhSc5wVP | instance : Neg (CauSeq β abv) | Mathlib_Data_Real_CauSeq |
α : Type ?u.93417
β : Type ?u.93420
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
i : ℕ
⊢ ↑(f + -g) i = (fun x => ↑f x - ↑g x) i | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simp [sub_eq_add_neg] | instance : Sub (CauSeq β abv) :=
⟨fun f g => ofEq (f + -g) (fun x => f x - g x) fun i => by | Mathlib.Data.Real.CauSeq.327_0.b72JnnMmhSc5wVP | instance : Sub (CauSeq β abv) | Mathlib_Data_Real_CauSeq |
α : Type ?u.115848
β : Type ?u.115851
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
⊢ ∀ (x : CauSeq β abv) (n : ℕ), ↑(n • x) = n • ↑x | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | intros | instance addGroupWithOne : AddGroupWithOne (CauSeq β abv) :=
Function.Injective.addGroupWithOne Subtype.val Subtype.val_injective rfl rfl
coe_add coe_neg coe_sub
(by | Mathlib.Data.Real.CauSeq.378_0.b72JnnMmhSc5wVP | instance addGroupWithOne : AddGroupWithOne (CauSeq β abv) | Mathlib_Data_Real_CauSeq |
α : Type ?u.115848
β : Type ?u.115851
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
x✝ : CauSeq β abv
n✝ : ℕ
⊢ ↑(n✝ • x✝) = n✝ • ↑x✝ | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rfl | instance addGroupWithOne : AddGroupWithOne (CauSeq β abv) :=
Function.Injective.addGroupWithOne Subtype.val Subtype.val_injective rfl rfl
coe_add coe_neg coe_sub
(by intros; | Mathlib.Data.Real.CauSeq.378_0.b72JnnMmhSc5wVP | instance addGroupWithOne : AddGroupWithOne (CauSeq β abv) | Mathlib_Data_Real_CauSeq |
α : Type ?u.115848
β : Type ?u.115851
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
⊢ ∀ (x : CauSeq β abv) (n : ℤ), ↑(n • x) = n • ↑x | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | intros | instance addGroupWithOne : AddGroupWithOne (CauSeq β abv) :=
Function.Injective.addGroupWithOne Subtype.val Subtype.val_injective rfl rfl
coe_add coe_neg coe_sub
(by intros; rfl)
(by | Mathlib.Data.Real.CauSeq.378_0.b72JnnMmhSc5wVP | instance addGroupWithOne : AddGroupWithOne (CauSeq β abv) | Mathlib_Data_Real_CauSeq |
α : Type ?u.115848
β : Type ?u.115851
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
x✝ : CauSeq β abv
n✝ : ℤ
⊢ ↑(n✝ • x✝) = n✝ • ↑x✝ | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rfl | instance addGroupWithOne : AddGroupWithOne (CauSeq β abv) :=
Function.Injective.addGroupWithOne Subtype.val Subtype.val_injective rfl rfl
coe_add coe_neg coe_sub
(by intros; rfl)
(by intros; | Mathlib.Data.Real.CauSeq.378_0.b72JnnMmhSc5wVP | instance addGroupWithOne : AddGroupWithOne (CauSeq β abv) | Mathlib_Data_Real_CauSeq |
α : Type ?u.115848
β : Type ?u.115851
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
⊢ ∀ (n : ℕ), ↑↑n = ↑n | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | intros | instance addGroupWithOne : AddGroupWithOne (CauSeq β abv) :=
Function.Injective.addGroupWithOne Subtype.val Subtype.val_injective rfl rfl
coe_add coe_neg coe_sub
(by intros; rfl)
(by intros; rfl)
(by | Mathlib.Data.Real.CauSeq.378_0.b72JnnMmhSc5wVP | instance addGroupWithOne : AddGroupWithOne (CauSeq β abv) | Mathlib_Data_Real_CauSeq |
α : Type ?u.115848
β : Type ?u.115851
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
n✝ : ℕ
⊢ ↑↑n✝ = ↑n✝ | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rfl | instance addGroupWithOne : AddGroupWithOne (CauSeq β abv) :=
Function.Injective.addGroupWithOne Subtype.val Subtype.val_injective rfl rfl
coe_add coe_neg coe_sub
(by intros; rfl)
(by intros; rfl)
(by intros; | Mathlib.Data.Real.CauSeq.378_0.b72JnnMmhSc5wVP | instance addGroupWithOne : AddGroupWithOne (CauSeq β abv) | Mathlib_Data_Real_CauSeq |
α : Type ?u.115848
β : Type ?u.115851
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
⊢ ∀ (n : ℤ), ↑↑n = ↑n | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | intros | instance addGroupWithOne : AddGroupWithOne (CauSeq β abv) :=
Function.Injective.addGroupWithOne Subtype.val Subtype.val_injective rfl rfl
coe_add coe_neg coe_sub
(by intros; rfl)
(by intros; rfl)
(by intros; rfl)
(by | Mathlib.Data.Real.CauSeq.378_0.b72JnnMmhSc5wVP | instance addGroupWithOne : AddGroupWithOne (CauSeq β abv) | Mathlib_Data_Real_CauSeq |
α : Type ?u.115848
β : Type ?u.115851
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
n✝ : ℤ
⊢ ↑↑n✝ = ↑n✝ | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rfl | instance addGroupWithOne : AddGroupWithOne (CauSeq β abv) :=
Function.Injective.addGroupWithOne Subtype.val Subtype.val_injective rfl rfl
coe_add coe_neg coe_sub
(by intros; rfl)
(by intros; rfl)
(by intros; rfl)
(by intros; | Mathlib.Data.Real.CauSeq.378_0.b72JnnMmhSc5wVP | instance addGroupWithOne : AddGroupWithOne (CauSeq β abv) | Mathlib_Data_Real_CauSeq |
α : Type ?u.117613
β : Type ?u.117616
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
n : ℕ
⊢ ∀ (i : ℕ), ↑(npowRec n f) i = ↑f i ^ n | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | induction n | instance : Pow (CauSeq β abv) ℕ :=
⟨fun f n =>
(ofEq (npowRec n f) fun i => f i ^ n) <| by | Mathlib.Data.Real.CauSeq.386_0.b72JnnMmhSc5wVP | instance : Pow (CauSeq β abv) ℕ | Mathlib_Data_Real_CauSeq |
case zero
α : Type ?u.117613
β : Type ?u.117616
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
⊢ ∀ (i : ℕ), ↑(npowRec Nat.zero f) i = ↑f i ^ Nat.zero | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simp [*, npowRec, pow_succ] | instance : Pow (CauSeq β abv) ℕ :=
⟨fun f n =>
(ofEq (npowRec n f) fun i => f i ^ n) <| by induction n <;> | Mathlib.Data.Real.CauSeq.386_0.b72JnnMmhSc5wVP | instance : Pow (CauSeq β abv) ℕ | Mathlib_Data_Real_CauSeq |
case succ
α : Type ?u.117613
β : Type ?u.117616
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
n✝ : ℕ
n_ih✝ : ∀ (i : ℕ), ↑(npowRec n✝ f) i = ↑f i ^ n✝
⊢ ∀ (i : ℕ), ↑(npowRec (Nat.succ n✝) f) i = ↑f i ^ Nat.succ n✝ | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simp [*, npowRec, pow_succ] | instance : Pow (CauSeq β abv) ℕ :=
⟨fun f n =>
(ofEq (npowRec n f) fun i => f i ^ n) <| by induction n <;> | Mathlib.Data.Real.CauSeq.386_0.b72JnnMmhSc5wVP | instance : Pow (CauSeq β abv) ℕ | Mathlib_Data_Real_CauSeq |
α : Type ?u.125395
β✝ : Type ?u.125398
inst✝⁴ : LinearOrderedField α
inst✝³ : Ring β✝
abv✝ : β✝ → α
inst✝² : IsAbsoluteValue abv✝
β : Type u_1
inst✝¹ : CommRing β
abv : β → α
inst✝ : IsAbsoluteValue abv
src✝ : Ring (CauSeq β abv) := ring
a b : CauSeq β abv
n : ℕ
⊢ ↑(a * b) n = ↑(b * a) n | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simp [mul_left_comm, mul_comm] | instance {β : Type*} [CommRing β] {abv : β → α} [IsAbsoluteValue abv] : CommRing (CauSeq β abv) :=
{ CauSeq.ring with
mul_comm := fun a b => ext $ fun n => by | Mathlib.Data.Real.CauSeq.408_0.b72JnnMmhSc5wVP | instance {β : Type*} [CommRing β] {abv : β → α} [IsAbsoluteValue abv] : CommRing (CauSeq β abv) | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : LimZero f
hg : LimZero g
ε : α
ε0 : ε > 0
i : ℕ
H : ∀ j ≥ i, abv (↑f j) < ε / 2 ∧ abv (↑g j) < ε / 2
j : ℕ
ij : j ≥ i
⊢ abv (↑(f + g) j) < ε | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | let ⟨H₁, H₂⟩ := H _ ij | theorem add_limZero {f g : CauSeq β abv} (hf : LimZero f) (hg : LimZero g) : LimZero (f + g)
| ε, ε0 =>
(exists_forall_ge_and (hf _ <| half_pos ε0) (hg _ <| half_pos ε0)).imp fun i H j ij => by
| Mathlib.Data.Real.CauSeq.417_0.b72JnnMmhSc5wVP | theorem add_limZero {f g : CauSeq β abv} (hf : LimZero f) (hg : LimZero g) : LimZero (f + g)
| ε, ε0 =>
(exists_forall_ge_and (hf _ <| half_pos ε0) (hg _ <| half_pos ε0)).imp fun i H j ij => by
let ⟨H₁, H₂⟩ | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : LimZero f
hg : LimZero g
ε : α
ε0 : ε > 0
i : ℕ
H : ∀ j ≥ i, abv (↑f j) < ε / 2 ∧ abv (↑g j) < ε / 2
j : ℕ
ij : j ≥ i
H₁ : abv (↑f j) < ε / 2
H₂ : abv (↑g j) < ε / 2
⊢ abv (↑(f + g) j)... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa [add_halves ε] using lt_of_le_of_lt (abv_add abv _ _) (add_lt_add H₁ H₂) | theorem add_limZero {f g : CauSeq β abv} (hf : LimZero f) (hg : LimZero g) : LimZero (f + g)
| ε, ε0 =>
(exists_forall_ge_and (hf _ <| half_pos ε0) (hg _ <| half_pos ε0)).imp fun i H j ij => by
let ⟨H₁, H₂⟩ := H _ ij
| Mathlib.Data.Real.CauSeq.417_0.b72JnnMmhSc5wVP | theorem add_limZero {f g : CauSeq β abv} (hf : LimZero f) (hg : LimZero g) : LimZero (f + g)
| ε, ε0 =>
(exists_forall_ge_and (hf _ <| half_pos ε0) (hg _ <| half_pos ε0)).imp fun i H j ij => by
let ⟨H₁, H₂⟩ | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hg : LimZero g
ε : α
ε0 : ε > 0
F : α
F0 : F > 0
hF : ∀ (i : ℕ), abv (↑f i) < F
i : ℕ
H : ∀ j ≥ i, abv (↑g j) < ε / F
j : ℕ
ij : j ≥ i
⊢ abv (↑(f * g) j) < ε | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | have := mul_lt_mul' (le_of_lt <| hF j) (H _ ij) (abv_nonneg abv _) F0 | theorem mul_limZero_right (f : CauSeq β abv) {g} (hg : LimZero g) : LimZero (f * g)
| ε, ε0 =>
let ⟨F, F0, hF⟩ := f.bounded' 0
(hg _ <| div_pos ε0 F0).imp fun i H j ij => by
| Mathlib.Data.Real.CauSeq.424_0.b72JnnMmhSc5wVP | theorem mul_limZero_right (f : CauSeq β abv) {g} (hg : LimZero g) : LimZero (f * g)
| ε, ε0 =>
let ⟨F, F0, hF⟩ | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hg : LimZero g
ε : α
ε0 : ε > 0
F : α
F0 : F > 0
hF : ∀ (i : ℕ), abv (↑f i) < F
i : ℕ
H : ∀ j ≥ i, abv (↑g j) < ε / F
j : ℕ
ij : j ≥ i
this : abv (↑f j) * abv (↑g j) < F * (ε / F)
⊢ abv (↑(... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rwa [mul_comm F, div_mul_cancel _ (ne_of_gt F0), ← abv_mul] at this | theorem mul_limZero_right (f : CauSeq β abv) {g} (hg : LimZero g) : LimZero (f * g)
| ε, ε0 =>
let ⟨F, F0, hF⟩ := f.bounded' 0
(hg _ <| div_pos ε0 F0).imp fun i H j ij => by
have := mul_lt_mul' (le_of_lt <| hF j) (H _ ij) (abv_nonneg abv _) F0
| Mathlib.Data.Real.CauSeq.424_0.b72JnnMmhSc5wVP | theorem mul_limZero_right (f : CauSeq β abv) {g} (hg : LimZero g) : LimZero (f * g)
| ε, ε0 =>
let ⟨F, F0, hF⟩ | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hg : LimZero f
ε : α
ε0 : ε > 0
G : α
G0 : G > 0
hG : ∀ (i : ℕ), abv (↑g i) < G
i : ℕ
H : ∀ j ≥ i, abv (↑f j) < ε / G
j : ℕ
ij : j ≥ i
⊢ abv (↑(f * g) j) < ε | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | have := mul_lt_mul'' (H _ ij) (hG j) (abv_nonneg abv _) (abv_nonneg abv _) | theorem mul_limZero_left {f} (g : CauSeq β abv) (hg : LimZero f) : LimZero (f * g)
| ε, ε0 =>
let ⟨G, G0, hG⟩ := g.bounded' 0
(hg _ <| div_pos ε0 G0).imp fun i H j ij => by
| Mathlib.Data.Real.CauSeq.432_0.b72JnnMmhSc5wVP | theorem mul_limZero_left {f} (g : CauSeq β abv) (hg : LimZero f) : LimZero (f * g)
| ε, ε0 =>
let ⟨G, G0, hG⟩ | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hg : LimZero f
ε : α
ε0 : ε > 0
G : α
G0 : G > 0
hG : ∀ (i : ℕ), abv (↑g i) < G
i : ℕ
H : ∀ j ≥ i, abv (↑f j) < ε / G
j : ℕ
ij : j ≥ i
this : abv (↑f j) * abv (↑g j) < ε / G * G
⊢ abv (↑(f ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rwa [div_mul_cancel _ (ne_of_gt G0), ← abv_mul] at this | theorem mul_limZero_left {f} (g : CauSeq β abv) (hg : LimZero f) : LimZero (f * g)
| ε, ε0 =>
let ⟨G, G0, hG⟩ := g.bounded' 0
(hg _ <| div_pos ε0 G0).imp fun i H j ij => by
have := mul_lt_mul'' (H _ ij) (hG j) (abv_nonneg abv _) (abv_nonneg abv _)
| Mathlib.Data.Real.CauSeq.432_0.b72JnnMmhSc5wVP | theorem mul_limZero_left {f} (g : CauSeq β abv) (hg : LimZero f) : LimZero (f * g)
| ε, ε0 =>
let ⟨G, G0, hG⟩ | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : LimZero f
⊢ LimZero (-f) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rw [← neg_one_mul f] | theorem neg_limZero {f : CauSeq β abv} (hf : LimZero f) : LimZero (-f) := by
| Mathlib.Data.Real.CauSeq.440_0.b72JnnMmhSc5wVP | theorem neg_limZero {f : CauSeq β abv} (hf : LimZero f) : LimZero (-f) | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : LimZero f
⊢ LimZero (-1 * f) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | exact mul_limZero_right _ hf | theorem neg_limZero {f : CauSeq β abv} (hf : LimZero f) : LimZero (-f) := by
rw [← neg_one_mul f]
| Mathlib.Data.Real.CauSeq.440_0.b72JnnMmhSc5wVP | theorem neg_limZero {f : CauSeq β abv} (hf : LimZero f) : LimZero (-f) | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : LimZero f
hg : LimZero g
⊢ LimZero (f - g) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa only [sub_eq_add_neg] using add_limZero hf (neg_limZero hg) | theorem sub_limZero {f g : CauSeq β abv} (hf : LimZero f) (hg : LimZero g) : LimZero (f - g) := by
| Mathlib.Data.Real.CauSeq.445_0.b72JnnMmhSc5wVP | theorem sub_limZero {f g : CauSeq β abv} (hf : LimZero f) (hg : LimZero g) : LimZero (f - g) | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hfg : LimZero (f - g)
⊢ LimZero (g - f) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa using neg_limZero hfg | theorem limZero_sub_rev {f g : CauSeq β abv} (hfg : LimZero (f - g)) : LimZero (g - f) := by
| Mathlib.Data.Real.CauSeq.449_0.b72JnnMmhSc5wVP | theorem limZero_sub_rev {f g : CauSeq β abv} (hfg : LimZero (f - g)) : LimZero (g - f) | Mathlib_Data_Real_CauSeq |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε : α
ε0 : ε > 0
j : ℕ
x✝ : j ≥ 0
⊢ abv (↑0 j) < ε | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa [abv_zero abv] using ε0 | theorem zero_limZero : LimZero (0 : CauSeq β abv)
| ε, ε0 => ⟨0, fun j _ => by | Mathlib.Data.Real.CauSeq.453_0.b72JnnMmhSc5wVP | theorem zero_limZero : LimZero (0 : CauSeq β abv)
| ε, ε0 => ⟨0, fun j _ => by simpa [abv_zero abv] using ε0⟩ | Mathlib_Data_Real_CauSeq |
α : Type ?u.139461
β : Type ?u.139464
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
⊢ LimZero (f - f) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simp [zero_limZero] | instance equiv : Setoid (CauSeq β abv) :=
⟨fun f g => LimZero (f - g),
⟨fun f => by | Mathlib.Data.Real.CauSeq.466_0.b72JnnMmhSc5wVP | instance equiv : Setoid (CauSeq β abv) | Mathlib_Data_Real_CauSeq |
α : Type ?u.139461
β : Type ?u.139464
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
x✝ y✝ : CauSeq β abv
f : LimZero (x✝ - y✝)
ε : α
hε : ε > 0
⊢ ∃ i, ∀ j ≥ i, abv (↑(y✝ - x✝) j) < ε | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa using neg_limZero f ε hε | instance equiv : Setoid (CauSeq β abv) :=
⟨fun f g => LimZero (f - g),
⟨fun f => by simp [zero_limZero],
fun f ε hε => by | Mathlib.Data.Real.CauSeq.466_0.b72JnnMmhSc5wVP | instance equiv : Setoid (CauSeq β abv) | Mathlib_Data_Real_CauSeq |
α : Type ?u.139461
β : Type ?u.139464
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
x✝ y✝ z✝ : CauSeq β abv
fg : LimZero (x✝ - y✝)
gh : LimZero (y✝ - z✝)
⊢ LimZero (x✝ - z✝) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa using add_limZero fg gh | instance equiv : Setoid (CauSeq β abv) :=
⟨fun f g => LimZero (f - g),
⟨fun f => by simp [zero_limZero],
fun f ε hε => by simpa using neg_limZero f ε hε,
fun fg gh => by | Mathlib.Data.Real.CauSeq.466_0.b72JnnMmhSc5wVP | instance equiv : Setoid (CauSeq β abv) | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f1 f2 g1 g2 : CauSeq β abv
hf : f1 ≈ f2
hg : g1 ≈ g2
⊢ f1 + g1 ≈ f2 + g2 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa only [← add_sub_add_comm] using add_limZero hf hg | theorem add_equiv_add {f1 f2 g1 g2 : CauSeq β abv} (hf : f1 ≈ f2) (hg : g1 ≈ g2) :
f1 + g1 ≈ f2 + g2 := by | Mathlib.Data.Real.CauSeq.473_0.b72JnnMmhSc5wVP | theorem add_equiv_add {f1 f2 g1 g2 : CauSeq β abv} (hf : f1 ≈ f2) (hg : g1 ≈ g2) :
f1 + g1 ≈ f2 + g2 | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : f ≈ g
⊢ -f ≈ -g | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa only [neg_sub'] using neg_limZero hf | theorem neg_equiv_neg {f g : CauSeq β abv} (hf : f ≈ g) : -f ≈ -g := by
| Mathlib.Data.Real.CauSeq.477_0.b72JnnMmhSc5wVP | theorem neg_equiv_neg {f g : CauSeq β abv} (hf : f ≈ g) : -f ≈ -g | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f1 f2 g1 g2 : CauSeq β abv
hf : f1 ≈ f2
hg : g1 ≈ g2
⊢ f1 - g1 ≈ f2 - g2 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa only [sub_eq_add_neg] using add_equiv_add hf (neg_equiv_neg hg) | theorem sub_equiv_sub {f1 f2 g1 g2 : CauSeq β abv} (hf : f1 ≈ f2) (hg : g1 ≈ g2) :
f1 - g1 ≈ f2 - g2 := by | Mathlib.Data.Real.CauSeq.481_0.b72JnnMmhSc5wVP | theorem sub_equiv_sub {f1 f2 g1 g2 : CauSeq β abv} (hf : f1 ≈ f2) (hg : g1 ≈ g2) :
f1 - g1 ≈ f2 - g2 | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
h : f ≈ g
ε : α
ε0 : 0 < ε
i : ℕ
H : ∀ j ≥ i, abv (↑(f - g) j) < ε / 2 ∧ ∀ k ≥ j, abv (↑f k - ↑f j) < ε / 2
j : ℕ
ij : j ≥ i
k : ℕ
jk : k ≥ j
⊢ abv (↑f k - ↑g j) < ε | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | let ⟨h₁, h₂⟩ := H _ ij | theorem equiv_def₃ {f g : CauSeq β abv} (h : f ≈ g) {ε : α} (ε0 : 0 < ε) :
∃ i, ∀ j ≥ i, ∀ k ≥ j, abv (f k - g j) < ε :=
(exists_forall_ge_and (h _ <| half_pos ε0) (f.cauchy₃ <| half_pos ε0)).imp fun i H j ij k jk => by
| Mathlib.Data.Real.CauSeq.485_0.b72JnnMmhSc5wVP | theorem equiv_def₃ {f g : CauSeq β abv} (h : f ≈ g) {ε : α} (ε0 : 0 < ε) :
∃ i, ∀ j ≥ i, ∀ k ≥ j, abv (f k - g j) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
h : f ≈ g
ε : α
ε0 : 0 < ε
i : ℕ
H : ∀ j ≥ i, abv (↑(f - g) j) < ε / 2 ∧ ∀ k ≥ j, abv (↑f k - ↑f j) < ε / 2
j : ℕ
ij : j ≥ i
k : ℕ
jk : k ≥ j
h₁ : abv (↑(f - g) j) < ε / 2
h₂ : ∀ k ≥ j, abv... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | have := lt_of_le_of_lt (abv_add abv (f j - g j) _) (add_lt_add h₁ (h₂ _ jk)) | theorem equiv_def₃ {f g : CauSeq β abv} (h : f ≈ g) {ε : α} (ε0 : 0 < ε) :
∃ i, ∀ j ≥ i, ∀ k ≥ j, abv (f k - g j) < ε :=
(exists_forall_ge_and (h _ <| half_pos ε0) (f.cauchy₃ <| half_pos ε0)).imp fun i H j ij k jk => by
let ⟨h₁, h₂⟩ := H _ ij
| Mathlib.Data.Real.CauSeq.485_0.b72JnnMmhSc5wVP | theorem equiv_def₃ {f g : CauSeq β abv} (h : f ≈ g) {ε : α} (ε0 : 0 < ε) :
∃ i, ∀ j ≥ i, ∀ k ≥ j, abv (f k - g j) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
h : f ≈ g
ε : α
ε0 : 0 < ε
i : ℕ
H : ∀ j ≥ i, abv (↑(f - g) j) < ε / 2 ∧ ∀ k ≥ j, abv (↑f k - ↑f j) < ε / 2
j : ℕ
ij : j ≥ i
k : ℕ
jk : k ≥ j
h₁ : abv (↑(f - g) j) < ε / 2
h₂ : ∀ k ≥ j, abv... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rwa [sub_add_sub_cancel', add_halves] at this | theorem equiv_def₃ {f g : CauSeq β abv} (h : f ≈ g) {ε : α} (ε0 : 0 < ε) :
∃ i, ∀ j ≥ i, ∀ k ≥ j, abv (f k - g j) < ε :=
(exists_forall_ge_and (h _ <| half_pos ε0) (f.cauchy₃ <| half_pos ε0)).imp fun i H j ij k jk => by
let ⟨h₁, h₂⟩ := H _ ij
have := lt_of_le_of_lt (abv_add abv (f j - g j) _) (add_lt_add ... | Mathlib.Data.Real.CauSeq.485_0.b72JnnMmhSc5wVP | theorem equiv_def₃ {f g : CauSeq β abv} (h : f ≈ g) {ε : α} (ε0 : 0 < ε) :
∃ i, ∀ j ≥ i, ∀ k ≥ j, abv (f k - g j) < ε | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
h : f ≈ g
l : LimZero f
⊢ LimZero g | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa using add_limZero (Setoid.symm h) l | theorem limZero_congr {f g : CauSeq β abv} (h : f ≈ g) : LimZero f ↔ LimZero g :=
⟨fun l => by | Mathlib.Data.Real.CauSeq.493_0.b72JnnMmhSc5wVP | theorem limZero_congr {f g : CauSeq β abv} (h : f ≈ g) : LimZero f ↔ LimZero g | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
h : f ≈ g
l : LimZero g
⊢ LimZero f | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa using add_limZero h l | theorem limZero_congr {f g : CauSeq β abv} (h : f ≈ g) : LimZero f ↔ LimZero g :=
⟨fun l => by simpa using add_limZero (Setoid.symm h) l, fun l => by | Mathlib.Data.Real.CauSeq.493_0.b72JnnMmhSc5wVP | theorem limZero_congr {f g : CauSeq β abv} (h : f ≈ g) : LimZero f ↔ LimZero g | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬LimZero f
⊢ ∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (↑f j) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | haveI := Classical.propDecidable | theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) := by
| Mathlib.Data.Real.CauSeq.497_0.b72JnnMmhSc5wVP | theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬LimZero f
this : (a : Prop) → Decidable a
⊢ ∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (↑f j) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | by_contra nk | theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) := by
haveI := Classical.propDecidable
| Mathlib.Data.Real.CauSeq.497_0.b72JnnMmhSc5wVP | theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬LimZero f
this : (a : Prop) → Decidable a
nk : ¬∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (↑f j)
⊢ False | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | refine' hf fun ε ε0 => _ | theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) := by
haveI := Classical.propDecidable
by_contra nk
| Mathlib.Data.Real.CauSeq.497_0.b72JnnMmhSc5wVP | theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬LimZero f
this : (a : Prop) → Decidable a
nk : ¬∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (↑f j)
ε : α
ε0 : ε > 0
⊢ ∃ i, ∀ j ≥ i, abv (↑f j) < ε | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simp? [not_forall] at nk says
simp only [gt_iff_lt, ge_iff_le, not_exists, not_and, not_forall, not_le, exists_prop] at nk | theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) := by
haveI := Classical.propDecidable
by_contra nk
refine' hf fun ε ε0 => _
| Mathlib.Data.Real.CauSeq.497_0.b72JnnMmhSc5wVP | theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬LimZero f
this : (a : Prop) → Decidable a
nk : ¬∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (↑f j)
ε : α
ε0 : ε > 0
⊢ ∃ i, ∀ j ≥ i, abv (↑f j) < ε | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simp only [gt_iff_lt, ge_iff_le, not_exists, not_and, not_forall, not_le, exists_prop] at nk | theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) := by
haveI := Classical.propDecidable
by_contra nk
refine' hf fun ε ε0 => _
simp? [not_forall] at nk says
| Mathlib.Data.Real.CauSeq.497_0.b72JnnMmhSc5wVP | theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬LimZero f
this : (a : Prop) → Decidable a
ε : α
ε0 : ε > 0
nk : ∀ (x : α), 0 < x → ∀ (x_1 : ℕ), ∃ x_2, x_1 ≤ x_2 ∧ abv (↑f x_2) < x
⊢ ∃ i, ∀ j ≥ i, abv (↑f j) < ε | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | cases' f.cauchy₃ (half_pos ε0) with i hi | theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) := by
haveI := Classical.propDecidable
by_contra nk
refine' hf fun ε ε0 => _
simp? [not_forall] at nk says
simp only [gt_iff_lt, ge_iff_le, not_exists, not_and, not_forall, not_le, exists_prop] at ... | Mathlib.Data.Real.CauSeq.497_0.b72JnnMmhSc5wVP | theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) | Mathlib_Data_Real_CauSeq |
case intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬LimZero f
this : (a : Prop) → Decidable a
ε : α
ε0 : ε > 0
nk : ∀ (x : α), 0 < x → ∀ (x_1 : ℕ), ∃ x_2, x_1 ≤ x_2 ∧ abv (↑f x_2) < x
i : ℕ
hi : ∀ j ≥ i, ∀ k ≥ j, abv (↑f k - ↑... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rcases nk _ (half_pos ε0) i with ⟨j, ij, hj⟩ | theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) := by
haveI := Classical.propDecidable
by_contra nk
refine' hf fun ε ε0 => _
simp? [not_forall] at nk says
simp only [gt_iff_lt, ge_iff_le, not_exists, not_and, not_forall, not_le, exists_prop] at ... | Mathlib.Data.Real.CauSeq.497_0.b72JnnMmhSc5wVP | theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) | Mathlib_Data_Real_CauSeq |
case intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬LimZero f
this : (a : Prop) → Decidable a
ε : α
ε0 : ε > 0
nk : ∀ (x : α), 0 < x → ∀ (x_1 : ℕ), ∃ x_2, x_1 ≤ x_2 ∧ abv (↑f x_2) < x
i : ℕ
hi : ∀ j ≥ i, ∀ k ≥ j, a... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | refine' ⟨j, fun k jk => _⟩ | theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) := by
haveI := Classical.propDecidable
by_contra nk
refine' hf fun ε ε0 => _
simp? [not_forall] at nk says
simp only [gt_iff_lt, ge_iff_le, not_exists, not_and, not_forall, not_le, exists_prop] at ... | Mathlib.Data.Real.CauSeq.497_0.b72JnnMmhSc5wVP | theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) | Mathlib_Data_Real_CauSeq |
case intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬LimZero f
this : (a : Prop) → Decidable a
ε : α
ε0 : ε > 0
nk : ∀ (x : α), 0 < x → ∀ (x_1 : ℕ), ∃ x_2, x_1 ≤ x_2 ∧ abv (↑f x_2) < x
i : ℕ
hi : ∀ j ≥ i, ∀ k ≥ j, a... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | have := lt_of_le_of_lt (abv_add abv _ _) (add_lt_add (hi j ij k jk) hj) | theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) := by
haveI := Classical.propDecidable
by_contra nk
refine' hf fun ε ε0 => _
simp? [not_forall] at nk says
simp only [gt_iff_lt, ge_iff_le, not_exists, not_and, not_forall, not_le, exists_prop] at ... | Mathlib.Data.Real.CauSeq.497_0.b72JnnMmhSc5wVP | theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) | Mathlib_Data_Real_CauSeq |
case intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬LimZero f
this✝ : (a : Prop) → Decidable a
ε : α
ε0 : ε > 0
nk : ∀ (x : α), 0 < x → ∀ (x_1 : ℕ), ∃ x_2, x_1 ≤ x_2 ∧ abv (↑f x_2) < x
i : ℕ
hi : ∀ j ≥ i, ∀ k ≥ j, ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rwa [sub_add_cancel, add_halves] at this | theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) := by
haveI := Classical.propDecidable
by_contra nk
refine' hf fun ε ε0 => _
simp? [not_forall] at nk says
simp only [gt_iff_lt, ge_iff_le, not_exists, not_and, not_forall, not_le, exists_prop] at ... | Mathlib.Data.Real.CauSeq.497_0.b72JnnMmhSc5wVP | theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : CauSeq β abv
h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - ↑g j) < ε
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abv (f j - ↑g j) < ε / 2 / 2 ∧ ∀ k ≥ j, abv (↑g k - ↑g j) < ε / 2
j : ℕ
ij : j ≥ i
⊢ abv (f j - ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | cases' hi _ le_rfl with h₁ h₂ | theorem of_near (f : ℕ → β) (g : CauSeq β abv) (h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - g j) < ε) :
IsCauSeq abv f
| ε, ε0 =>
let ⟨i, hi⟩ := exists_forall_ge_and (h _ (half_pos <| half_pos ε0)) (g.cauchy₃ <| half_pos ε0)
⟨i, fun j ij => by
| Mathlib.Data.Real.CauSeq.511_0.b72JnnMmhSc5wVP | theorem of_near (f : ℕ → β) (g : CauSeq β abv) (h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - g j) < ε) :
IsCauSeq abv f
| ε, ε0 =>
let ⟨i, hi⟩ | Mathlib_Data_Real_CauSeq |
case intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : CauSeq β abv
h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - ↑g j) < ε
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abv (f j - ↑g j) < ε / 2 / 2 ∧ ∀ k ≥ j, abv (↑g k - ↑g j) < ε / 2
j : ℕ
ij : j ≥ i
h₁... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rw [abv_sub abv] at h₁ | theorem of_near (f : ℕ → β) (g : CauSeq β abv) (h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - g j) < ε) :
IsCauSeq abv f
| ε, ε0 =>
let ⟨i, hi⟩ := exists_forall_ge_and (h _ (half_pos <| half_pos ε0)) (g.cauchy₃ <| half_pos ε0)
⟨i, fun j ij => by
cases' hi _ le_rfl with h₁ h₂; | Mathlib.Data.Real.CauSeq.511_0.b72JnnMmhSc5wVP | theorem of_near (f : ℕ → β) (g : CauSeq β abv) (h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - g j) < ε) :
IsCauSeq abv f
| ε, ε0 =>
let ⟨i, hi⟩ | Mathlib_Data_Real_CauSeq |
case intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : CauSeq β abv
h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - ↑g j) < ε
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abv (f j - ↑g j) < ε / 2 / 2 ∧ ∀ k ≥ j, abv (↑g k - ↑g j) < ε / 2
j : ℕ
ij : j ≥ i
h₁... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | have := lt_of_le_of_lt (abv_add abv _ _) (add_lt_add (hi _ ij).1 h₁) | theorem of_near (f : ℕ → β) (g : CauSeq β abv) (h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - g j) < ε) :
IsCauSeq abv f
| ε, ε0 =>
let ⟨i, hi⟩ := exists_forall_ge_and (h _ (half_pos <| half_pos ε0)) (g.cauchy₃ <| half_pos ε0)
⟨i, fun j ij => by
cases' hi _ le_rfl with h₁ h₂; rw [abv_sub abv] at h₁
| Mathlib.Data.Real.CauSeq.511_0.b72JnnMmhSc5wVP | theorem of_near (f : ℕ → β) (g : CauSeq β abv) (h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - g j) < ε) :
IsCauSeq abv f
| ε, ε0 =>
let ⟨i, hi⟩ | Mathlib_Data_Real_CauSeq |
case intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : CauSeq β abv
h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - ↑g j) < ε
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abv (f j - ↑g j) < ε / 2 / 2 ∧ ∀ k ≥ j, abv (↑g k - ↑g j) < ε / 2
j : ℕ
ij : j ≥ i
h₁... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | have := lt_of_le_of_lt (abv_add abv _ _) (add_lt_add this (h₂ _ ij)) | theorem of_near (f : ℕ → β) (g : CauSeq β abv) (h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - g j) < ε) :
IsCauSeq abv f
| ε, ε0 =>
let ⟨i, hi⟩ := exists_forall_ge_and (h _ (half_pos <| half_pos ε0)) (g.cauchy₃ <| half_pos ε0)
⟨i, fun j ij => by
cases' hi _ le_rfl with h₁ h₂; rw [abv_sub abv] at h₁
ha... | Mathlib.Data.Real.CauSeq.511_0.b72JnnMmhSc5wVP | theorem of_near (f : ℕ → β) (g : CauSeq β abv) (h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - g j) < ε) :
IsCauSeq abv f
| ε, ε0 =>
let ⟨i, hi⟩ | Mathlib_Data_Real_CauSeq |
case intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : CauSeq β abv
h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - ↑g j) < ε
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abv (f j - ↑g j) < ε / 2 / 2 ∧ ∀ k ≥ j, abv (↑g k - ↑g j) < ε / 2
j : ℕ
ij : j ≥ i
h₁... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rwa [add_halves, add_halves, add_right_comm, sub_add_sub_cancel, sub_add_sub_cancel] at this | theorem of_near (f : ℕ → β) (g : CauSeq β abv) (h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - g j) < ε) :
IsCauSeq abv f
| ε, ε0 =>
let ⟨i, hi⟩ := exists_forall_ge_and (h _ (half_pos <| half_pos ε0)) (g.cauchy₃ <| half_pos ε0)
⟨i, fun j ij => by
cases' hi _ le_rfl with h₁ h₂; rw [abv_sub abv] at h₁
ha... | Mathlib.Data.Real.CauSeq.511_0.b72JnnMmhSc5wVP | theorem of_near (f : ℕ → β) (g : CauSeq β abv) (h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - g j) < ε) :
IsCauSeq abv f
| ε, ε0 =>
let ⟨i, hi⟩ | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬f ≈ 0
⊢ ¬LimZero f | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | intro h | theorem not_limZero_of_not_congr_zero {f : CauSeq _ abv} (hf : ¬f ≈ 0) : ¬LimZero f := by
| Mathlib.Data.Real.CauSeq.522_0.b72JnnMmhSc5wVP | theorem not_limZero_of_not_congr_zero {f : CauSeq _ abv} (hf : ¬f ≈ 0) : ¬LimZero f | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬f ≈ 0
h : LimZero f
⊢ False | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | have : LimZero (f - 0) := by simp [h] | theorem not_limZero_of_not_congr_zero {f : CauSeq _ abv} (hf : ¬f ≈ 0) : ¬LimZero f := by
intro h
| Mathlib.Data.Real.CauSeq.522_0.b72JnnMmhSc5wVP | theorem not_limZero_of_not_congr_zero {f : CauSeq _ abv} (hf : ¬f ≈ 0) : ¬LimZero f | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬f ≈ 0
h : LimZero f
⊢ LimZero (f - 0) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simp [h] | theorem not_limZero_of_not_congr_zero {f : CauSeq _ abv} (hf : ¬f ≈ 0) : ¬LimZero f := by
intro h
have : LimZero (f - 0) := by | Mathlib.Data.Real.CauSeq.522_0.b72JnnMmhSc5wVP | theorem not_limZero_of_not_congr_zero {f : CauSeq _ abv} (hf : ¬f ≈ 0) : ¬LimZero f | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬f ≈ 0
h : LimZero f
this : LimZero (f - 0)
⊢ False | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | exact hf this | theorem not_limZero_of_not_congr_zero {f : CauSeq _ abv} (hf : ¬f ≈ 0) : ¬LimZero f := by
intro h
have : LimZero (f - 0) := by simp [h]
| Mathlib.Data.Real.CauSeq.522_0.b72JnnMmhSc5wVP | theorem not_limZero_of_not_congr_zero {f : CauSeq _ abv} (hf : ¬f ≈ 0) : ¬LimZero f | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
g f : CauSeq β abv
hf : f ≈ 0
this : LimZero (f - 0)
⊢ LimZero f | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa | theorem mul_equiv_zero (g : CauSeq _ abv) {f : CauSeq _ abv} (hf : f ≈ 0) : g * f ≈ 0 :=
have : LimZero (f - 0) := hf
have : LimZero (g * f) := mul_limZero_right _ <| by | Mathlib.Data.Real.CauSeq.528_0.b72JnnMmhSc5wVP | theorem mul_equiv_zero (g : CauSeq _ abv) {f : CauSeq _ abv} (hf : f ≈ 0) : g * f ≈ 0 | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
g f : CauSeq β abv
hf : f ≈ 0
this✝ : LimZero (f - 0)
this : LimZero (g * f)
⊢ LimZero (g * f - 0) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa | theorem mul_equiv_zero (g : CauSeq _ abv) {f : CauSeq _ abv} (hf : f ≈ 0) : g * f ≈ 0 :=
have : LimZero (f - 0) := hf
have : LimZero (g * f) := mul_limZero_right _ <| by simpa
show LimZero (g * f - 0) by | Mathlib.Data.Real.CauSeq.528_0.b72JnnMmhSc5wVP | theorem mul_equiv_zero (g : CauSeq _ abv) {f : CauSeq _ abv} (hf : f ≈ 0) : g * f ≈ 0 | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
g f : CauSeq β abv
hf : f ≈ 0
this : LimZero (f - 0)
⊢ LimZero f | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa | theorem mul_equiv_zero' (g : CauSeq _ abv) {f : CauSeq _ abv} (hf : f ≈ 0) : f * g ≈ 0 :=
have : LimZero (f - 0) := hf
have : LimZero (f * g) := mul_limZero_left _ <| by | Mathlib.Data.Real.CauSeq.534_0.b72JnnMmhSc5wVP | theorem mul_equiv_zero' (g : CauSeq _ abv) {f : CauSeq _ abv} (hf : f ≈ 0) : f * g ≈ 0 | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
g f : CauSeq β abv
hf : f ≈ 0
this✝ : LimZero (f - 0)
this : LimZero (f * g)
⊢ LimZero (f * g - 0) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa | theorem mul_equiv_zero' (g : CauSeq _ abv) {f : CauSeq _ abv} (hf : f ≈ 0) : f * g ≈ 0 :=
have : LimZero (f - 0) := hf
have : LimZero (f * g) := mul_limZero_left _ <| by simpa
show LimZero (f * g - 0) by | Mathlib.Data.Real.CauSeq.534_0.b72JnnMmhSc5wVP | theorem mul_equiv_zero' (g : CauSeq _ abv) {f : CauSeq _ abv} (hf : f ≈ 0) : f * g ≈ 0 | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this : LimZero (f * g - 0)
⊢ False | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | have hlz : LimZero (f * g) := by simpa | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
fun (this : LimZero (f * g - 0)) => by
| Mathlib.Data.Real.CauSeq.540_0.b72JnnMmhSc5wVP | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this : LimZero (f * g - 0)
⊢ LimZero (f * g) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
fun (this : LimZero (f * g - 0)) => by
have hlz : LimZero (f * g) := by | Mathlib.Data.Real.CauSeq.540_0.b72JnnMmhSc5wVP | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this : LimZero (f * g - 0)
hlz : LimZero (f * g)
⊢ False | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
fun (this : LimZero (f * g - 0)) => by
have hlz : LimZero (f * g) := by simpa
| Mathlib.Data.Real.CauSeq.540_0.b72JnnMmhSc5wVP | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this : LimZero (f * g - 0)
hlz : LimZero (f * g)
⊢ ¬LimZero f | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa using show ¬LimZero (f - 0) from hf | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
fun (this : LimZero (f * g - 0)) => by
have hlz : LimZero (f * g) := by simpa
have hf' : ¬LimZero f := by | Mathlib.Data.Real.CauSeq.540_0.b72JnnMmhSc5wVP | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
⊢ False | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
fun (this : LimZero (f * g - 0)) => by
have hlz : LimZero (f * g) := by simpa
have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
| Mathlib.Data.Real.CauSeq.540_0.b72JnnMmhSc5wVP | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
⊢ ¬LimZero g | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | simpa using show ¬LimZero (g - 0) from hg | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
fun (this : LimZero (f * g - 0)) => by
have hlz : LimZero (f * g) := by simpa
have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
have hg' : ¬LimZero g := by | Mathlib.Data.Real.CauSeq.540_0.b72JnnMmhSc5wVP | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
hg' : ¬LimZero g
⊢ False | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rcases abv_pos_of_not_limZero hf' with ⟨a1, ha1, N1, hN1⟩ | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
fun (this : LimZero (f * g - 0)) => by
have hlz : LimZero (f * g) := by simpa
have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg
| Mathlib.Data.Real.CauSeq.540_0.b72JnnMmhSc5wVP | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 | Mathlib_Data_Real_CauSeq |
case intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
hg' : ¬LimZero g
a1 : α
ha1 : a1 > 0
N1 : ℕ
hN1 : ∀ j ≥ N1, a1 ≤ abv (↑f j)
... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rcases abv_pos_of_not_limZero hg' with ⟨a2, ha2, N2, hN2⟩ | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
fun (this : LimZero (f * g - 0)) => by
have hlz : LimZero (f * g) := by simpa
have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg
... | Mathlib.Data.Real.CauSeq.540_0.b72JnnMmhSc5wVP | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 | Mathlib_Data_Real_CauSeq |
case intro.intro.intro.intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
hg' : ¬LimZero g
a1 : α
ha1 : a1 > 0
N1 : ℕ
hN1 : ∀ j ≥ N1... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | have : 0 < a1 * a2 := mul_pos ha1 ha2 | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
fun (this : LimZero (f * g - 0)) => by
have hlz : LimZero (f * g) := by simpa
have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg
... | Mathlib.Data.Real.CauSeq.540_0.b72JnnMmhSc5wVP | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 | Mathlib_Data_Real_CauSeq |
case intro.intro.intro.intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this✝ : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
hg' : ¬LimZero g
a1 : α
ha1 : a1 > 0
N1 : ℕ
hN1 : ∀ j ≥ N... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | cases' hlz _ this with N hN | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
fun (this : LimZero (f * g - 0)) => by
have hlz : LimZero (f * g) := by simpa
have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg
... | Mathlib.Data.Real.CauSeq.540_0.b72JnnMmhSc5wVP | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 | Mathlib_Data_Real_CauSeq |
case intro.intro.intro.intro.intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this✝ : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
hg' : ¬LimZero g
a1 : α
ha1 : a1 > 0
N1 : ℕ
hN1 : ∀... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | let i := max N (max N1 N2) | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
fun (this : LimZero (f * g - 0)) => by
have hlz : LimZero (f * g) := by simpa
have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg
... | Mathlib.Data.Real.CauSeq.540_0.b72JnnMmhSc5wVP | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 | Mathlib_Data_Real_CauSeq |
case intro.intro.intro.intro.intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this✝ : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
hg' : ¬LimZero g
a1 : α
ha1 : a1 > 0
N1 : ℕ
hN1 : ∀... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | have hN' := hN i (le_max_left _ _) | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
fun (this : LimZero (f * g - 0)) => by
have hlz : LimZero (f * g) := by simpa
have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg
... | Mathlib.Data.Real.CauSeq.540_0.b72JnnMmhSc5wVP | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 | Mathlib_Data_Real_CauSeq |
case intro.intro.intro.intro.intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this✝ : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
hg' : ¬LimZero g
a1 : α
ha1 : a1 > 0
N1 : ℕ
hN1 : ∀... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | have hN1' := hN1 i (le_trans (le_max_left _ _) (le_max_right _ _)) | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
fun (this : LimZero (f * g - 0)) => by
have hlz : LimZero (f * g) := by simpa
have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg
... | Mathlib.Data.Real.CauSeq.540_0.b72JnnMmhSc5wVP | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 | Mathlib_Data_Real_CauSeq |
case intro.intro.intro.intro.intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this✝ : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
hg' : ¬LimZero g
a1 : α
ha1 : a1 > 0
N1 : ℕ
hN1 : ∀... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | have hN1' := hN2 i (le_trans (le_max_right _ _) (le_max_right _ _)) | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
fun (this : LimZero (f * g - 0)) => by
have hlz : LimZero (f * g) := by simpa
have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg
... | Mathlib.Data.Real.CauSeq.540_0.b72JnnMmhSc5wVP | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 | Mathlib_Data_Real_CauSeq |
case intro.intro.intro.intro.intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this✝ : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
hg' : ¬LimZero g
a1 : α
ha1 : a1 > 0
N1 : ℕ
hN1 : ∀... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | apply not_le_of_lt hN' | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
fun (this : LimZero (f * g - 0)) => by
have hlz : LimZero (f * g) := by simpa
have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg
... | Mathlib.Data.Real.CauSeq.540_0.b72JnnMmhSc5wVP | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 | Mathlib_Data_Real_CauSeq |
case intro.intro.intro.intro.intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this✝ : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
hg' : ¬LimZero g
a1 : α
ha1 : a1 > 0
N1 : ℕ
hN1 : ∀... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | change _ ≤ abv (_ * _) | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
fun (this : LimZero (f * g - 0)) => by
have hlz : LimZero (f * g) := by simpa
have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg
... | Mathlib.Data.Real.CauSeq.540_0.b72JnnMmhSc5wVP | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 | Mathlib_Data_Real_CauSeq |
case intro.intro.intro.intro.intro.intro.intro
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this✝ : LimZero (f * g - 0)
hlz : LimZero (f * g)
hf' : ¬LimZero f
hg' : ¬LimZero g
a1 : α
ha1 : a1 > 0
N1 : ℕ
hN1 : ∀... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupPower.Lemmas
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Field.Basic... | rw [abv_mul abv] | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
fun (this : LimZero (f * g - 0)) => by
have hlz : LimZero (f * g) := by simpa
have hf' : ¬LimZero f := by simpa using show ¬LimZero (f - 0) from hf
have hg' : ¬LimZero g := by simpa using show ¬LimZero (g - 0) from hg
... | Mathlib.Data.Real.CauSeq.540_0.b72JnnMmhSc5wVP | theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 | Mathlib_Data_Real_CauSeq |
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