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 |
|---|---|---|---|---|---|---|
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... | gcongr | 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 |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
x y : β
⊢ LimZero (const x - const y) ↔ x = y | /-
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 [← const_sub, const_limZero, sub_eq_zero] | theorem const_equiv {x y : β} : const x ≈ const y ↔ x = y :=
show LimZero _ ↔ _ by | Mathlib.Data.Real.CauSeq.559_0.b72JnnMmhSc5wVP | theorem const_equiv {x y : β} : const x ≈ const y ↔ x = y | 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... | change LimZero (f1 * g1 - f2 * g2) | theorem mul_equiv_mul {f1 f2 g1 g2 : CauSeq β abv} (hf : f1 ≈ f2) (hg : g1 ≈ g2) :
f1 * g1 ≈ f2 * g2 := by
| Mathlib.Data.Real.CauSeq.563_0.b72JnnMmhSc5wVP | theorem mul_equiv_mul {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
f1 f2 g1 g2 : CauSeq β abv
hf : f1 ≈ f2
hg : g1 ≈ g2
⊢ LimZero (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... | convert add_limZero (mul_limZero_left g1 hf) (mul_limZero_right f2 hg) using 1 | theorem mul_equiv_mul {f1 f2 g1 g2 : CauSeq β abv} (hf : f1 ≈ f2) (hg : g1 ≈ g2) :
f1 * g1 ≈ f2 * g2 := by
change LimZero (f1 * g1 - f2 * g2)
| Mathlib.Data.Real.CauSeq.563_0.b72JnnMmhSc5wVP | theorem mul_equiv_mul {f1 f2 g1 g2 : CauSeq β abv} (hf : f1 ≈ f2) (hg : g1 ≈ g2) :
f1 * g1 ≈ f2 * g2 | Mathlib_Data_Real_CauSeq |
case h.e'_6
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 = (f1 - f2) * g1 + f2 * (g1 - 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... | rw [mul_sub, sub_mul] | theorem mul_equiv_mul {f1 f2 g1 g2 : CauSeq β abv} (hf : f1 ≈ f2) (hg : g1 ≈ g2) :
f1 * g1 ≈ f2 * g2 := by
change LimZero (f1 * g1 - f2 * g2)
convert add_limZero (mul_limZero_left g1 hf) (mul_limZero_right f2 hg) using 1
| Mathlib.Data.Real.CauSeq.563_0.b72JnnMmhSc5wVP | theorem mul_equiv_mul {f1 f2 g1 g2 : CauSeq β abv} (hf : f1 ≈ f2) (hg : g1 ≈ g2) :
f1 * g1 ≈ f2 * g2 | Mathlib_Data_Real_CauSeq |
case h.e'_6
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 = f1 * g1 - f2 * g1 + (f2 * 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... | exact (sub_add_sub_cancel (f1*g1) (f2*g1) (f2*g2)).symm | theorem mul_equiv_mul {f1 f2 g1 g2 : CauSeq β abv} (hf : f1 ≈ f2) (hg : g1 ≈ g2) :
f1 * g1 ≈ f2 * g2 := by
change LimZero (f1 * g1 - f2 * g2)
convert add_limZero (mul_limZero_left g1 hf) (mul_limZero_right f2 hg) using 1
rw [mul_sub, sub_mul]
-- Porting note: doesn't work with `rw`, but did in Lean 3
| Mathlib.Data.Real.CauSeq.563_0.b72JnnMmhSc5wVP | theorem mul_equiv_mul {f1 f2 g1 g2 : CauSeq β abv} (hf : f1 ≈ f2) (hg : g1 ≈ g2) :
f1 * g1 ≈ f2 * g2 | Mathlib_Data_Real_CauSeq |
α : Type u_3
β : Type u_2
inst✝⁴ : LinearOrderedField α
inst✝³ : Ring β
abv : β → α
inst✝² : IsAbsoluteValue abv
G : Type u_1
inst✝¹ : SMul G β
inst✝ : IsScalarTower G β β
f1 f2 : CauSeq β abv
c : G
hf : f1 ≈ f2
⊢ c • f1 ≈ c • f2 | /-
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 [const_smul, smul_one_mul _ _] using
mul_equiv_mul (const_equiv.mpr <| Eq.refl <| c • (1 : β)) hf | theorem smul_equiv_smul [SMul G β] [IsScalarTower G β β] {f1 f2 : CauSeq β abv} (c : G)
(hf : f1 ≈ f2) : c • f1 ≈ c • f2 := by
| Mathlib.Data.Real.CauSeq.577_0.b72JnnMmhSc5wVP | theorem smul_equiv_smul [SMul G β] [IsScalarTower G β β] {f1 f2 : CauSeq β abv} (c : G)
(hf : f1 ≈ f2) : c • f1 ≈ c • f2 | 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 : CauSeq β abv
hf : f1 ≈ f2
n : ℕ
⊢ f1 ^ n ≈ f2 ^ 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 with n ih | theorem pow_equiv_pow {f1 f2 : CauSeq β abv} (hf : f1 ≈ f2) (n : ℕ) : f1 ^ n ≈ f2 ^ n := by
| Mathlib.Data.Real.CauSeq.583_0.b72JnnMmhSc5wVP | theorem pow_equiv_pow {f1 f2 : CauSeq β abv} (hf : f1 ≈ f2) (n : ℕ) : f1 ^ n ≈ f2 ^ n | Mathlib_Data_Real_CauSeq |
case zero
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f1 f2 : CauSeq β abv
hf : f1 ≈ f2
⊢ f1 ^ Nat.zero ≈ f2 ^ 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 only [Nat.zero_eq, pow_zero, Setoid.refl] | theorem pow_equiv_pow {f1 f2 : CauSeq β abv} (hf : f1 ≈ f2) (n : ℕ) : f1 ^ n ≈ f2 ^ n := by
induction' n with n ih
· | Mathlib.Data.Real.CauSeq.583_0.b72JnnMmhSc5wVP | theorem pow_equiv_pow {f1 f2 : CauSeq β abv} (hf : f1 ≈ f2) (n : ℕ) : f1 ^ n ≈ f2 ^ n | Mathlib_Data_Real_CauSeq |
case succ
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f1 f2 : CauSeq β abv
hf : f1 ≈ f2
n : ℕ
ih : f1 ^ n ≈ f2 ^ n
⊢ f1 ^ Nat.succ n ≈ f2 ^ 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... | simpa only [pow_succ] using mul_equiv_mul hf ih | theorem pow_equiv_pow {f1 f2 : CauSeq β abv} (hf : f1 ≈ f2) (n : ℕ) : f1 ^ n ≈ f2 ^ n := by
induction' n with n ih
· simp only [Nat.zero_eq, pow_zero, Setoid.refl]
· | Mathlib.Data.Real.CauSeq.583_0.b72JnnMmhSc5wVP | theorem pow_equiv_pow {f1 f2 : CauSeq β abv} (hf : f1 ≈ f2) (n : ℕ) : f1 ^ n ≈ f2 ^ n | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Ring β
inst✝¹ : IsDomain β
abv : β → α
inst✝ : IsAbsoluteValue abv
h : const abv 1 ≈ const abv 0
this : ∀ ε > 0, ∃ i, ∀ (k : ℕ), i ≤ k → abv (1 - 0) < ε
h2 : 0 < abv 1
i : ℕ
hi : ∀ (k : ℕ), i ≤ k → abv (1 - 0) < abv 1
⊢ abv 1 < abv 1 | /-
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 hi _ le_rfl | theorem one_not_equiv_zero : ¬const abv 1 ≈ const abv 0 := fun h =>
have : ∀ ε > 0, ∃ i, ∀ k, i ≤ k → abv (1 - 0) < ε := h
have h1 : abv 1 ≤ 0 :=
le_of_not_gt fun h2 : 0 < abv 1 =>
(Exists.elim (this _ h2)) fun i hi => lt_irrefl (abv 1) <| by | Mathlib.Data.Real.CauSeq.595_0.b72JnnMmhSc5wVP | theorem one_not_equiv_zero : ¬const abv 1 ≈ const abv 0 | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : DivisionRing β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬LimZero f
ε : α
ε0 : ε > 0
K : α
K0 : K > 0
i : ℕ
H : ∀ j ≥ i, K ≤ abv (↑f j)
j : ℕ
ij : j ≥ i
⊢ abv (↑(inv f hf * f - 1) 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_pos abv).1 (lt_of_lt_of_le K0 (H _ ij)), abv_zero abv] using ε0 | theorem inv_mul_cancel {f : CauSeq β abv} (hf) : inv f hf * f ≈ 1 := fun ε ε0 =>
let ⟨K, K0, i, H⟩ := abv_pos_of_not_limZero hf
⟨i, fun j ij => by | Mathlib.Data.Real.CauSeq.639_0.b72JnnMmhSc5wVP | theorem inv_mul_cancel {f : CauSeq β abv} (hf) : inv f hf * f ≈ 1 | Mathlib_Data_Real_CauSeq |
α : Type u_2
β : Type u_1
inst✝² : LinearOrderedField α
inst✝¹ : DivisionRing β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬LimZero f
ε : α
ε0 : ε > 0
K : α
K0 : K > 0
i : ℕ
H : ∀ j ≥ i, K ≤ abv (↑f j)
j : ℕ
ij : j ≥ i
⊢ abv (↑(f * inv f hf - 1) 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_pos abv).1 (lt_of_lt_of_le K0 (H _ ij)), abv_zero abv] using ε0 | theorem mul_inv_cancel {f : CauSeq β abv} (hf) : f * inv f hf ≈ 1 := fun ε ε0 =>
let ⟨K, K0, i, H⟩ := abv_pos_of_not_limZero hf
⟨i, fun j ij => by | Mathlib.Data.Real.CauSeq.644_0.b72JnnMmhSc5wVP | theorem mul_inv_cancel {f : CauSeq β abv} (hf) : f * inv f hf ≈ 1 | Mathlib_Data_Real_CauSeq |
α : Type ?u.197134
β : Type ?u.197137
inst✝² : LinearOrderedField α
inst✝¹ : DivisionRing β
abv : β → α
inst✝ : IsAbsoluteValue abv
x : β
hx : x ≠ 0
⊢ ¬LimZero (const abv 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... | rwa [const_limZero] | theorem const_inv {x : β} (hx : x ≠ 0) :
const abv x⁻¹ = inv (const abv x) (by | Mathlib.Data.Real.CauSeq.649_0.b72JnnMmhSc5wVP | theorem const_inv {x : β} (hx : x ≠ 0) :
const abv x⁻¹ = inv (const abv x) (by rwa [const_limZero]) | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
f g : CauSeq α abs
F : α
F0 : F > 0
hF : ∃ i, ∀ j ≥ i, F ≤ ↑f j
H : LimZero g
i : ℕ
h : ∀ j ≥ i, F ≤ ↑f j ∧ |↑g j| < F / 2
j : ℕ
ij : j ≥ i
⊢ F / 2 ≤ ↑(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... | cases' h j ij with h₁ h₂ | theorem pos_add_limZero {f g : CauSeq α abs} : Pos f → LimZero g → Pos (f + g)
| ⟨F, F0, hF⟩, H =>
let ⟨i, h⟩ := exists_forall_ge_and hF (H _ (half_pos F0))
⟨_, half_pos F0, i, fun j ij => by
| Mathlib.Data.Real.CauSeq.685_0.b72JnnMmhSc5wVP | theorem pos_add_limZero {f g : CauSeq α abs} : Pos f → LimZero g → Pos (f + g)
| ⟨F, F0, hF⟩, H =>
let ⟨i, h⟩ | Mathlib_Data_Real_CauSeq |
case intro
α : Type u_1
inst✝ : LinearOrderedField α
f g : CauSeq α abs
F : α
F0 : F > 0
hF : ∃ i, ∀ j ≥ i, F ≤ ↑f j
H : LimZero g
i : ℕ
h : ∀ j ≥ i, F ≤ ↑f j ∧ |↑g j| < F / 2
j : ℕ
ij : j ≥ i
h₁ : F ≤ ↑f j
h₂ : |↑g j| < F / 2
⊢ F / 2 ≤ ↑(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... | have := add_le_add h₁ (le_of_lt (abs_lt.1 h₂).1) | theorem pos_add_limZero {f g : CauSeq α abs} : Pos f → LimZero g → Pos (f + g)
| ⟨F, F0, hF⟩, H =>
let ⟨i, h⟩ := exists_forall_ge_and hF (H _ (half_pos F0))
⟨_, half_pos F0, i, fun j ij => by
cases' h j ij with h₁ h₂
| Mathlib.Data.Real.CauSeq.685_0.b72JnnMmhSc5wVP | theorem pos_add_limZero {f g : CauSeq α abs} : Pos f → LimZero g → Pos (f + g)
| ⟨F, F0, hF⟩, H =>
let ⟨i, h⟩ | Mathlib_Data_Real_CauSeq |
case intro
α : Type u_1
inst✝ : LinearOrderedField α
f g : CauSeq α abs
F : α
F0 : F > 0
hF : ∃ i, ∀ j ≥ i, F ≤ ↑f j
H : LimZero g
i : ℕ
h : ∀ j ≥ i, F ≤ ↑f j ∧ |↑g j| < F / 2
j : ℕ
ij : j ≥ i
h₁ : F ≤ ↑f j
h₂ : |↑g j| < F / 2
this : F + -(F / 2) ≤ ↑f j + ↑g j
⊢ F / 2 ≤ ↑(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... | rwa [← sub_eq_add_neg, sub_self_div_two] at this | theorem pos_add_limZero {f g : CauSeq α abs} : Pos f → LimZero g → Pos (f + g)
| ⟨F, F0, hF⟩, H =>
let ⟨i, h⟩ := exists_forall_ge_and hF (H _ (half_pos F0))
⟨_, half_pos F0, i, fun j ij => by
cases' h j ij with h₁ h₂
have := add_le_add h₁ (le_of_lt (abs_lt.1 h₂).1)
| Mathlib.Data.Real.CauSeq.685_0.b72JnnMmhSc5wVP | theorem pos_add_limZero {f g : CauSeq α abs} : Pos f → LimZero g → Pos (f + g)
| ⟨F, F0, hF⟩, H =>
let ⟨i, h⟩ | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
⊢ Pos f ∨ LimZero f ∨ Pos (-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... | cases' Classical.em (LimZero f) with h h | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) := by
| Mathlib.Data.Real.CauSeq.702_0.b72JnnMmhSc5wVP | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) | Mathlib_Data_Real_CauSeq |
case inl
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
h : LimZero f
⊢ Pos f ∨ LimZero f ∨ Pos (-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 [*] | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) := by
cases' Classical.em (LimZero f) with h h <;> | Mathlib.Data.Real.CauSeq.702_0.b72JnnMmhSc5wVP | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) | Mathlib_Data_Real_CauSeq |
case inr
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
h : ¬LimZero f
⊢ Pos f ∨ LimZero f ∨ Pos (-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 [*] | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) := by
cases' Classical.em (LimZero f) with h h <;> | Mathlib.Data.Real.CauSeq.702_0.b72JnnMmhSc5wVP | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) | Mathlib_Data_Real_CauSeq |
case inr
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
h : ¬LimZero f
⊢ Pos f ∨ Pos (-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... | rcases abv_pos_of_not_limZero h with ⟨K, K0, hK⟩ | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) := by
cases' Classical.em (LimZero f) with h h <;> simp [*]
| Mathlib.Data.Real.CauSeq.702_0.b72JnnMmhSc5wVP | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) | Mathlib_Data_Real_CauSeq |
case inr.intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
h : ¬LimZero f
K : α
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ |↑f j|
⊢ Pos f ∨ Pos (-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... | rcases exists_forall_ge_and hK (f.cauchy₃ K0) with ⟨i, hi⟩ | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) := by
cases' Classical.em (LimZero f) with h h <;> simp [*]
rcases abv_pos_of_not_limZero h with ⟨K, K0, hK⟩
| Mathlib.Data.Real.CauSeq.702_0.b72JnnMmhSc5wVP | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) | Mathlib_Data_Real_CauSeq |
case inr.intro.intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
h : ¬LimZero f
K : α
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ |↑f j|
i : ℕ
hi : ∀ j ≥ i, K ≤ |↑f j| ∧ ∀ k ≥ j, |↑f k - ↑f j| < K
⊢ Pos f ∨ Pos (-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... | refine' (le_total 0 (f i)).imp _ _ | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) := by
cases' Classical.em (LimZero f) with h h <;> simp [*]
rcases abv_pos_of_not_limZero h with ⟨K, K0, hK⟩
rcases exists_forall_ge_and hK (f.cauchy₃ K0) with ⟨i, hi⟩
| Mathlib.Data.Real.CauSeq.702_0.b72JnnMmhSc5wVP | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) | Mathlib_Data_Real_CauSeq |
case inr.intro.intro.intro.refine'_1
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
h : ¬LimZero f
K : α
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ |↑f j|
i : ℕ
hi : ∀ j ≥ i, K ≤ |↑f j| ∧ ∀ k ≥ j, |↑f k - ↑f j| < K
⊢ 0 ≤ ↑f i → Pos 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... | refine' fun h => ⟨K, K0, i, fun j ij => _⟩ | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) := by
cases' Classical.em (LimZero f) with h h <;> simp [*]
rcases abv_pos_of_not_limZero h with ⟨K, K0, hK⟩
rcases exists_forall_ge_and hK (f.cauchy₃ K0) with ⟨i, hi⟩
refine' (le_total 0 (f i)).imp _ _ <;>
| Mathlib.Data.Real.CauSeq.702_0.b72JnnMmhSc5wVP | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) | Mathlib_Data_Real_CauSeq |
case inr.intro.intro.intro.refine'_2
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
h : ¬LimZero f
K : α
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ |↑f j|
i : ℕ
hi : ∀ j ≥ i, K ≤ |↑f j| ∧ ∀ k ≥ j, |↑f k - ↑f j| < K
⊢ ↑f i ≤ 0 → Pos (-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... | refine' fun h => ⟨K, K0, i, fun j ij => _⟩ | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) := by
cases' Classical.em (LimZero f) with h h <;> simp [*]
rcases abv_pos_of_not_limZero h with ⟨K, K0, hK⟩
rcases exists_forall_ge_and hK (f.cauchy₃ K0) with ⟨i, hi⟩
refine' (le_total 0 (f i)).imp _ _ <;>
| Mathlib.Data.Real.CauSeq.702_0.b72JnnMmhSc5wVP | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) | Mathlib_Data_Real_CauSeq |
case inr.intro.intro.intro.refine'_1
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
h✝ : ¬LimZero f
K : α
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ |↑f j|
i : ℕ
hi : ∀ j ≥ i, K ≤ |↑f j| ∧ ∀ k ≥ j, |↑f k - ↑f j| < K
h : 0 ≤ ↑f i
j : ℕ
ij : j ≥ i
⊢ K ≤ ↑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... | have := (hi _ ij).1 | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) := by
cases' Classical.em (LimZero f) with h h <;> simp [*]
rcases abv_pos_of_not_limZero h with ⟨K, K0, hK⟩
rcases exists_forall_ge_and hK (f.cauchy₃ K0) with ⟨i, hi⟩
refine' (le_total 0 (f i)).imp _ _ <;>
refine' fun h => ⟨K, K0, i, fun ... | Mathlib.Data.Real.CauSeq.702_0.b72JnnMmhSc5wVP | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) | Mathlib_Data_Real_CauSeq |
case inr.intro.intro.intro.refine'_2
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
h✝ : ¬LimZero f
K : α
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ |↑f j|
i : ℕ
hi : ∀ j ≥ i, K ≤ |↑f j| ∧ ∀ k ≥ j, |↑f k - ↑f j| < K
h : ↑f i ≤ 0
j : ℕ
ij : j ≥ i
⊢ K ≤ ↑(-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... | have := (hi _ ij).1 | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) := by
cases' Classical.em (LimZero f) with h h <;> simp [*]
rcases abv_pos_of_not_limZero h with ⟨K, K0, hK⟩
rcases exists_forall_ge_and hK (f.cauchy₃ K0) with ⟨i, hi⟩
refine' (le_total 0 (f i)).imp _ _ <;>
refine' fun h => ⟨K, K0, i, fun ... | Mathlib.Data.Real.CauSeq.702_0.b72JnnMmhSc5wVP | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) | Mathlib_Data_Real_CauSeq |
case inr.intro.intro.intro.refine'_1
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
h✝ : ¬LimZero f
K : α
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ |↑f j|
i : ℕ
hi : ∀ j ≥ i, K ≤ |↑f j| ∧ ∀ k ≥ j, |↑f k - ↑f j| < K
h : 0 ≤ ↑f i
j : ℕ
ij : j ≥ i
this : K ≤ |↑f j|
⊢ K ≤ ↑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 trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) := by
cases' Classical.em (LimZero f) with h h <;> simp [*]
rcases abv_pos_of_not_limZero h with ⟨K, K0, hK⟩
rcases exists_forall_ge_and hK (f.cauchy₃ K0) with ⟨i, hi⟩
refine' (le_total 0 (f i)).imp _ _ <;>
refine' fun h => ⟨K, K0, i, fun ... | Mathlib.Data.Real.CauSeq.702_0.b72JnnMmhSc5wVP | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) | Mathlib_Data_Real_CauSeq |
case inr.intro.intro.intro.refine'_2
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
h✝ : ¬LimZero f
K : α
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ |↑f j|
i : ℕ
hi : ∀ j ≥ i, K ≤ |↑f j| ∧ ∀ k ≥ j, |↑f k - ↑f j| < K
h : ↑f i ≤ 0
j : ℕ
ij : j ≥ i
this : K ≤ |↑f j|
⊢ K ≤ ↑(-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 trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) := by
cases' Classical.em (LimZero f) with h h <;> simp [*]
rcases abv_pos_of_not_limZero h with ⟨K, K0, hK⟩
rcases exists_forall_ge_and hK (f.cauchy₃ K0) with ⟨i, hi⟩
refine' (le_total 0 (f i)).imp _ _ <;>
refine' fun h => ⟨K, K0, i, fun ... | Mathlib.Data.Real.CauSeq.702_0.b72JnnMmhSc5wVP | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) | Mathlib_Data_Real_CauSeq |
case inr.intro.intro.intro.refine'_1.intro
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
h✝ : ¬LimZero f
K : α
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ |↑f j|
i : ℕ
hi : ∀ j ≥ i, K ≤ |↑f j| ∧ ∀ k ≥ j, |↑f k - ↑f j| < K
h : 0 ≤ ↑f i
j : ℕ
ij : j ≥ i
this : K ≤ |↑f j|
h₁ : K ≤ |↑f i|
h₂ : ∀ k ≥ i, |↑f k - ↑f i| < 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... | rwa [abs_of_nonneg] at this | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) := by
cases' Classical.em (LimZero f) with h h <;> simp [*]
rcases abv_pos_of_not_limZero h with ⟨K, K0, hK⟩
rcases exists_forall_ge_and hK (f.cauchy₃ K0) with ⟨i, hi⟩
refine' (le_total 0 (f i)).imp _ _ <;>
refine' fun h => ⟨K, K0, i, fun ... | Mathlib.Data.Real.CauSeq.702_0.b72JnnMmhSc5wVP | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) | Mathlib_Data_Real_CauSeq |
case inr.intro.intro.intro.refine'_1.intro
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
h✝ : ¬LimZero f
K : α
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ |↑f j|
i : ℕ
hi : ∀ j ≥ i, K ≤ |↑f j| ∧ ∀ k ≥ j, |↑f k - ↑f j| < K
h : 0 ≤ ↑f i
j : ℕ
ij : j ≥ i
this : K ≤ |↑f j|
h₁ : K ≤ |↑f i|
h₂ : ∀ k ≥ i, |↑f k - ↑f i| < 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... | rw [abs_of_nonneg h] at h₁ | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) := by
cases' Classical.em (LimZero f) with h h <;> simp [*]
rcases abv_pos_of_not_limZero h with ⟨K, K0, hK⟩
rcases exists_forall_ge_and hK (f.cauchy₃ K0) with ⟨i, hi⟩
refine' (le_total 0 (f i)).imp _ _ <;>
refine' fun h => ⟨K, K0, i, fun ... | Mathlib.Data.Real.CauSeq.702_0.b72JnnMmhSc5wVP | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) | Mathlib_Data_Real_CauSeq |
case inr.intro.intro.intro.refine'_1.intro
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
h✝ : ¬LimZero f
K : α
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ |↑f j|
i : ℕ
hi : ∀ j ≥ i, K ≤ |↑f j| ∧ ∀ k ≥ j, |↑f k - ↑f j| < K
h : 0 ≤ ↑f i
j : ℕ
ij : j ≥ i
this : K ≤ |↑f j|
h₁ : K ≤ ↑f i
h₂ : ∀ k ≥ i, |↑f k - ↑f i| < 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... | exact
(le_add_iff_nonneg_right _).1
(le_trans h₁ <| neg_le_sub_iff_le_add'.1 <| le_of_lt (abs_lt.1 <| h₂ _ ij).1) | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) := by
cases' Classical.em (LimZero f) with h h <;> simp [*]
rcases abv_pos_of_not_limZero h with ⟨K, K0, hK⟩
rcases exists_forall_ge_and hK (f.cauchy₃ K0) with ⟨i, hi⟩
refine' (le_total 0 (f i)).imp _ _ <;>
refine' fun h => ⟨K, K0, i, fun ... | Mathlib.Data.Real.CauSeq.702_0.b72JnnMmhSc5wVP | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) | Mathlib_Data_Real_CauSeq |
case inr.intro.intro.intro.refine'_2.intro
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
h✝ : ¬LimZero f
K : α
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ |↑f j|
i : ℕ
hi : ∀ j ≥ i, K ≤ |↑f j| ∧ ∀ k ≥ j, |↑f k - ↑f j| < K
h : ↑f i ≤ 0
j : ℕ
ij : j ≥ i
this : K ≤ |↑f j|
h₁ : K ≤ |↑f i|
h₂ : ∀ k ≥ i, |↑f k - ↑f i| < 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... | rwa [abs_of_nonpos] at this | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) := by
cases' Classical.em (LimZero f) with h h <;> simp [*]
rcases abv_pos_of_not_limZero h with ⟨K, K0, hK⟩
rcases exists_forall_ge_and hK (f.cauchy₃ K0) with ⟨i, hi⟩
refine' (le_total 0 (f i)).imp _ _ <;>
refine' fun h => ⟨K, K0, i, fun ... | Mathlib.Data.Real.CauSeq.702_0.b72JnnMmhSc5wVP | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) | Mathlib_Data_Real_CauSeq |
case inr.intro.intro.intro.refine'_2.intro
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
h✝ : ¬LimZero f
K : α
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ |↑f j|
i : ℕ
hi : ∀ j ≥ i, K ≤ |↑f j| ∧ ∀ k ≥ j, |↑f k - ↑f j| < K
h : ↑f i ≤ 0
j : ℕ
ij : j ≥ i
this : K ≤ |↑f j|
h₁ : K ≤ |↑f i|
h₂ : ∀ k ≥ i, |↑f k - ↑f i| < 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... | rw [abs_of_nonpos h] at h₁ | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) := by
cases' Classical.em (LimZero f) with h h <;> simp [*]
rcases abv_pos_of_not_limZero h with ⟨K, K0, hK⟩
rcases exists_forall_ge_and hK (f.cauchy₃ K0) with ⟨i, hi⟩
refine' (le_total 0 (f i)).imp _ _ <;>
refine' fun h => ⟨K, K0, i, fun ... | Mathlib.Data.Real.CauSeq.702_0.b72JnnMmhSc5wVP | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) | Mathlib_Data_Real_CauSeq |
case inr.intro.intro.intro.refine'_2.intro
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
h✝ : ¬LimZero f
K : α
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ |↑f j|
i : ℕ
hi : ∀ j ≥ i, K ≤ |↑f j| ∧ ∀ k ≥ j, |↑f k - ↑f j| < K
h : ↑f i ≤ 0
j : ℕ
ij : j ≥ i
this : K ≤ |↑f j|
h₁ : K ≤ -↑f i
h₂ : ∀ k ≥ i, |↑f k - ↑f i| < 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... | rw [← sub_le_sub_iff_right, zero_sub] | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) := by
cases' Classical.em (LimZero f) with h h <;> simp [*]
rcases abv_pos_of_not_limZero h with ⟨K, K0, hK⟩
rcases exists_forall_ge_and hK (f.cauchy₃ K0) with ⟨i, hi⟩
refine' (le_total 0 (f i)).imp _ _ <;>
refine' fun h => ⟨K, K0, i, fun ... | Mathlib.Data.Real.CauSeq.702_0.b72JnnMmhSc5wVP | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) | Mathlib_Data_Real_CauSeq |
case inr.intro.intro.intro.refine'_2.intro
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
h✝ : ¬LimZero f
K : α
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ |↑f j|
i : ℕ
hi : ∀ j ≥ i, K ≤ |↑f j| ∧ ∀ k ≥ j, |↑f k - ↑f j| < K
h : ↑f i ≤ 0
j : ℕ
ij : j ≥ i
this : K ≤ |↑f j|
h₁ : K ≤ -↑f i
h₂ : ∀ k ≥ i, |↑f k - ↑f i| < 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... | exact le_trans (le_of_lt (abs_lt.1 <| h₂ _ ij).2) h₁ | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) := by
cases' Classical.em (LimZero f) with h h <;> simp [*]
rcases abv_pos_of_not_limZero h with ⟨K, K0, hK⟩
rcases exists_forall_ge_and hK (f.cauchy₃ K0) with ⟨i, hi⟩
refine' (le_total 0 (f i)).imp _ _ <;>
refine' fun h => ⟨K, K0, i, fun ... | Mathlib.Data.Real.CauSeq.702_0.b72JnnMmhSc5wVP | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
f g h : CauSeq α abs
fg : f < g
gh : g ≈ h
⊢ Pos (h - 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... | convert pos_add_limZero fg (neg_limZero gh) using 1 | theorem lt_of_lt_of_eq {f g h : CauSeq α abs} (fg : f < g) (gh : g ≈ h) : f < h :=
show Pos (h - f) by
| Mathlib.Data.Real.CauSeq.727_0.b72JnnMmhSc5wVP | theorem lt_of_lt_of_eq {f g h : CauSeq α abs} (fg : f < g) (gh : g ≈ h) : f < h | Mathlib_Data_Real_CauSeq |
case h.e'_3
α : Type u_1
inst✝ : LinearOrderedField α
f g h : CauSeq α abs
fg : f < g
gh : g ≈ h
⊢ h - f = g - f + -(g - h) | /-
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 lt_of_lt_of_eq {f g h : CauSeq α abs} (fg : f < g) (gh : g ≈ h) : f < h :=
show Pos (h - f) by
convert pos_add_limZero fg (neg_limZero gh) using 1
| Mathlib.Data.Real.CauSeq.727_0.b72JnnMmhSc5wVP | theorem lt_of_lt_of_eq {f g h : CauSeq α abs} (fg : f < g) (gh : g ≈ h) : f < h | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
f g h : CauSeq α abs
fg : f ≈ g
gh : g < h
⊢ f < h | /-
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 := pos_add_limZero gh (neg_limZero fg) | theorem lt_of_eq_of_lt {f g h : CauSeq α abs} (fg : f ≈ g) (gh : g < h) : f < h := by
| Mathlib.Data.Real.CauSeq.733_0.b72JnnMmhSc5wVP | theorem lt_of_eq_of_lt {f g h : CauSeq α abs} (fg : f ≈ g) (gh : g < h) : f < h | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
f g h : CauSeq α abs
fg : f ≈ g
gh : g < h
this : Pos (h - g + -(f - g))
⊢ f < h | /-
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_eq_add_neg, sub_sub_sub_cancel_right] at this | theorem lt_of_eq_of_lt {f g h : CauSeq α abs} (fg : f ≈ g) (gh : g < h) : f < h := by
have := pos_add_limZero gh (neg_limZero fg)
| Mathlib.Data.Real.CauSeq.733_0.b72JnnMmhSc5wVP | theorem lt_of_eq_of_lt {f g h : CauSeq α abs} (fg : f ≈ g) (gh : g < h) : f < h | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
f g h : CauSeq α abs
fg : f < g
gh : g < h
⊢ Pos (h - 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... | convert add_pos fg gh using 1 | theorem lt_trans {f g h : CauSeq α abs} (fg : f < g) (gh : g < h) : f < h :=
show Pos (h - f) by
| Mathlib.Data.Real.CauSeq.738_0.b72JnnMmhSc5wVP | theorem lt_trans {f g h : CauSeq α abs} (fg : f < g) (gh : g < h) : f < h | Mathlib_Data_Real_CauSeq |
case h.e'_3
α : Type u_1
inst✝ : LinearOrderedField α
f g h : CauSeq α abs
fg : f < g
gh : g < h
⊢ h - f = g - f + (h - 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... | simp | theorem lt_trans {f g h : CauSeq α abs} (fg : f < g) (gh : g < h) : f < h :=
show Pos (h - f) by
convert add_pos fg gh using 1
| Mathlib.Data.Real.CauSeq.738_0.b72JnnMmhSc5wVP | theorem lt_trans {f g h : CauSeq α abs} (fg : f < g) (gh : g < h) : f < h | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
x✝ : f < f
h : f < f := x✝
⊢ 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] | theorem lt_irrefl {f : CauSeq α abs} : ¬f < f
| h => not_limZero_of_pos h (by | Mathlib.Data.Real.CauSeq.744_0.b72JnnMmhSc5wVP | theorem lt_irrefl {f : CauSeq α abs} : ¬f < f
| h => not_limZero_of_pos h (by simp [zero_limZero]) | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
f g : CauSeq α abs
h✝ : LimZero (g - f) ∨ Pos (-(g - f))
h : Pos (-(g - f))
⊢ 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... | rwa [neg_sub] at h | theorem lt_total (f g : CauSeq α abs) : f < g ∨ f ≈ g ∨ g < f :=
(trichotomy (g - f)).imp_right fun h =>
h.imp (fun h => Setoid.symm h) fun h => by | Mathlib.Data.Real.CauSeq.774_0.b72JnnMmhSc5wVP | theorem lt_total (f g : CauSeq α abs) : f < g ∨ f ≈ g ∨ g < f | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
x y : α
⊢ Pos (const y - const x) ↔ x < y | /-
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 [← const_sub, const_pos, sub_pos] | theorem const_lt {x y : α} : const x < const y ↔ x < y :=
show Pos _ ↔ _ by | Mathlib.Data.Real.CauSeq.783_0.b72JnnMmhSc5wVP | theorem const_lt {x y : α} : const x < const y ↔ x < y | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
x y : α
⊢ const x ≤ const y ↔ x ≤ y | /-
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 [le_iff_lt_or_eq] | theorem const_le {x y : α} : const x ≤ const y ↔ x ≤ y := by
| Mathlib.Data.Real.CauSeq.787_0.b72JnnMmhSc5wVP | theorem const_le {x y : α} : const x ≤ const y ↔ x ≤ y | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
x y : α
⊢ const x ≤ const y ↔ x < y ∨ x = y | /-
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 or_congr const_lt const_equiv | theorem const_le {x y : α} : const x ≤ const y ↔ x ≤ y := by
rw [le_iff_lt_or_eq]; | Mathlib.Data.Real.CauSeq.787_0.b72JnnMmhSc5wVP | theorem const_le {x y : α} : const x ≤ const y ↔ x ≤ y | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
K : α
H : ∀ (i : ℕ), |↑f i| < K
i : ℕ
x✝ : i ≥ 0
⊢ 1 ≤ ↑(const (K + 1) - f) 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... | rw [sub_apply, const_apply, le_sub_iff_add_le', add_le_add_iff_right] | theorem exists_gt (f : CauSeq α abs) : ∃ a : α, f < const a :=
let ⟨K, H⟩ := f.bounded
⟨K + 1, 1, zero_lt_one, 0, fun i _ => by
| Mathlib.Data.Real.CauSeq.800_0.b72JnnMmhSc5wVP | theorem exists_gt (f : CauSeq α abs) : ∃ a : α, f < const a | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
K : α
H : ∀ (i : ℕ), |↑f i| < K
i : ℕ
x✝ : i ≥ 0
⊢ ↑f i ≤ 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... | exact le_of_lt (abs_lt.1 (H _)).2 | theorem exists_gt (f : CauSeq α abs) : ∃ a : α, f < const a :=
let ⟨K, H⟩ := f.bounded
⟨K + 1, 1, zero_lt_one, 0, fun i _ => by
rw [sub_apply, const_apply, le_sub_iff_add_le', add_le_add_iff_right]
| Mathlib.Data.Real.CauSeq.800_0.b72JnnMmhSc5wVP | theorem exists_gt (f : CauSeq α abs) : ∃ a : α, f < const a | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
a : α
h : -f < const a
⊢ Pos (f - const (-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... | rwa [const_neg, sub_neg_eq_add, add_comm, ← sub_neg_eq_add] | theorem exists_lt (f : CauSeq α abs) : ∃ a : α, const a < f :=
let ⟨a, h⟩ := (-f).exists_gt
⟨-a, show Pos _ by | Mathlib.Data.Real.CauSeq.807_0.b72JnnMmhSc5wVP | theorem exists_lt (f : CauSeq α abs) : ∃ a : α, const a < f | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
f g : CauSeq α abs
hf : LimZero f
hg : LimZero g
ε : α
ε0 : ε > 0
i : ℕ
H : ∀ j ≥ i, |↑f j| < ε ∧ |↑g j| < ε
j : ℕ
ij : j ≥ i
⊢ |↑(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 sup_limZero {f g : CauSeq α abs} (hf : LimZero f) (hg : LimZero g) : LimZero (f ⊔ g)
| ε, ε0 =>
(exists_forall_ge_and (hf _ ε0) (hg _ ε0)).imp fun i H j ij => by
| Mathlib.Data.Real.CauSeq.848_0.b72JnnMmhSc5wVP | theorem sup_limZero {f g : CauSeq α abs} (hf : LimZero f) (hg : LimZero g) : LimZero (f ⊔ g)
| ε, ε0 =>
(exists_forall_ge_and (hf _ ε0) (hg _ ε0)).imp fun i H j ij => by
let ⟨H₁, H₂⟩ | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
f g : CauSeq α abs
hf : LimZero f
hg : LimZero g
ε : α
ε0 : ε > 0
i : ℕ
H : ∀ j ≥ i, |↑f j| < ε ∧ |↑g j| < ε
j : ℕ
ij : j ≥ i
H₁ : |↑f j| < ε
H₂ : |↑g j| < ε
⊢ |↑(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... | rw [abs_lt] at H₁ H₂ ⊢ | theorem sup_limZero {f g : CauSeq α abs} (hf : LimZero f) (hg : LimZero g) : LimZero (f ⊔ g)
| ε, ε0 =>
(exists_forall_ge_and (hf _ ε0) (hg _ ε0)).imp fun i H j ij => by
let ⟨H₁, H₂⟩ := H _ ij
| Mathlib.Data.Real.CauSeq.848_0.b72JnnMmhSc5wVP | theorem sup_limZero {f g : CauSeq α abs} (hf : LimZero f) (hg : LimZero g) : LimZero (f ⊔ g)
| ε, ε0 =>
(exists_forall_ge_and (hf _ ε0) (hg _ ε0)).imp fun i H j ij => by
let ⟨H₁, H₂⟩ | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
f g : CauSeq α abs
hf : LimZero f
hg : LimZero g
ε : α
ε0 : ε > 0
i : ℕ
H : ∀ j ≥ i, |↑f j| < ε ∧ |↑g j| < ε
j : ℕ
ij : j ≥ i
H₁ : -ε < ↑f j ∧ ↑f j < ε
H₂ : -ε < ↑g j ∧ ↑g j < ε
⊢ -ε < ↑(f ⊔ g) j ∧ ↑(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... | exact ⟨lt_sup_iff.mpr (Or.inl H₁.1), sup_lt_iff.mpr ⟨H₁.2, H₂.2⟩⟩ | theorem sup_limZero {f g : CauSeq α abs} (hf : LimZero f) (hg : LimZero g) : LimZero (f ⊔ g)
| ε, ε0 =>
(exists_forall_ge_and (hf _ ε0) (hg _ ε0)).imp fun i H j ij => by
let ⟨H₁, H₂⟩ := H _ ij
rw [abs_lt] at H₁ H₂ ⊢
| Mathlib.Data.Real.CauSeq.848_0.b72JnnMmhSc5wVP | theorem sup_limZero {f g : CauSeq α abs} (hf : LimZero f) (hg : LimZero g) : LimZero (f ⊔ g)
| ε, ε0 =>
(exists_forall_ge_and (hf _ ε0) (hg _ ε0)).imp fun i H j ij => by
let ⟨H₁, H₂⟩ | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
f g : CauSeq α abs
hf : LimZero f
hg : LimZero g
ε : α
ε0 : ε > 0
i : ℕ
H : ∀ j ≥ i, |↑f j| < ε ∧ |↑g j| < ε
j : ℕ
ij : j ≥ i
⊢ |↑(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 inf_limZero {f g : CauSeq α abs} (hf : LimZero f) (hg : LimZero g) : LimZero (f ⊓ g)
| ε, ε0 =>
(exists_forall_ge_and (hf _ ε0) (hg _ ε0)).imp fun i H j ij => by
| Mathlib.Data.Real.CauSeq.856_0.b72JnnMmhSc5wVP | theorem inf_limZero {f g : CauSeq α abs} (hf : LimZero f) (hg : LimZero g) : LimZero (f ⊓ g)
| ε, ε0 =>
(exists_forall_ge_and (hf _ ε0) (hg _ ε0)).imp fun i H j ij => by
let ⟨H₁, H₂⟩ | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
f g : CauSeq α abs
hf : LimZero f
hg : LimZero g
ε : α
ε0 : ε > 0
i : ℕ
H : ∀ j ≥ i, |↑f j| < ε ∧ |↑g j| < ε
j : ℕ
ij : j ≥ i
H₁ : |↑f j| < ε
H₂ : |↑g j| < ε
⊢ |↑(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... | rw [abs_lt] at H₁ H₂ ⊢ | theorem inf_limZero {f g : CauSeq α abs} (hf : LimZero f) (hg : LimZero g) : LimZero (f ⊓ g)
| ε, ε0 =>
(exists_forall_ge_and (hf _ ε0) (hg _ ε0)).imp fun i H j ij => by
let ⟨H₁, H₂⟩ := H _ ij
| Mathlib.Data.Real.CauSeq.856_0.b72JnnMmhSc5wVP | theorem inf_limZero {f g : CauSeq α abs} (hf : LimZero f) (hg : LimZero g) : LimZero (f ⊓ g)
| ε, ε0 =>
(exists_forall_ge_and (hf _ ε0) (hg _ ε0)).imp fun i H j ij => by
let ⟨H₁, H₂⟩ | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
f g : CauSeq α abs
hf : LimZero f
hg : LimZero g
ε : α
ε0 : ε > 0
i : ℕ
H : ∀ j ≥ i, |↑f j| < ε ∧ |↑g j| < ε
j : ℕ
ij : j ≥ i
H₁ : -ε < ↑f j ∧ ↑f j < ε
H₂ : -ε < ↑g j ∧ ↑g j < ε
⊢ -ε < ↑(f ⊓ g) j ∧ ↑(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... | exact ⟨lt_inf_iff.mpr ⟨H₁.1, H₂.1⟩, inf_lt_iff.mpr (Or.inl H₁.2)⟩ | theorem inf_limZero {f g : CauSeq α abs} (hf : LimZero f) (hg : LimZero g) : LimZero (f ⊓ g)
| ε, ε0 =>
(exists_forall_ge_and (hf _ ε0) (hg _ ε0)).imp fun i H j ij => by
let ⟨H₁, H₂⟩ := H _ ij
rw [abs_lt] at H₁ H₂ ⊢
| Mathlib.Data.Real.CauSeq.856_0.b72JnnMmhSc5wVP | theorem inf_limZero {f g : CauSeq α abs} (hf : LimZero f) (hg : LimZero g) : LimZero (f ⊓ g)
| ε, ε0 =>
(exists_forall_ge_and (hf _ ε0) (hg _ ε0)).imp fun i H j ij => by
let ⟨H₁, H₂⟩ | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
a₁ b₁ a₂ b₂ : CauSeq α abs
ha : a₁ ≈ a₂
hb : b₁ ≈ b₂
⊢ a₁ ⊔ b₁ ≈ a₂ ⊔ b₂ | /-
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 ε ε0 | theorem sup_equiv_sup {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :
a₁ ⊔ b₁ ≈ a₂ ⊔ b₂ := by
| Mathlib.Data.Real.CauSeq.864_0.b72JnnMmhSc5wVP | theorem sup_equiv_sup {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :
a₁ ⊔ b₁ ≈ a₂ ⊔ b₂ | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
a₁ b₁ a₂ b₂ : CauSeq α abs
ha : a₁ ≈ a₂
hb : b₁ ≈ b₂
ε : α
ε0 : ε > 0
⊢ ∃ i, ∀ j ≥ i, |↑(a₁ ⊔ b₁ - a₂ ⊔ b₂) 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... | obtain ⟨ai, hai⟩ := ha ε ε0 | theorem sup_equiv_sup {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :
a₁ ⊔ b₁ ≈ a₂ ⊔ b₂ := by
intro ε ε0
| Mathlib.Data.Real.CauSeq.864_0.b72JnnMmhSc5wVP | theorem sup_equiv_sup {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :
a₁ ⊔ b₁ ≈ a₂ ⊔ b₂ | Mathlib_Data_Real_CauSeq |
case intro
α : Type u_1
inst✝ : LinearOrderedField α
a₁ b₁ a₂ b₂ : CauSeq α abs
ha : a₁ ≈ a₂
hb : b₁ ≈ b₂
ε : α
ε0 : ε > 0
ai : ℕ
hai : ∀ j ≥ ai, |↑(a₁ - a₂) j| < ε
⊢ ∃ i, ∀ j ≥ i, |↑(a₁ ⊔ b₁ - a₂ ⊔ b₂) 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... | obtain ⟨bi, hbi⟩ := hb ε ε0 | theorem sup_equiv_sup {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :
a₁ ⊔ b₁ ≈ a₂ ⊔ b₂ := by
intro ε ε0
obtain ⟨ai, hai⟩ := ha ε ε0
| Mathlib.Data.Real.CauSeq.864_0.b72JnnMmhSc5wVP | theorem sup_equiv_sup {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :
a₁ ⊔ b₁ ≈ a₂ ⊔ b₂ | Mathlib_Data_Real_CauSeq |
case intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
a₁ b₁ a₂ b₂ : CauSeq α abs
ha : a₁ ≈ a₂
hb : b₁ ≈ b₂
ε : α
ε0 : ε > 0
ai : ℕ
hai : ∀ j ≥ ai, |↑(a₁ - a₂) j| < ε
bi : ℕ
hbi : ∀ j ≥ bi, |↑(b₁ - b₂) j| < ε
⊢ ∃ i, ∀ j ≥ i, |↑(a₁ ⊔ b₁ - a₂ ⊔ b₂) 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... | exact
⟨ai ⊔ bi, fun i hi =>
(abs_max_sub_max_le_max (a₁ i) (b₁ i) (a₂ i) (b₂ i)).trans_lt
(max_lt (hai i (sup_le_iff.mp hi).1) (hbi i (sup_le_iff.mp hi).2))⟩ | theorem sup_equiv_sup {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :
a₁ ⊔ b₁ ≈ a₂ ⊔ b₂ := by
intro ε ε0
obtain ⟨ai, hai⟩ := ha ε ε0
obtain ⟨bi, hbi⟩ := hb ε ε0
| Mathlib.Data.Real.CauSeq.864_0.b72JnnMmhSc5wVP | theorem sup_equiv_sup {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :
a₁ ⊔ b₁ ≈ a₂ ⊔ b₂ | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
a₁ b₁ a₂ b₂ : CauSeq α abs
ha : a₁ ≈ a₂
hb : b₁ ≈ b₂
⊢ a₁ ⊓ b₁ ≈ a₂ ⊓ b₂ | /-
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 ε ε0 | theorem inf_equiv_inf {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :
a₁ ⊓ b₁ ≈ a₂ ⊓ b₂ := by
| Mathlib.Data.Real.CauSeq.875_0.b72JnnMmhSc5wVP | theorem inf_equiv_inf {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :
a₁ ⊓ b₁ ≈ a₂ ⊓ b₂ | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
a₁ b₁ a₂ b₂ : CauSeq α abs
ha : a₁ ≈ a₂
hb : b₁ ≈ b₂
ε : α
ε0 : ε > 0
⊢ ∃ i, ∀ j ≥ i, |↑(a₁ ⊓ b₁ - a₂ ⊓ b₂) 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... | obtain ⟨ai, hai⟩ := ha ε ε0 | theorem inf_equiv_inf {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :
a₁ ⊓ b₁ ≈ a₂ ⊓ b₂ := by
intro ε ε0
| Mathlib.Data.Real.CauSeq.875_0.b72JnnMmhSc5wVP | theorem inf_equiv_inf {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :
a₁ ⊓ b₁ ≈ a₂ ⊓ b₂ | Mathlib_Data_Real_CauSeq |
case intro
α : Type u_1
inst✝ : LinearOrderedField α
a₁ b₁ a₂ b₂ : CauSeq α abs
ha : a₁ ≈ a₂
hb : b₁ ≈ b₂
ε : α
ε0 : ε > 0
ai : ℕ
hai : ∀ j ≥ ai, |↑(a₁ - a₂) j| < ε
⊢ ∃ i, ∀ j ≥ i, |↑(a₁ ⊓ b₁ - a₂ ⊓ b₂) 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... | obtain ⟨bi, hbi⟩ := hb ε ε0 | theorem inf_equiv_inf {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :
a₁ ⊓ b₁ ≈ a₂ ⊓ b₂ := by
intro ε ε0
obtain ⟨ai, hai⟩ := ha ε ε0
| Mathlib.Data.Real.CauSeq.875_0.b72JnnMmhSc5wVP | theorem inf_equiv_inf {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :
a₁ ⊓ b₁ ≈ a₂ ⊓ b₂ | Mathlib_Data_Real_CauSeq |
case intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
a₁ b₁ a₂ b₂ : CauSeq α abs
ha : a₁ ≈ a₂
hb : b₁ ≈ b₂
ε : α
ε0 : ε > 0
ai : ℕ
hai : ∀ j ≥ ai, |↑(a₁ - a₂) j| < ε
bi : ℕ
hbi : ∀ j ≥ bi, |↑(b₁ - b₂) j| < ε
⊢ ∃ i, ∀ j ≥ i, |↑(a₁ ⊓ b₁ - a₂ ⊓ b₂) 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... | exact
⟨ai ⊔ bi, fun i hi =>
(abs_min_sub_min_le_max (a₁ i) (b₁ i) (a₂ i) (b₂ i)).trans_lt
(max_lt (hai i (sup_le_iff.mp hi).1) (hbi i (sup_le_iff.mp hi).2))⟩ | theorem inf_equiv_inf {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :
a₁ ⊓ b₁ ≈ a₂ ⊓ b₂ := by
intro ε ε0
obtain ⟨ai, hai⟩ := ha ε ε0
obtain ⟨bi, hbi⟩ := hb ε ε0
| Mathlib.Data.Real.CauSeq.875_0.b72JnnMmhSc5wVP | theorem inf_equiv_inf {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :
a₁ ⊓ b₁ ≈ a₂ ⊓ b₂ | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
a b c : CauSeq α abs
ha : a < c
hb : b < c
⊢ a ⊔ b < c | /-
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... | obtain ⟨⟨εa, εa0, ia, ha⟩, ⟨εb, εb0, ib, hb⟩⟩ := ha, hb | protected theorem sup_lt {a b c : CauSeq α abs} (ha : a < c) (hb : b < c) : a ⊔ b < c := by
| Mathlib.Data.Real.CauSeq.886_0.b72JnnMmhSc5wVP | protected theorem sup_lt {a b c : CauSeq α abs} (ha : a < c) (hb : b < c) : a ⊔ b < c | Mathlib_Data_Real_CauSeq |
case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
a b c : CauSeq α abs
εa : α
εa0 : εa > 0
ia : ℕ
ha : ∀ j ≥ ia, εa ≤ ↑(c - a) j
εb : α
εb0 : εb > 0
ib : ℕ
hb : ∀ j ≥ ib, εb ≤ ↑(c - b) j
⊢ a ⊔ b < c | /-
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' ⟨εa ⊓ εb, lt_inf_iff.mpr ⟨εa0, εb0⟩, ia ⊔ ib, fun i hi => _⟩ | protected theorem sup_lt {a b c : CauSeq α abs} (ha : a < c) (hb : b < c) : a ⊔ b < c := by
obtain ⟨⟨εa, εa0, ia, ha⟩, ⟨εb, εb0, ib, hb⟩⟩ := ha, hb
| Mathlib.Data.Real.CauSeq.886_0.b72JnnMmhSc5wVP | protected theorem sup_lt {a b c : CauSeq α abs} (ha : a < c) (hb : b < c) : a ⊔ b < c | Mathlib_Data_Real_CauSeq |
case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
a b c : CauSeq α abs
εa : α
εa0 : εa > 0
ia : ℕ
ha : ∀ j ≥ ia, εa ≤ ↑(c - a) j
εb : α
εb0 : εb > 0
ib : ℕ
hb : ∀ j ≥ ib, εb ≤ ↑(c - b) j
i : ℕ
hi : i ≥ ia ⊔ ib
⊢ εa ⊓ εb ≤ ↑(c - a ⊔ b) 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... | have := min_le_min (ha _ (sup_le_iff.mp hi).1) (hb _ (sup_le_iff.mp hi).2) | protected theorem sup_lt {a b c : CauSeq α abs} (ha : a < c) (hb : b < c) : a ⊔ b < c := by
obtain ⟨⟨εa, εa0, ia, ha⟩, ⟨εb, εb0, ib, hb⟩⟩ := ha, hb
refine' ⟨εa ⊓ εb, lt_inf_iff.mpr ⟨εa0, εb0⟩, ia ⊔ ib, fun i hi => _⟩
| Mathlib.Data.Real.CauSeq.886_0.b72JnnMmhSc5wVP | protected theorem sup_lt {a b c : CauSeq α abs} (ha : a < c) (hb : b < c) : a ⊔ b < c | Mathlib_Data_Real_CauSeq |
case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
a b c : CauSeq α abs
εa : α
εa0 : εa > 0
ia : ℕ
ha : ∀ j ≥ ia, εa ≤ ↑(c - a) j
εb : α
εb0 : εb > 0
ib : ℕ
hb : ∀ j ≥ ib, εb ≤ ↑(c - b) j
i : ℕ
hi : i ≥ ia ⊔ ib
this : min εa εb ≤ min (↑(c - a) i) (↑(c - b) i)
⊢ εa ⊓ εb ≤ ↑(c - a ⊔ b) 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... | exact this.trans_eq (min_sub_sub_left _ _ _) | protected theorem sup_lt {a b c : CauSeq α abs} (ha : a < c) (hb : b < c) : a ⊔ b < c := by
obtain ⟨⟨εa, εa0, ia, ha⟩, ⟨εb, εb0, ib, hb⟩⟩ := ha, hb
refine' ⟨εa ⊓ εb, lt_inf_iff.mpr ⟨εa0, εb0⟩, ia ⊔ ib, fun i hi => _⟩
have := min_le_min (ha _ (sup_le_iff.mp hi).1) (hb _ (sup_le_iff.mp hi).2)
| Mathlib.Data.Real.CauSeq.886_0.b72JnnMmhSc5wVP | protected theorem sup_lt {a b c : CauSeq α abs} (ha : a < c) (hb : b < c) : a ⊔ b < c | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
a b c : CauSeq α abs
hb : a < b
hc : a < c
⊢ a < b ⊓ c | /-
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... | obtain ⟨⟨εb, εb0, ib, hb⟩, ⟨εc, εc0, ic, hc⟩⟩ := hb, hc | protected theorem lt_inf {a b c : CauSeq α abs} (hb : a < b) (hc : a < c) : a < b ⊓ c := by
| Mathlib.Data.Real.CauSeq.893_0.b72JnnMmhSc5wVP | protected theorem lt_inf {a b c : CauSeq α abs} (hb : a < b) (hc : a < c) : a < b ⊓ c | Mathlib_Data_Real_CauSeq |
case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
a b c : CauSeq α abs
εb : α
εb0 : εb > 0
ib : ℕ
hb : ∀ j ≥ ib, εb ≤ ↑(b - a) j
εc : α
εc0 : εc > 0
ic : ℕ
hc : ∀ j ≥ ic, εc ≤ ↑(c - a) j
⊢ a < b ⊓ c | /-
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' ⟨εb ⊓ εc, lt_inf_iff.mpr ⟨εb0, εc0⟩, ib ⊔ ic, fun i hi => _⟩ | protected theorem lt_inf {a b c : CauSeq α abs} (hb : a < b) (hc : a < c) : a < b ⊓ c := by
obtain ⟨⟨εb, εb0, ib, hb⟩, ⟨εc, εc0, ic, hc⟩⟩ := hb, hc
| Mathlib.Data.Real.CauSeq.893_0.b72JnnMmhSc5wVP | protected theorem lt_inf {a b c : CauSeq α abs} (hb : a < b) (hc : a < c) : a < b ⊓ c | Mathlib_Data_Real_CauSeq |
case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
a b c : CauSeq α abs
εb : α
εb0 : εb > 0
ib : ℕ
hb : ∀ j ≥ ib, εb ≤ ↑(b - a) j
εc : α
εc0 : εc > 0
ic : ℕ
hc : ∀ j ≥ ic, εc ≤ ↑(c - a) j
i : ℕ
hi : i ≥ ib ⊔ ic
⊢ εb ⊓ εc ≤ ↑(b ⊓ c - a) 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... | have := min_le_min (hb _ (sup_le_iff.mp hi).1) (hc _ (sup_le_iff.mp hi).2) | protected theorem lt_inf {a b c : CauSeq α abs} (hb : a < b) (hc : a < c) : a < b ⊓ c := by
obtain ⟨⟨εb, εb0, ib, hb⟩, ⟨εc, εc0, ic, hc⟩⟩ := hb, hc
refine' ⟨εb ⊓ εc, lt_inf_iff.mpr ⟨εb0, εc0⟩, ib ⊔ ic, fun i hi => _⟩
| Mathlib.Data.Real.CauSeq.893_0.b72JnnMmhSc5wVP | protected theorem lt_inf {a b c : CauSeq α abs} (hb : a < b) (hc : a < c) : a < b ⊓ c | Mathlib_Data_Real_CauSeq |
case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
a b c : CauSeq α abs
εb : α
εb0 : εb > 0
ib : ℕ
hb : ∀ j ≥ ib, εb ≤ ↑(b - a) j
εc : α
εc0 : εc > 0
ic : ℕ
hc : ∀ j ≥ ic, εc ≤ ↑(c - a) j
i : ℕ
hi : i ≥ ib ⊔ ic
this : min εb εc ≤ min (↑(b - a) i) (↑(c - a) i)
⊢ εb ⊓ εc ≤ ↑(b ⊓ c - a) 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... | exact this.trans_eq (min_sub_sub_right _ _ _) | protected theorem lt_inf {a b c : CauSeq α abs} (hb : a < b) (hc : a < c) : a < b ⊓ c := by
obtain ⟨⟨εb, εb0, ib, hb⟩, ⟨εc, εc0, ic, hc⟩⟩ := hb, hc
refine' ⟨εb ⊓ εc, lt_inf_iff.mpr ⟨εb0, εc0⟩, ib ⊔ ic, fun i hi => _⟩
have := min_le_min (hb _ (sup_le_iff.mp hi).1) (hc _ (sup_le_iff.mp hi).2)
| Mathlib.Data.Real.CauSeq.893_0.b72JnnMmhSc5wVP | protected theorem lt_inf {a b c : CauSeq α abs} (hb : a < b) (hc : a < c) : a < b ⊓ c | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
a b : CauSeq α abs
h : a ≤ b
⊢ a ⊔ b ≈ b | /-
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... | obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h | protected theorem sup_eq_right {a b : CauSeq α abs} (h : a ≤ b) : a ⊔ b ≈ b := by
| Mathlib.Data.Real.CauSeq.918_0.b72JnnMmhSc5wVP | protected theorem sup_eq_right {a b : CauSeq α abs} (h : a ≤ b) : a ⊔ b ≈ b | Mathlib_Data_Real_CauSeq |
case inl.intro.intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
a b : CauSeq α abs
ε : α
ε0 : 0 < ε
i : ℕ
h : ∀ j ≥ i, ε ≤ ↑(b - a) j
⊢ a ⊔ b ≈ b | /-
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 _ _ | protected theorem sup_eq_right {a b : CauSeq α abs} (h : a ≤ b) : a ⊔ b ≈ b := by
obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h
· | Mathlib.Data.Real.CauSeq.918_0.b72JnnMmhSc5wVP | protected theorem sup_eq_right {a b : CauSeq α abs} (h : a ≤ b) : a ⊔ b ≈ b | Mathlib_Data_Real_CauSeq |
case inl.intro.intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
a b : CauSeq α abs
ε : α
ε0 : 0 < ε
i : ℕ
h : ∀ j ≥ i, ε ≤ ↑(b - a) j
ε✝ : α
a✝ : ε✝ > 0
⊢ ∃ i, ∀ j ≥ i, |↑(a ⊔ b - b) 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... | refine' ⟨i, fun j hj => _⟩ | protected theorem sup_eq_right {a b : CauSeq α abs} (h : a ≤ b) : a ⊔ b ≈ b := by
obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h
· intro _ _
| Mathlib.Data.Real.CauSeq.918_0.b72JnnMmhSc5wVP | protected theorem sup_eq_right {a b : CauSeq α abs} (h : a ≤ b) : a ⊔ b ≈ b | Mathlib_Data_Real_CauSeq |
case inl.intro.intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
a b : CauSeq α abs
ε : α
ε0 : 0 < ε
i : ℕ
h : ∀ j ≥ i, ε ≤ ↑(b - a) j
ε✝ : α
a✝ : ε✝ > 0
j : ℕ
hj : j ≥ i
⊢ |↑(a ⊔ b - b) 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... | dsimp | protected theorem sup_eq_right {a b : CauSeq α abs} (h : a ≤ b) : a ⊔ b ≈ b := by
obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h
· intro _ _
refine' ⟨i, fun j hj => _⟩
| Mathlib.Data.Real.CauSeq.918_0.b72JnnMmhSc5wVP | protected theorem sup_eq_right {a b : CauSeq α abs} (h : a ≤ b) : a ⊔ b ≈ b | Mathlib_Data_Real_CauSeq |
case inl.intro.intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
a b : CauSeq α abs
ε : α
ε0 : 0 < ε
i : ℕ
h : ∀ j ≥ i, ε ≤ ↑(b - a) j
ε✝ : α
a✝ : ε✝ > 0
j : ℕ
hj : j ≥ i
⊢ |↑a j ⊔ ↑b j - ↑b 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... | erw [← max_sub_sub_right] | protected theorem sup_eq_right {a b : CauSeq α abs} (h : a ≤ b) : a ⊔ b ≈ b := by
obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h
· intro _ _
refine' ⟨i, fun j hj => _⟩
dsimp
| Mathlib.Data.Real.CauSeq.918_0.b72JnnMmhSc5wVP | protected theorem sup_eq_right {a b : CauSeq α abs} (h : a ≤ b) : a ⊔ b ≈ b | Mathlib_Data_Real_CauSeq |
case inl.intro.intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
a b : CauSeq α abs
ε : α
ε0 : 0 < ε
i : ℕ
h : ∀ j ≥ i, ε ≤ ↑(b - a) j
ε✝ : α
a✝ : ε✝ > 0
j : ℕ
hj : j ≥ i
⊢ |max (↑a j - ↑b j) (↑b j - ↑b 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... | rwa [sub_self, max_eq_right, abs_zero] | protected theorem sup_eq_right {a b : CauSeq α abs} (h : a ≤ b) : a ⊔ b ≈ b := by
obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h
· intro _ _
refine' ⟨i, fun j hj => _⟩
dsimp
erw [← max_sub_sub_right]
| Mathlib.Data.Real.CauSeq.918_0.b72JnnMmhSc5wVP | protected theorem sup_eq_right {a b : CauSeq α abs} (h : a ≤ b) : a ⊔ b ≈ b | Mathlib_Data_Real_CauSeq |
case inl.intro.intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
a b : CauSeq α abs
ε : α
ε0 : 0 < ε
i : ℕ
h : ∀ j ≥ i, ε ≤ ↑(b - a) j
ε✝ : α
a✝ : ε✝ > 0
j : ℕ
hj : j ≥ i
⊢ ↑a j - ↑b j ≤ 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... | rw [sub_nonpos, ← sub_nonneg] | protected theorem sup_eq_right {a b : CauSeq α abs} (h : a ≤ b) : a ⊔ b ≈ b := by
obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h
· intro _ _
refine' ⟨i, fun j hj => _⟩
dsimp
erw [← max_sub_sub_right]
rwa [sub_self, max_eq_right, abs_zero]
| Mathlib.Data.Real.CauSeq.918_0.b72JnnMmhSc5wVP | protected theorem sup_eq_right {a b : CauSeq α abs} (h : a ≤ b) : a ⊔ b ≈ b | Mathlib_Data_Real_CauSeq |
case inl.intro.intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
a b : CauSeq α abs
ε : α
ε0 : 0 < ε
i : ℕ
h : ∀ j ≥ i, ε ≤ ↑(b - a) j
ε✝ : α
a✝ : ε✝ > 0
j : ℕ
hj : j ≥ i
⊢ 0 ≤ ↑b j - ↑a 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... | exact ε0.le.trans (h _ hj) | protected theorem sup_eq_right {a b : CauSeq α abs} (h : a ≤ b) : a ⊔ b ≈ b := by
obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h
· intro _ _
refine' ⟨i, fun j hj => _⟩
dsimp
erw [← max_sub_sub_right]
rwa [sub_self, max_eq_right, abs_zero]
rw [sub_nonpos, ← sub_nonneg]
| Mathlib.Data.Real.CauSeq.918_0.b72JnnMmhSc5wVP | protected theorem sup_eq_right {a b : CauSeq α abs} (h : a ≤ b) : a ⊔ b ≈ b | Mathlib_Data_Real_CauSeq |
case inr
α : Type u_1
inst✝ : LinearOrderedField α
a b : CauSeq α abs
h : a ≈ b
⊢ a ⊔ b ≈ b | /-
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' Setoid.trans (sup_equiv_sup h (Setoid.refl _)) _ | protected theorem sup_eq_right {a b : CauSeq α abs} (h : a ≤ b) : a ⊔ b ≈ b := by
obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h
· intro _ _
refine' ⟨i, fun j hj => _⟩
dsimp
erw [← max_sub_sub_right]
rwa [sub_self, max_eq_right, abs_zero]
rw [sub_nonpos, ← sub_nonneg]
exact ε0.le.trans (h _ hj)
· | Mathlib.Data.Real.CauSeq.918_0.b72JnnMmhSc5wVP | protected theorem sup_eq_right {a b : CauSeq α abs} (h : a ≤ b) : a ⊔ b ≈ b | Mathlib_Data_Real_CauSeq |
case inr
α : Type u_1
inst✝ : LinearOrderedField α
a b : CauSeq α abs
h : a ≈ b
⊢ b ⊔ b ≈ b | /-
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 [CauSeq.sup_idem] | protected theorem sup_eq_right {a b : CauSeq α abs} (h : a ≤ b) : a ⊔ b ≈ b := by
obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h
· intro _ _
refine' ⟨i, fun j hj => _⟩
dsimp
erw [← max_sub_sub_right]
rwa [sub_self, max_eq_right, abs_zero]
rw [sub_nonpos, ← sub_nonneg]
exact ε0.le.trans (h _ hj)
· r... | Mathlib.Data.Real.CauSeq.918_0.b72JnnMmhSc5wVP | protected theorem sup_eq_right {a b : CauSeq α abs} (h : a ≤ b) : a ⊔ b ≈ b | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
a b : CauSeq α abs
h : b ≤ a
⊢ a ⊓ b ≈ b | /-
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... | obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h | protected theorem inf_eq_right {a b : CauSeq α abs} (h : b ≤ a) : a ⊓ b ≈ b := by
| Mathlib.Data.Real.CauSeq.931_0.b72JnnMmhSc5wVP | protected theorem inf_eq_right {a b : CauSeq α abs} (h : b ≤ a) : a ⊓ b ≈ b | Mathlib_Data_Real_CauSeq |
case inl.intro.intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
a b : CauSeq α abs
ε : α
ε0 : 0 < ε
i : ℕ
h : ∀ j ≥ i, ε ≤ ↑(a - b) j
⊢ a ⊓ b ≈ b | /-
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 _ _ | protected theorem inf_eq_right {a b : CauSeq α abs} (h : b ≤ a) : a ⊓ b ≈ b := by
obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h
· | Mathlib.Data.Real.CauSeq.931_0.b72JnnMmhSc5wVP | protected theorem inf_eq_right {a b : CauSeq α abs} (h : b ≤ a) : a ⊓ b ≈ b | Mathlib_Data_Real_CauSeq |
case inl.intro.intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
a b : CauSeq α abs
ε : α
ε0 : 0 < ε
i : ℕ
h : ∀ j ≥ i, ε ≤ ↑(a - b) j
ε✝ : α
a✝ : ε✝ > 0
⊢ ∃ i, ∀ j ≥ i, |↑(a ⊓ b - b) 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... | refine' ⟨i, fun j hj => _⟩ | protected theorem inf_eq_right {a b : CauSeq α abs} (h : b ≤ a) : a ⊓ b ≈ b := by
obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h
· intro _ _
| Mathlib.Data.Real.CauSeq.931_0.b72JnnMmhSc5wVP | protected theorem inf_eq_right {a b : CauSeq α abs} (h : b ≤ a) : a ⊓ b ≈ b | Mathlib_Data_Real_CauSeq |
case inl.intro.intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
a b : CauSeq α abs
ε : α
ε0 : 0 < ε
i : ℕ
h : ∀ j ≥ i, ε ≤ ↑(a - b) j
ε✝ : α
a✝ : ε✝ > 0
j : ℕ
hj : j ≥ i
⊢ |↑(a ⊓ b - b) 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... | dsimp | protected theorem inf_eq_right {a b : CauSeq α abs} (h : b ≤ a) : a ⊓ b ≈ b := by
obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h
· intro _ _
refine' ⟨i, fun j hj => _⟩
| Mathlib.Data.Real.CauSeq.931_0.b72JnnMmhSc5wVP | protected theorem inf_eq_right {a b : CauSeq α abs} (h : b ≤ a) : a ⊓ b ≈ b | Mathlib_Data_Real_CauSeq |
case inl.intro.intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
a b : CauSeq α abs
ε : α
ε0 : 0 < ε
i : ℕ
h : ∀ j ≥ i, ε ≤ ↑(a - b) j
ε✝ : α
a✝ : ε✝ > 0
j : ℕ
hj : j ≥ i
⊢ |↑a j ⊓ ↑b j - ↑b 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... | erw [← min_sub_sub_right] | protected theorem inf_eq_right {a b : CauSeq α abs} (h : b ≤ a) : a ⊓ b ≈ b := by
obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h
· intro _ _
refine' ⟨i, fun j hj => _⟩
dsimp
| Mathlib.Data.Real.CauSeq.931_0.b72JnnMmhSc5wVP | protected theorem inf_eq_right {a b : CauSeq α abs} (h : b ≤ a) : a ⊓ b ≈ b | Mathlib_Data_Real_CauSeq |
case inl.intro.intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
a b : CauSeq α abs
ε : α
ε0 : 0 < ε
i : ℕ
h : ∀ j ≥ i, ε ≤ ↑(a - b) j
ε✝ : α
a✝ : ε✝ > 0
j : ℕ
hj : j ≥ i
⊢ |min (↑a j - ↑b j) (↑b j - ↑b 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... | rwa [sub_self, min_eq_right, abs_zero] | protected theorem inf_eq_right {a b : CauSeq α abs} (h : b ≤ a) : a ⊓ b ≈ b := by
obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h
· intro _ _
refine' ⟨i, fun j hj => _⟩
dsimp
erw [← min_sub_sub_right]
| Mathlib.Data.Real.CauSeq.931_0.b72JnnMmhSc5wVP | protected theorem inf_eq_right {a b : CauSeq α abs} (h : b ≤ a) : a ⊓ b ≈ b | Mathlib_Data_Real_CauSeq |
case inl.intro.intro.intro
α : Type u_1
inst✝ : LinearOrderedField α
a b : CauSeq α abs
ε : α
ε0 : 0 < ε
i : ℕ
h : ∀ j ≥ i, ε ≤ ↑(a - b) j
ε✝ : α
a✝ : ε✝ > 0
j : ℕ
hj : j ≥ i
⊢ 0 ≤ ↑a j - ↑b 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... | exact ε0.le.trans (h _ hj) | protected theorem inf_eq_right {a b : CauSeq α abs} (h : b ≤ a) : a ⊓ b ≈ b := by
obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h
· intro _ _
refine' ⟨i, fun j hj => _⟩
dsimp
erw [← min_sub_sub_right]
rwa [sub_self, min_eq_right, abs_zero]
| Mathlib.Data.Real.CauSeq.931_0.b72JnnMmhSc5wVP | protected theorem inf_eq_right {a b : CauSeq α abs} (h : b ≤ a) : a ⊓ b ≈ b | Mathlib_Data_Real_CauSeq |
case inr
α : Type u_1
inst✝ : LinearOrderedField α
a b : CauSeq α abs
h : b ≈ a
⊢ a ⊓ b ≈ b | /-
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' Setoid.trans (inf_equiv_inf (Setoid.symm h) (Setoid.refl _)) _ | protected theorem inf_eq_right {a b : CauSeq α abs} (h : b ≤ a) : a ⊓ b ≈ b := by
obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h
· intro _ _
refine' ⟨i, fun j hj => _⟩
dsimp
erw [← min_sub_sub_right]
rwa [sub_self, min_eq_right, abs_zero]
exact ε0.le.trans (h _ hj)
· | Mathlib.Data.Real.CauSeq.931_0.b72JnnMmhSc5wVP | protected theorem inf_eq_right {a b : CauSeq α abs} (h : b ≤ a) : a ⊓ b ≈ b | Mathlib_Data_Real_CauSeq |
case inr
α : Type u_1
inst✝ : LinearOrderedField α
a b : CauSeq α abs
h : b ≈ a
⊢ b ⊓ b ≈ b | /-
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 [CauSeq.inf_idem] | protected theorem inf_eq_right {a b : CauSeq α abs} (h : b ≤ a) : a ⊓ b ≈ b := by
obtain ⟨ε, ε0 : _ < _, i, h⟩ | h := h
· intro _ _
refine' ⟨i, fun j hj => _⟩
dsimp
erw [← min_sub_sub_right]
rwa [sub_self, min_eq_right, abs_zero]
exact ε0.le.trans (h _ hj)
· refine' Setoid.trans (inf_equiv_inf... | Mathlib.Data.Real.CauSeq.931_0.b72JnnMmhSc5wVP | protected theorem inf_eq_right {a b : CauSeq α abs} (h : b ≤ a) : a ⊓ b ≈ b | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
a b : CauSeq α abs
h : b ≤ a
⊢ a ⊔ b ≈ 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... | simpa only [CauSeq.sup_comm] using CauSeq.sup_eq_right h | protected theorem sup_eq_left {a b : CauSeq α abs} (h : b ≤ a) : a ⊔ b ≈ a := by
| Mathlib.Data.Real.CauSeq.943_0.b72JnnMmhSc5wVP | protected theorem sup_eq_left {a b : CauSeq α abs} (h : b ≤ a) : a ⊔ b ≈ a | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
a b : CauSeq α abs
h : a ≤ b
⊢ a ⊓ b ≈ 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... | simpa only [CauSeq.inf_comm] using CauSeq.inf_eq_right h | protected theorem inf_eq_left {a b : CauSeq α abs} (h : a ≤ b) : a ⊓ b ≈ a := by
| Mathlib.Data.Real.CauSeq.947_0.b72JnnMmhSc5wVP | protected theorem inf_eq_left {a b : CauSeq α abs} (h : a ≤ b) : a ⊓ b ≈ a | Mathlib_Data_Real_CauSeq |
α : Type u_1
inst✝ : LinearOrderedField α
a b c : CauSeq α abs
ha : a ≤ c
hb : b ≤ c
⊢ a ⊔ b ≤ c | /-
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' ha with ha ha | protected theorem sup_le {a b c : CauSeq α abs} (ha : a ≤ c) (hb : b ≤ c) : a ⊔ b ≤ c := by
| Mathlib.Data.Real.CauSeq.967_0.b72JnnMmhSc5wVP | protected theorem sup_le {a b c : CauSeq α abs} (ha : a ≤ c) (hb : b ≤ c) : a ⊔ b ≤ c | Mathlib_Data_Real_CauSeq |
case inl
α : Type u_1
inst✝ : LinearOrderedField α
a b c : CauSeq α abs
hb : b ≤ c
ha : a < c
⊢ a ⊔ b ≤ c | /-
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' hb with hb hb | protected theorem sup_le {a b c : CauSeq α abs} (ha : a ≤ c) (hb : b ≤ c) : a ⊔ b ≤ c := by
cases' ha with ha ha
· | Mathlib.Data.Real.CauSeq.967_0.b72JnnMmhSc5wVP | protected theorem sup_le {a b c : CauSeq α abs} (ha : a ≤ c) (hb : b ≤ c) : a ⊔ b ≤ c | Mathlib_Data_Real_CauSeq |
case inl.inl
α : Type u_1
inst✝ : LinearOrderedField α
a b c : CauSeq α abs
ha : a < c
hb : b < c
⊢ a ⊔ b ≤ c | /-
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 Or.inl (CauSeq.sup_lt ha hb) | protected theorem sup_le {a b c : CauSeq α abs} (ha : a ≤ c) (hb : b ≤ c) : a ⊔ b ≤ c := by
cases' ha with ha ha
· cases' hb with hb hb
· | Mathlib.Data.Real.CauSeq.967_0.b72JnnMmhSc5wVP | protected theorem sup_le {a b c : CauSeq α abs} (ha : a ≤ c) (hb : b ≤ c) : a ⊔ b ≤ c | Mathlib_Data_Real_CauSeq |
case inl.inr
α : Type u_1
inst✝ : LinearOrderedField α
a b c : CauSeq α abs
ha : a < c
hb : b ≈ c
⊢ a ⊔ b ≤ c | /-
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... | replace ha := le_of_le_of_eq ha.le (Setoid.symm hb) | protected theorem sup_le {a b c : CauSeq α abs} (ha : a ≤ c) (hb : b ≤ c) : a ⊔ b ≤ c := by
cases' ha with ha ha
· cases' hb with hb hb
· exact Or.inl (CauSeq.sup_lt ha hb)
· | Mathlib.Data.Real.CauSeq.967_0.b72JnnMmhSc5wVP | protected theorem sup_le {a b c : CauSeq α abs} (ha : a ≤ c) (hb : b ≤ c) : a ⊔ b ≤ c | Mathlib_Data_Real_CauSeq |
case inl.inr
α : Type u_1
inst✝ : LinearOrderedField α
a b c : CauSeq α abs
hb : b ≈ c
ha : a ≤ b
⊢ a ⊔ b ≤ c | /-
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' le_of_le_of_eq (Or.inr _) hb | protected theorem sup_le {a b c : CauSeq α abs} (ha : a ≤ c) (hb : b ≤ c) : a ⊔ b ≤ c := by
cases' ha with ha ha
· cases' hb with hb hb
· exact Or.inl (CauSeq.sup_lt ha hb)
· replace ha := le_of_le_of_eq ha.le (Setoid.symm hb)
| Mathlib.Data.Real.CauSeq.967_0.b72JnnMmhSc5wVP | protected theorem sup_le {a b c : CauSeq α abs} (ha : a ≤ c) (hb : b ≤ c) : a ⊔ b ≤ c | Mathlib_Data_Real_CauSeq |
case inl.inr
α : Type u_1
inst✝ : LinearOrderedField α
a b c : CauSeq α abs
hb : b ≈ c
ha : a ≤ b
⊢ a ⊔ b ≈ b | /-
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 CauSeq.sup_eq_right ha | protected theorem sup_le {a b c : CauSeq α abs} (ha : a ≤ c) (hb : b ≤ c) : a ⊔ b ≤ c := by
cases' ha with ha ha
· cases' hb with hb hb
· exact Or.inl (CauSeq.sup_lt ha hb)
· replace ha := le_of_le_of_eq ha.le (Setoid.symm hb)
refine' le_of_le_of_eq (Or.inr _) hb
| Mathlib.Data.Real.CauSeq.967_0.b72JnnMmhSc5wVP | protected theorem sup_le {a b c : CauSeq α abs} (ha : a ≤ c) (hb : b ≤ c) : a ⊔ b ≤ c | Mathlib_Data_Real_CauSeq |
case inr
α : Type u_1
inst✝ : LinearOrderedField α
a b c : CauSeq α abs
hb : b ≤ c
ha : a ≈ c
⊢ a ⊔ b ≤ c | /-
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... | replace hb := le_of_le_of_eq hb (Setoid.symm ha) | protected theorem sup_le {a b c : CauSeq α abs} (ha : a ≤ c) (hb : b ≤ c) : a ⊔ b ≤ c := by
cases' ha with ha ha
· cases' hb with hb hb
· exact Or.inl (CauSeq.sup_lt ha hb)
· replace ha := le_of_le_of_eq ha.le (Setoid.symm hb)
refine' le_of_le_of_eq (Or.inr _) hb
exact CauSeq.sup_eq_right ha
·... | Mathlib.Data.Real.CauSeq.967_0.b72JnnMmhSc5wVP | protected theorem sup_le {a b c : CauSeq α abs} (ha : a ≤ c) (hb : b ≤ c) : a ⊔ b ≤ c | Mathlib_Data_Real_CauSeq |
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