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 zero
P : ℝ → Prop
x₀ r : ℝ
hr : 1 < r
hx₀ : 0 < x₀
base : ∀ x ∈ Ico x₀ (r * x₀), P x
step : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z
⊢ ∀ x ∈ Ico x₀ (r ^ (Nat.zero + 1) * x₀), P x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | | zero => simpa using base | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib.Analysis.SpecialFunctions.Log.Base.445_0.egNyp4fdqSCAE7f | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib_Analysis_SpecialFunctions_Log_Base |
case zero
P : ℝ → Prop
x₀ r : ℝ
hr : 1 < r
hx₀ : 0 < x₀
base : ∀ x ∈ Ico x₀ (r * x₀), P x
step : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z
⊢ ∀ x ∈ Ico x₀ (r ^ (Nat.zero + 1) * x₀), P x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | simpa using base | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib.Analysis.SpecialFunctions.Log.Base.445_0.egNyp4fdqSCAE7f | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib_Analysis_SpecialFunctions_Log_Base |
case succ
P : ℝ → Prop
x₀ r : ℝ
hr : 1 < r
hx₀ : 0 < x₀
base : ∀ x ∈ Ico x₀ (r * x₀), P x
step : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z
n : ℕ
ih : ∀ x ∈ Ico x₀ (r ^ (n + 1) * x₀), P x
⊢ ∀ x ∈ Ico x₀ (r ^ (Nat.succ n + 1) * x₀), P x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | | succ n ih =>
exact fun x hx => (Ico_subset_Ico_union_Ico hx).elim (ih x) (step (n + 1) (by simp) ih _) | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib.Analysis.SpecialFunctions.Log.Base.445_0.egNyp4fdqSCAE7f | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib_Analysis_SpecialFunctions_Log_Base |
case succ
P : ℝ → Prop
x₀ r : ℝ
hr : 1 < r
hx₀ : 0 < x₀
base : ∀ x ∈ Ico x₀ (r * x₀), P x
step : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z
n : ℕ
ih : ∀ x ∈ Ico x₀ (r ^ (n + 1) * x₀), P x
⊢ ∀ x ∈ Ico x₀ (r ^ (Nat.succ n + 1) * x₀), P x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | exact fun x hx => (Ico_subset_Ico_union_Ico hx).elim (ih x) (step (n + 1) (by simp) ih _) | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib.Analysis.SpecialFunctions.Log.Base.445_0.egNyp4fdqSCAE7f | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib_Analysis_SpecialFunctions_Log_Base |
P : ℝ → Prop
x₀ r : ℝ
hr : 1 < r
hx₀ : 0 < x₀
base : ∀ x ∈ Ico x₀ (r * x₀), P x
step : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z
n : ℕ
ih : ∀ x ∈ Ico x₀ (r ^ (n + 1) * x₀), P x
x : ℝ
hx : x ∈ Ico x₀ (r ^ (Nat.succ n + 1) * x₀)
⊢ n + 1 ≥ 1 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | simp | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib.Analysis.SpecialFunctions.Log.Base.445_0.egNyp4fdqSCAE7f | /-- Induction principle for intervals of real numbers: if a proposition `P` is true
on `[x₀, r * x₀)` and if `P` for `[x₀, r^n * x₀)` implies `P` for `[r^n * x₀, r^(n+1) * x₀)`,
then `P` is true for all `x ≥ x₀`. -/
lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx₀ : 0 < x₀)
(base : ∀ x ∈ Set... | Mathlib_Analysis_SpecialFunctions_Log_Base |
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ : β
a₂ : α
b₂ : β
⊢ (a₁, b₁) < (a₂, b₂) ↔ (a₁, b₁) ≤ (a₂, b₂) ∧ ¬(a₂, b₂) ≤ (a₁, b₁) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | constructor | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mp
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ : β
a₂ : α
b₂ : β
⊢ (a₁, b₁) < (a₂, b₂) → (a₁, b₁) ≤ (a₂, b₂) ∧ ¬(a₂, b₂) ≤ (a₁, b₁) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | rintro (⟨_, _, hlt⟩ | ⟨_, hlt⟩) | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mp.left
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ : β
a₂ : α
b₂ : β
hlt : a₁ < a₂
⊢ (a₁, b₁) ≤ (a₂, b₂) ∧ ¬(a₂, b₂) ≤ (a₁, b₁) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | constructor | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mp.left.left
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ : β
a₂ : α
b₂ : β
hlt : a₁ < a₂
⊢ (a₁, b₁) ≤ (a₂, b₂) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | exact left _ _ hlt | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mp.left.right
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ : β
a₂ : α
b₂ : β
hlt : a₁ < a₂
⊢ ¬(a₂, b₂) ≤ (a₁, b₁) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | rintro ⟨⟩ | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mp.left.right.left
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ : β
a₂ : α
b₂ : β
hlt : a₁ < a₂
h✝ : a₂ < a₁
⊢ False | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | apply lt_asymm hlt | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mp.left.right.left
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ : β
a₂ : α
b₂ : β
hlt : a₁ < a₂
h✝ : a₂ < a₁
⊢ a₂ < a₁ | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | assumption | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mp.left.right.right
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ b₂ : β
hlt : a₁ < a₁
h✝ : b₂ ≤ b₁
⊢ False | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | exact lt_irrefl _ hlt | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mp.right
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ b₂ : β
hlt : b₁ < b₂
⊢ (a₁, b₁) ≤ (a₁, b₂) ∧ ¬(a₁, b₂) ≤ (a₁, b₁) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | constructor | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mp.right.left
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ b₂ : β
hlt : b₁ < b₂
⊢ (a₁, b₁) ≤ (a₁, b₂) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | right | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mp.right.left.h
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ b₂ : β
hlt : b₁ < b₂
⊢ b₁ ≤ b₂ | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | rw [lt_iff_le_not_le] at hlt | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mp.right.left.h
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ b₂ : β
hlt : b₁ ≤ b₂ ∧ ¬b₂ ≤ b₁
⊢ b₁ ≤ b₂ | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | exact hlt.1 | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mp.right.right
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ b₂ : β
hlt : b₁ < b₂
⊢ ¬(a₁, b₂) ≤ (a₁, b₁) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | rintro ⟨⟩ | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mp.right.right.left
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ b₂ : β
hlt : b₁ < b₂
h✝ : a₁ < a₁
⊢ False | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | apply lt_irrefl a₁ | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mp.right.right.left
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ b₂ : β
hlt : b₁ < b₂
h✝ : a₁ < a₁
⊢ a₁ < a₁ | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | assumption | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mp.right.right.right
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ b₂ : β
hlt : b₁ < b₂
h✝ : b₂ ≤ b₁
⊢ False | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | rw [lt_iff_le_not_le] at hlt | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mp.right.right.right
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ b₂ : β
hlt : b₁ ≤ b₂ ∧ ¬b₂ ≤ b₁
h✝ : b₂ ≤ b₁
⊢ False | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | apply hlt.2 | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mp.right.right.right
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ b₂ : β
hlt : b₁ ≤ b₂ ∧ ¬b₂ ≤ b₁
h✝ : b₂ ≤ b₁
⊢ b₂ ≤ b₁ | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | assumption | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mpr
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ : β
a₂ : α
b₂ : β
⊢ (a₁, b₁) ≤ (a₂, b₂) ∧ ¬(a₂, b₂) ≤ (a₁, b₁) → (a₁, b₁) < (a₂, b₂) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | rintro ⟨⟨⟩, h₂r⟩ | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mpr.intro.left
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ : β
a₂ : α
b₂ : β
h₂r : ¬(a₂, b₂) ≤ (a₁, b₁)
h✝ : a₁ < a₂
⊢ (a₁, b₁) < (a₂, b₂) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | left | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mpr.intro.left.h
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ : β
a₂ : α
b₂ : β
h₂r : ¬(a₂, b₂) ≤ (a₁, b₁)
h✝ : a₁ < a₂
⊢ a₁ < a₂ | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | assumption | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mpr.intro.right
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ b₂ : β
h₂r : ¬(a₁, b₂) ≤ (a₁, b₁)
h✝ : b₁ ≤ b₂
⊢ (a₁, b₁) < (a₁, b₂) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | right | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mpr.intro.right.h
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ b₂ : β
h₂r : ¬(a₁, b₂) ≤ (a₁, b₁)
h✝ : b₁ ≤ b₂
⊢ b₁ < b₂ | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | rw [lt_iff_le_not_le] | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mpr.intro.right.h
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ b₂ : β
h₂r : ¬(a₁, b₂) ≤ (a₁, b₁)
h✝ : b₁ ≤ b₂
⊢ b₁ ≤ b₂ ∧ ¬b₂ ≤ b₁ | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | constructor | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mpr.intro.right.h.left
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ b₂ : β
h₂r : ¬(a₁, b₂) ≤ (a₁, b₁)
h✝ : b₁ ≤ b₂
⊢ b₁ ≤ b₂ | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | assumption | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mpr.intro.right.h.right
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ b₂ : β
h₂r : ¬(a₁, b₂) ≤ (a₁, b₁)
h✝ : b₁ ≤ b₂
⊢ ¬b₂ ≤ b₁ | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | intro h | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mpr.intro.right.h.right
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ b₂ : β
h₂r : ¬(a₁, b₂) ≤ (a₁, b₁)
h✝ : b₁ ≤ b₂
h : b₂ ≤ b₁
⊢ False | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | apply h₂r | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mpr.intro.right.h.right
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ b₂ : β
h₂r : ¬(a₁, b₂) ≤ (a₁, b₁)
h✝ : b₁ ≤ b₂
h : b₂ ≤ b₁
⊢ (a₁, b₂) ≤ (a₁, b... | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | right | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mpr.intro.right.h.right.h
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
src✝¹ : LE (Lex (α × β)) := instLE α β
src✝ : LT (Lex (α × β)) := instLT α β
x₁ x₂ : Lex (α × β)
a₁ : α
b₁ b₂ : β
h₂r : ¬(a₁, b₂) ≤ (a₁, b₁)
h✝ : b₁ ≤ b₂
h : b₂ ≤ b₁
⊢ b₂ ≤ b₁ | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | exact h | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) :=
{ Prod.Lex.instLE α β, Prod.Lex.instLT α β with
le_refl := refl_of <| Prod.Lex _ _,
le_trans := fun _ _ _ => trans_of <| Prod.Lex _ _,
lt_iff_le_not_le := fun x₁ x₂ =>
... | Mathlib.Data.Prod.Lex.69_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic preorder for pairs. -/
instance preorder (α β : Type*) [Preorder α] [Preorder β] : Preorder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : PartialOrder α
inst✝ : Preorder β
⊢ Monotone ⇑toLex | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | rintro ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ ⟨ha, hb⟩ | theorem toLex_mono : @Monotone _ _ _ (Prod.Lex.preorder α β) (toLex : α × β → α ×ₗ β) := by
| Mathlib.Data.Prod.Lex.115_0.6Yc4sDJ4nVbbQgh | theorem toLex_mono : @Monotone _ _ _ (Prod.Lex.preorder α β) (toLex : α × β → α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mk.mk.intro
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : PartialOrder α
inst✝ : Preorder β
a₁ : α
b₁ : β
a₂ : α
b₂ : β
ha : (a₁, b₁).1 ≤ (a₂, b₂).1
hb : (a₁, b₁).2 ≤ (a₂, b₂).2
⊢ toLex (a₁, b₁) ≤ toLex (a₂, b₂) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | obtain rfl | ha : a₁ = a₂ ∨ _ := ha.eq_or_lt | theorem toLex_mono : @Monotone _ _ _ (Prod.Lex.preorder α β) (toLex : α × β → α ×ₗ β) := by
rintro ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ ⟨ha, hb⟩
| Mathlib.Data.Prod.Lex.115_0.6Yc4sDJ4nVbbQgh | theorem toLex_mono : @Monotone _ _ _ (Prod.Lex.preorder α β) (toLex : α × β → α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mk.mk.intro.inl
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : PartialOrder α
inst✝ : Preorder β
a₁ : α
b₁ b₂ : β
ha : (a₁, b₁).1 ≤ (a₁, b₂).1
hb : (a₁, b₁).2 ≤ (a₁, b₂).2
⊢ toLex (a₁, b₁) ≤ toLex (a₁, b₂) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | exact right _ hb | theorem toLex_mono : @Monotone _ _ _ (Prod.Lex.preorder α β) (toLex : α × β → α ×ₗ β) := by
rintro ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ ⟨ha, hb⟩
obtain rfl | ha : a₁ = a₂ ∨ _ := ha.eq_or_lt
· | Mathlib.Data.Prod.Lex.115_0.6Yc4sDJ4nVbbQgh | theorem toLex_mono : @Monotone _ _ _ (Prod.Lex.preorder α β) (toLex : α × β → α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mk.mk.intro.inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : PartialOrder α
inst✝ : Preorder β
a₁ : α
b₁ : β
a₂ : α
b₂ : β
ha✝ : (a₁, b₁).1 ≤ (a₂, b₂).1
hb : (a₁, b₁).2 ≤ (a₂, b₂).2
ha : (a₁, b₁).1 < (a₂, b₂).1
⊢ toLex (a₁, b₁) ≤ toLex (a₂, b₂) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | exact left _ _ ha | theorem toLex_mono : @Monotone _ _ _ (Prod.Lex.preorder α β) (toLex : α × β → α ×ₗ β) := by
rintro ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ ⟨ha, hb⟩
obtain rfl | ha : a₁ = a₂ ∨ _ := ha.eq_or_lt
· exact right _ hb
· | Mathlib.Data.Prod.Lex.115_0.6Yc4sDJ4nVbbQgh | theorem toLex_mono : @Monotone _ _ _ (Prod.Lex.preorder α β) (toLex : α × β → α ×ₗ β) | Mathlib_Data_Prod_Lex |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : PartialOrder α
inst✝ : Preorder β
⊢ StrictMono ⇑toLex | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | rintro ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ h | theorem toLex_strictMono : @StrictMono _ _ _ (Prod.Lex.preorder α β) (toLex : α × β → α ×ₗ β) := by
| Mathlib.Data.Prod.Lex.126_0.6Yc4sDJ4nVbbQgh | theorem toLex_strictMono : @StrictMono _ _ _ (Prod.Lex.preorder α β) (toLex : α × β → α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mk.mk
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : PartialOrder α
inst✝ : Preorder β
a₁ : α
b₁ : β
a₂ : α
b₂ : β
h : (a₁, b₁) < (a₂, b₂)
⊢ toLex (a₁, b₁) < toLex (a₂, b₂) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | obtain rfl | ha : a₁ = a₂ ∨ _ := h.le.1.eq_or_lt | theorem toLex_strictMono : @StrictMono _ _ _ (Prod.Lex.preorder α β) (toLex : α × β → α ×ₗ β) := by
rintro ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ h
| Mathlib.Data.Prod.Lex.126_0.6Yc4sDJ4nVbbQgh | theorem toLex_strictMono : @StrictMono _ _ _ (Prod.Lex.preorder α β) (toLex : α × β → α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mk.mk.inl
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : PartialOrder α
inst✝ : Preorder β
a₁ : α
b₁ b₂ : β
h : (a₁, b₁) < (a₁, b₂)
⊢ toLex (a₁, b₁) < toLex (a₁, b₂) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | exact right _ (Prod.mk_lt_mk_iff_right.1 h) | theorem toLex_strictMono : @StrictMono _ _ _ (Prod.Lex.preorder α β) (toLex : α × β → α ×ₗ β) := by
rintro ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ h
obtain rfl | ha : a₁ = a₂ ∨ _ := h.le.1.eq_or_lt
· | Mathlib.Data.Prod.Lex.126_0.6Yc4sDJ4nVbbQgh | theorem toLex_strictMono : @StrictMono _ _ _ (Prod.Lex.preorder α β) (toLex : α × β → α ×ₗ β) | Mathlib_Data_Prod_Lex |
case mk.mk.inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : PartialOrder α
inst✝ : Preorder β
a₁ : α
b₁ : β
a₂ : α
b₂ : β
h : (a₁, b₁) < (a₂, b₂)
ha : (a₁, b₁).1 < (a₂, b₂).1
⊢ toLex (a₁, b₁) < toLex (a₂, b₂) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | exact left _ _ ha | theorem toLex_strictMono : @StrictMono _ _ _ (Prod.Lex.preorder α β) (toLex : α × β → α ×ₗ β) := by
rintro ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ h
obtain rfl | ha : a₁ = a₂ ∨ _ := h.le.1.eq_or_lt
· exact right _ (Prod.mk_lt_mk_iff_right.1 h)
· | Mathlib.Data.Prod.Lex.126_0.6Yc4sDJ4nVbbQgh | theorem toLex_strictMono : @StrictMono _ _ _ (Prod.Lex.preorder α β) (toLex : α × β → α ×ₗ β) | Mathlib_Data_Prod_Lex |
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : PartialOrder α
inst✝ : PartialOrder β
src✝ : Preorder (Lex (α × β)) := preorder α β
⊢ ∀ (a b : Lex (α × β)), a ≤ b → b ≤ a → a = b | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | haveI : IsStrictOrder α (· < ·) := { irrefl := lt_irrefl, trans := fun _ _ _ => lt_trans } | /-- Dictionary / lexicographic partial order for pairs. -/
instance partialOrder (α β : Type*) [PartialOrder α] [PartialOrder β] : PartialOrder (α ×ₗ β) :=
{ Prod.Lex.preorder α β with
le_antisymm := by
| Mathlib.Data.Prod.Lex.135_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic partial order for pairs. -/
instance partialOrder (α β : Type*) [PartialOrder α] [PartialOrder β] : PartialOrder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : PartialOrder α
inst✝ : PartialOrder β
src✝ : Preorder (Lex (α × β)) := preorder α β
this : IsStrictOrder α fun x x_1 => x < x_1
⊢ ∀ (a b : Lex (α × β)), a ≤ b → b ≤ a → a = b | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | haveI : IsAntisymm β (· ≤ ·) := ⟨fun _ _ => le_antisymm⟩ | /-- Dictionary / lexicographic partial order for pairs. -/
instance partialOrder (α β : Type*) [PartialOrder α] [PartialOrder β] : PartialOrder (α ×ₗ β) :=
{ Prod.Lex.preorder α β with
le_antisymm := by
haveI : IsStrictOrder α (· < ·) := { irrefl := lt_irrefl, trans := fun _ _ _ => lt_trans }
| Mathlib.Data.Prod.Lex.135_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic partial order for pairs. -/
instance partialOrder (α β : Type*) [PartialOrder α] [PartialOrder β] : PartialOrder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : PartialOrder α
inst✝ : PartialOrder β
src✝ : Preorder (Lex (α × β)) := preorder α β
this✝ : IsStrictOrder α fun x x_1 => x < x_1
this : IsAntisymm β fun x x_1 => x ≤ x_1
⊢ ∀ (a b : Lex (α × β)), a ≤ b → b ≤ a → a = b | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | exact @antisymm _ (Prod.Lex _ _) _ | /-- Dictionary / lexicographic partial order for pairs. -/
instance partialOrder (α β : Type*) [PartialOrder α] [PartialOrder β] : PartialOrder (α ×ₗ β) :=
{ Prod.Lex.preorder α β with
le_antisymm := by
haveI : IsStrictOrder α (· < ·) := { irrefl := lt_irrefl, trans := fun _ _ _ => lt_trans }
haveI : ... | Mathlib.Data.Prod.Lex.135_0.6Yc4sDJ4nVbbQgh | /-- Dictionary / lexicographic partial order for pairs. -/
instance partialOrder (α β : Type*) [PartialOrder α] [PartialOrder β] : PartialOrder (α ×ₗ β) | Mathlib_Data_Prod_Lex |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : DenselyOrdered α
inst✝ : DenselyOrdered β
⊢ ∀ (a₁ a₂ : Lex (α × β)), a₁ < a₂ → ∃ a, a₁ < a ∧ a < a₂ | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | rintro _ _ (@⟨a₁, b₁, a₂, b₂, h⟩ | @⟨a, b₁, b₂, h⟩) | instance [Preorder α] [Preorder β] [DenselyOrdered α] [DenselyOrdered β] :
DenselyOrdered (α ×ₗ β) where
dense := by
| Mathlib.Data.Prod.Lex.171_0.6Yc4sDJ4nVbbQgh | instance [Preorder α] [Preorder β] [DenselyOrdered α] [DenselyOrdered β] :
DenselyOrdered (α ×ₗ β) where
dense | Mathlib_Data_Prod_Lex |
case left
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : DenselyOrdered α
inst✝ : DenselyOrdered β
a₁ : α
b₁ : β
a₂ : α
b₂ : β
h : a₁ < a₂
⊢ ∃ a, (a₁, b₁) < a ∧ a < (a₂, b₂) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | obtain ⟨c, h₁, h₂⟩ := exists_between h | instance [Preorder α] [Preorder β] [DenselyOrdered α] [DenselyOrdered β] :
DenselyOrdered (α ×ₗ β) where
dense := by
rintro _ _ (@⟨a₁, b₁, a₂, b₂, h⟩ | @⟨a, b₁, b₂, h⟩)
· | Mathlib.Data.Prod.Lex.171_0.6Yc4sDJ4nVbbQgh | instance [Preorder α] [Preorder β] [DenselyOrdered α] [DenselyOrdered β] :
DenselyOrdered (α ×ₗ β) where
dense | Mathlib_Data_Prod_Lex |
case left.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : DenselyOrdered α
inst✝ : DenselyOrdered β
a₁ : α
b₁ : β
a₂ : α
b₂ : β
h : a₁ < a₂
c : α
h₁ : a₁ < c
h₂ : c < a₂
⊢ ∃ a, (a₁, b₁) < a ∧ a < (a₂, b₂) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | exact ⟨(c, b₁), left _ _ h₁, left _ _ h₂⟩ | instance [Preorder α] [Preorder β] [DenselyOrdered α] [DenselyOrdered β] :
DenselyOrdered (α ×ₗ β) where
dense := by
rintro _ _ (@⟨a₁, b₁, a₂, b₂, h⟩ | @⟨a, b₁, b₂, h⟩)
· obtain ⟨c, h₁, h₂⟩ := exists_between h
| Mathlib.Data.Prod.Lex.171_0.6Yc4sDJ4nVbbQgh | instance [Preorder α] [Preorder β] [DenselyOrdered α] [DenselyOrdered β] :
DenselyOrdered (α ×ₗ β) where
dense | Mathlib_Data_Prod_Lex |
case right
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : DenselyOrdered α
inst✝ : DenselyOrdered β
a : α
b₁ b₂ : β
h : b₁ < b₂
⊢ ∃ a_1, (a, b₁) < a_1 ∧ a_1 < (a, b₂) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | obtain ⟨c, h₁, h₂⟩ := exists_between h | instance [Preorder α] [Preorder β] [DenselyOrdered α] [DenselyOrdered β] :
DenselyOrdered (α ×ₗ β) where
dense := by
rintro _ _ (@⟨a₁, b₁, a₂, b₂, h⟩ | @⟨a, b₁, b₂, h⟩)
· obtain ⟨c, h₁, h₂⟩ := exists_between h
exact ⟨(c, b₁), left _ _ h₁, left _ _ h₂⟩
· | Mathlib.Data.Prod.Lex.171_0.6Yc4sDJ4nVbbQgh | instance [Preorder α] [Preorder β] [DenselyOrdered α] [DenselyOrdered β] :
DenselyOrdered (α ×ₗ β) where
dense | Mathlib_Data_Prod_Lex |
case right.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : DenselyOrdered α
inst✝ : DenselyOrdered β
a : α
b₁ b₂ : β
h : b₁ < b₂
c : β
h₁ : b₁ < c
h₂ : c < b₂
⊢ ∃ a_1, (a, b₁) < a_1 ∧ a_1 < (a, b₂) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | exact ⟨(a, c), right _ h₁, right _ h₂⟩ | instance [Preorder α] [Preorder β] [DenselyOrdered α] [DenselyOrdered β] :
DenselyOrdered (α ×ₗ β) where
dense := by
rintro _ _ (@⟨a₁, b₁, a₂, b₂, h⟩ | @⟨a, b₁, b₂, h⟩)
· obtain ⟨c, h₁, h₂⟩ := exists_between h
exact ⟨(c, b₁), left _ _ h₁, left _ _ h₂⟩
· obtain ⟨c, h₁, h₂⟩ := exists_between h
... | Mathlib.Data.Prod.Lex.171_0.6Yc4sDJ4nVbbQgh | instance [Preorder α] [Preorder β] [DenselyOrdered α] [DenselyOrdered β] :
DenselyOrdered (α ×ₗ β) where
dense | Mathlib_Data_Prod_Lex |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : NoMaxOrder α
⊢ ∀ (a : Lex (α × β)), ∃ b, a < b | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | rintro ⟨a, b⟩ | instance noMaxOrder_of_left [Preorder α] [Preorder β] [NoMaxOrder α] : NoMaxOrder (α ×ₗ β) where
exists_gt := by
| Mathlib.Data.Prod.Lex.180_0.6Yc4sDJ4nVbbQgh | instance noMaxOrder_of_left [Preorder α] [Preorder β] [NoMaxOrder α] : NoMaxOrder (α ×ₗ β) where
exists_gt | Mathlib_Data_Prod_Lex |
case mk
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : NoMaxOrder α
a : α
b : β
⊢ ∃ b_1, (a, b) < b_1 | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | obtain ⟨c, h⟩ := exists_gt a | instance noMaxOrder_of_left [Preorder α] [Preorder β] [NoMaxOrder α] : NoMaxOrder (α ×ₗ β) where
exists_gt := by
rintro ⟨a, b⟩
| Mathlib.Data.Prod.Lex.180_0.6Yc4sDJ4nVbbQgh | instance noMaxOrder_of_left [Preorder α] [Preorder β] [NoMaxOrder α] : NoMaxOrder (α ×ₗ β) where
exists_gt | Mathlib_Data_Prod_Lex |
case mk.intro
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : NoMaxOrder α
a : α
b : β
c : α
h : a < c
⊢ ∃ b_1, (a, b) < b_1 | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | exact ⟨⟨c, b⟩, left _ _ h⟩ | instance noMaxOrder_of_left [Preorder α] [Preorder β] [NoMaxOrder α] : NoMaxOrder (α ×ₗ β) where
exists_gt := by
rintro ⟨a, b⟩
obtain ⟨c, h⟩ := exists_gt a
| Mathlib.Data.Prod.Lex.180_0.6Yc4sDJ4nVbbQgh | instance noMaxOrder_of_left [Preorder α] [Preorder β] [NoMaxOrder α] : NoMaxOrder (α ×ₗ β) where
exists_gt | Mathlib_Data_Prod_Lex |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : NoMinOrder α
⊢ ∀ (a : Lex (α × β)), ∃ b, b < a | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | rintro ⟨a, b⟩ | instance noMinOrder_of_left [Preorder α] [Preorder β] [NoMinOrder α] : NoMinOrder (α ×ₗ β) where
exists_lt := by
| Mathlib.Data.Prod.Lex.187_0.6Yc4sDJ4nVbbQgh | instance noMinOrder_of_left [Preorder α] [Preorder β] [NoMinOrder α] : NoMinOrder (α ×ₗ β) where
exists_lt | Mathlib_Data_Prod_Lex |
case mk
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : NoMinOrder α
a : α
b : β
⊢ ∃ b_1, b_1 < (a, b) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | obtain ⟨c, h⟩ := exists_lt a | instance noMinOrder_of_left [Preorder α] [Preorder β] [NoMinOrder α] : NoMinOrder (α ×ₗ β) where
exists_lt := by
rintro ⟨a, b⟩
| Mathlib.Data.Prod.Lex.187_0.6Yc4sDJ4nVbbQgh | instance noMinOrder_of_left [Preorder α] [Preorder β] [NoMinOrder α] : NoMinOrder (α ×ₗ β) where
exists_lt | Mathlib_Data_Prod_Lex |
case mk.intro
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : NoMinOrder α
a : α
b : β
c : α
h : c < a
⊢ ∃ b_1, b_1 < (a, b) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | exact ⟨⟨c, b⟩, left _ _ h⟩ | instance noMinOrder_of_left [Preorder α] [Preorder β] [NoMinOrder α] : NoMinOrder (α ×ₗ β) where
exists_lt := by
rintro ⟨a, b⟩
obtain ⟨c, h⟩ := exists_lt a
| Mathlib.Data.Prod.Lex.187_0.6Yc4sDJ4nVbbQgh | instance noMinOrder_of_left [Preorder α] [Preorder β] [NoMinOrder α] : NoMinOrder (α ×ₗ β) where
exists_lt | Mathlib_Data_Prod_Lex |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : NoMaxOrder β
⊢ ∀ (a : Lex (α × β)), ∃ b, a < b | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | rintro ⟨a, b⟩ | instance noMaxOrder_of_right [Preorder α] [Preorder β] [NoMaxOrder β] : NoMaxOrder (α ×ₗ β) where
exists_gt := by
| Mathlib.Data.Prod.Lex.194_0.6Yc4sDJ4nVbbQgh | instance noMaxOrder_of_right [Preorder α] [Preorder β] [NoMaxOrder β] : NoMaxOrder (α ×ₗ β) where
exists_gt | Mathlib_Data_Prod_Lex |
case mk
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : NoMaxOrder β
a : α
b : β
⊢ ∃ b_1, (a, b) < b_1 | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | obtain ⟨c, h⟩ := exists_gt b | instance noMaxOrder_of_right [Preorder α] [Preorder β] [NoMaxOrder β] : NoMaxOrder (α ×ₗ β) where
exists_gt := by
rintro ⟨a, b⟩
| Mathlib.Data.Prod.Lex.194_0.6Yc4sDJ4nVbbQgh | instance noMaxOrder_of_right [Preorder α] [Preorder β] [NoMaxOrder β] : NoMaxOrder (α ×ₗ β) where
exists_gt | Mathlib_Data_Prod_Lex |
case mk.intro
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : NoMaxOrder β
a : α
b c : β
h : b < c
⊢ ∃ b_1, (a, b) < b_1 | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | exact ⟨⟨a, c⟩, right _ h⟩ | instance noMaxOrder_of_right [Preorder α] [Preorder β] [NoMaxOrder β] : NoMaxOrder (α ×ₗ β) where
exists_gt := by
rintro ⟨a, b⟩
obtain ⟨c, h⟩ := exists_gt b
| Mathlib.Data.Prod.Lex.194_0.6Yc4sDJ4nVbbQgh | instance noMaxOrder_of_right [Preorder α] [Preorder β] [NoMaxOrder β] : NoMaxOrder (α ×ₗ β) where
exists_gt | Mathlib_Data_Prod_Lex |
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : NoMinOrder β
⊢ ∀ (a : Lex (α × β)), ∃ b, b < a | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | rintro ⟨a, b⟩ | instance noMinOrder_of_right [Preorder α] [Preorder β] [NoMinOrder β] : NoMinOrder (α ×ₗ β) where
exists_lt := by
| Mathlib.Data.Prod.Lex.201_0.6Yc4sDJ4nVbbQgh | instance noMinOrder_of_right [Preorder α] [Preorder β] [NoMinOrder β] : NoMinOrder (α ×ₗ β) where
exists_lt | Mathlib_Data_Prod_Lex |
case mk
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : NoMinOrder β
a : α
b : β
⊢ ∃ b_1, b_1 < (a, b) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | obtain ⟨c, h⟩ := exists_lt b | instance noMinOrder_of_right [Preorder α] [Preorder β] [NoMinOrder β] : NoMinOrder (α ×ₗ β) where
exists_lt := by
rintro ⟨a, b⟩
| Mathlib.Data.Prod.Lex.201_0.6Yc4sDJ4nVbbQgh | instance noMinOrder_of_right [Preorder α] [Preorder β] [NoMinOrder β] : NoMinOrder (α ×ₗ β) where
exists_lt | Mathlib_Data_Prod_Lex |
case mk.intro
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : NoMinOrder β
a : α
b c : β
h : c < b
⊢ ∃ b_1, b_1 < (a, b) | /-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Minchao Wu
-/
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Lexic... | exact ⟨⟨a, c⟩, right _ h⟩ | instance noMinOrder_of_right [Preorder α] [Preorder β] [NoMinOrder β] : NoMinOrder (α ×ₗ β) where
exists_lt := by
rintro ⟨a, b⟩
obtain ⟨c, h⟩ := exists_lt b
| Mathlib.Data.Prod.Lex.201_0.6Yc4sDJ4nVbbQgh | instance noMinOrder_of_right [Preorder α] [Preorder β] [NoMinOrder β] : NoMinOrder (α ×ₗ β) where
exists_lt | Mathlib_Data_Prod_Lex |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : E → ℝ
add_le : ∀ (x y : E), f (x + y) ≤ f x + f y
smul : ∀ (a : 𝕜) (x : E), ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [← zero_smul 𝕜 (0 : E), smul, norm_zero, zero_mul] | /-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing 𝕜`. -/
def Seminorm.of [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ)
(add_le : ∀ x y : E, f (x + y) ≤ f x + f y) (smul : ∀ (a : 𝕜) (x : E), f (a • x) = ‖a‖ * f x) :
Seminorm 𝕜 E where
... | Mathlib.Analysis.Seminorm.75_0.ywwMCgoKeIFKDZ3 | /-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing 𝕜`. -/
def Seminorm.of [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ)
(add_le : ∀ x y : E, f (x + y) ≤ f x + f y) (smul : ∀ (a : 𝕜) (x : E), f (a • x) = ‖a‖ * f x) :
Seminorm 𝕜 E where
... | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : E → ℝ
add_le : ∀ (x y : E), f (x + y) ≤ f x + f y
smul : ∀ (a : 𝕜) (x : E), ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [← neg_one_smul 𝕜, smul, norm_neg, ← smul, one_smul] | /-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing 𝕜`. -/
def Seminorm.of [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ)
(add_le : ∀ x y : E, f (x + y) ≤ f x + f y) (smul : ∀ (a : 𝕜) (x : E), f (a • x) = ‖a‖ * f x) :
Seminorm 𝕜 E where
... | Mathlib.Analysis.Seminorm.75_0.ywwMCgoKeIFKDZ3 | /-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing 𝕜`. -/
def Seminorm.of [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ)
(add_le : ∀ x y : E, f (x + y) ≤ f x + f y) (smul : ∀ (a : 𝕜) (x : E), f (a • x) = ‖a‖ * f x) :
Seminorm 𝕜 E where
... | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : E → ℝ
map_zero : f 0 = 0
add_le : ∀ (x y : E), f (x + y) ≤ f x + f y
smul_le : ∀... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | refine' le_antisymm (smul_le r x) _ | /-- Alternative constructor for a `Seminorm` over a normed field `𝕜` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ) (map_zero : f 0 = 0)
(add_le : ∀ x y, f (x + y) ≤ f x + f y) (smul_le : ∀ (r : 𝕜) (... | Mathlib.Analysis.Seminorm.87_0.ywwMCgoKeIFKDZ3 | /-- Alternative constructor for a `Seminorm` over a normed field `𝕜` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ) (map_zero : f 0 = 0)
(add_le : ∀ x y, f (x + y) ≤ f x + f y) (smul_le : ∀ (r : 𝕜) (... | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : E → ℝ
map_zero : f 0 = 0
add_le : ∀ (x y : E), f (x + y) ≤ f x + f y
smul_le : ∀... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | by_cases h : r = 0 | /-- Alternative constructor for a `Seminorm` over a normed field `𝕜` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ) (map_zero : f 0 = 0)
(add_le : ∀ x y, f (x + y) ≤ f x + f y) (smul_le : ∀ (r : 𝕜) (... | Mathlib.Analysis.Seminorm.87_0.ywwMCgoKeIFKDZ3 | /-- Alternative constructor for a `Seminorm` over a normed field `𝕜` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ) (map_zero : f 0 = 0)
(add_le : ∀ x y, f (x + y) ≤ f x + f y) (smul_le : ∀ (r : 𝕜) (... | Mathlib_Analysis_Seminorm |
case pos
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : E → ℝ
map_zero : f 0 = 0
add_le : ∀ (x y : E), f (x + y) ≤ f x + f y
sm... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp [h, map_zero] | /-- Alternative constructor for a `Seminorm` over a normed field `𝕜` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ) (map_zero : f 0 = 0)
(add_le : ∀ x y, f (x + y) ≤ f x + f y) (smul_le : ∀ (r : 𝕜) (... | Mathlib.Analysis.Seminorm.87_0.ywwMCgoKeIFKDZ3 | /-- Alternative constructor for a `Seminorm` over a normed field `𝕜` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ) (map_zero : f 0 = 0)
(add_le : ∀ x y, f (x + y) ≤ f x + f y) (smul_le : ∀ (r : 𝕜) (... | Mathlib_Analysis_Seminorm |
case neg
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : E → ℝ
map_zero : f 0 = 0
add_le : ∀ (x y : E), f (x + y) ≤ f x + f y
sm... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [← mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))] | /-- Alternative constructor for a `Seminorm` over a normed field `𝕜` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ) (map_zero : f 0 = 0)
(add_le : ∀ x y, f (x + y) ≤ f x + f y) (smul_le : ∀ (r : 𝕜) (... | Mathlib.Analysis.Seminorm.87_0.ywwMCgoKeIFKDZ3 | /-- Alternative constructor for a `Seminorm` over a normed field `𝕜` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ) (map_zero : f 0 = 0)
(add_le : ∀ x y, f (x + y) ≤ f x + f y) (smul_le : ∀ (r : 𝕜) (... | Mathlib_Analysis_Seminorm |
case neg
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : E → ℝ
map_zero : f 0 = 0
add_le : ∀ (x y : E), f (x + y) ≤ f x + f y
sm... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [inv_mul_cancel_left₀ (norm_ne_zero_iff.mpr h)] | /-- Alternative constructor for a `Seminorm` over a normed field `𝕜` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ) (map_zero : f 0 = 0)
(add_le : ∀ x y, f (x + y) ≤ f x + f y) (smul_le : ∀ (r : 𝕜) (... | Mathlib.Analysis.Seminorm.87_0.ywwMCgoKeIFKDZ3 | /-- Alternative constructor for a `Seminorm` over a normed field `𝕜` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ) (map_zero : f 0 = 0)
(add_le : ∀ x y, f (x + y) ≤ f x + f y) (smul_le : ∀ (r : 𝕜) (... | Mathlib_Analysis_Seminorm |
case neg
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : E → ℝ
map_zero : f 0 = 0
add_le : ∀ (x y : E), f (x + y) ≤ f x + f y
sm... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | specialize smul_le r⁻¹ (r • x) | /-- Alternative constructor for a `Seminorm` over a normed field `𝕜` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ) (map_zero : f 0 = 0)
(add_le : ∀ x y, f (x + y) ≤ f x + f y) (smul_le : ∀ (r : 𝕜) (... | Mathlib.Analysis.Seminorm.87_0.ywwMCgoKeIFKDZ3 | /-- Alternative constructor for a `Seminorm` over a normed field `𝕜` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ) (map_zero : f 0 = 0)
(add_le : ∀ x y, f (x + y) ≤ f x + f y) (smul_le : ∀ (r : 𝕜) (... | Mathlib_Analysis_Seminorm |
case neg
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : E → ℝ
map_zero : f 0 = 0
add_le : ∀ (x y : E), f (x + y) ≤ f x + f y
r ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [norm_inv] at smul_le | /-- Alternative constructor for a `Seminorm` over a normed field `𝕜` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ) (map_zero : f 0 = 0)
(add_le : ∀ x y, f (x + y) ≤ f x + f y) (smul_le : ∀ (r : 𝕜) (... | Mathlib.Analysis.Seminorm.87_0.ywwMCgoKeIFKDZ3 | /-- Alternative constructor for a `Seminorm` over a normed field `𝕜` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ) (map_zero : f 0 = 0)
(add_le : ∀ x y, f (x + y) ≤ f x + f y) (smul_le : ∀ (r : 𝕜) (... | Mathlib_Analysis_Seminorm |
case neg
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : E → ℝ
map_zero : f 0 = 0
add_le : ∀ (x y : E), f (x + y) ≤ f x + f y
r ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | convert smul_le | /-- Alternative constructor for a `Seminorm` over a normed field `𝕜` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ) (map_zero : f 0 = 0)
(add_le : ∀ x y, f (x + y) ≤ f x + f y) (smul_le : ∀ (r : 𝕜) (... | Mathlib.Analysis.Seminorm.87_0.ywwMCgoKeIFKDZ3 | /-- Alternative constructor for a `Seminorm` over a normed field `𝕜` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ) (map_zero : f 0 = 0)
(add_le : ∀ x y, f (x + y) ≤ f x + f y) (smul_le : ∀ (r : 𝕜) (... | Mathlib_Analysis_Seminorm |
case h.e'_3.h.e'_1
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : E → ℝ
map_zero : f 0 = 0
add_le : ∀ (x y : E), f (x + y) ≤ f ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp [h] | /-- Alternative constructor for a `Seminorm` over a normed field `𝕜` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ) (map_zero : f 0 = 0)
(add_le : ∀ x y, f (x + y) ≤ f x + f y) (smul_le : ∀ (r : 𝕜) (... | Mathlib.Analysis.Seminorm.87_0.ywwMCgoKeIFKDZ3 | /-- Alternative constructor for a `Seminorm` over a normed field `𝕜` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ) (map_zero : f 0 = 0)
(add_le : ∀ x y, f (x + y) ≤ f x + f y) (smul_le : ∀ (r : 𝕜) (... | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddGroup E
inst✝ : SMul 𝕜 E
f g : Seminorm 𝕜 E
h : (fun f => f.toFun) f = (fun f => f.toFun) g
⊢ f = g | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rcases f with ⟨⟨_⟩⟩ | instance instSeminormClass : SeminormClass (Seminorm 𝕜 E) 𝕜 E where
coe f := f.toFun
coe_injective' f g h := by
| Mathlib.Analysis.Seminorm.120_0.ywwMCgoKeIFKDZ3 | instance instSeminormClass : SeminormClass (Seminorm 𝕜 E) 𝕜 E where
coe f | Mathlib_Analysis_Seminorm |
case mk.mk
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddGroup E
inst✝ : SMul 𝕜 E
g : Seminorm 𝕜 E
toFun✝ : E → ℝ
map_zero'✝ : toFun✝ 0 = 0
add_le'✝ : ∀ (r s... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rcases g with ⟨⟨_⟩⟩ | instance instSeminormClass : SeminormClass (Seminorm 𝕜 E) 𝕜 E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with ⟨⟨_⟩⟩
| Mathlib.Analysis.Seminorm.120_0.ywwMCgoKeIFKDZ3 | instance instSeminormClass : SeminormClass (Seminorm 𝕜 E) 𝕜 E where
coe f | Mathlib_Analysis_Seminorm |
case mk.mk.mk.mk
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddGroup E
inst✝ : SMul 𝕜 E
toFun✝¹ : E → ℝ
map_zero'✝¹ : toFun✝¹ 0 = 0
add_le'✝¹ : ∀ (r s : E), t... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | congr | instance instSeminormClass : SeminormClass (Seminorm 𝕜 E) 𝕜 E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with ⟨⟨_⟩⟩
rcases g with ⟨⟨_⟩⟩
| Mathlib.Analysis.Seminorm.120_0.ywwMCgoKeIFKDZ3 | instance instSeminormClass : SeminormClass (Seminorm 𝕜 E) 𝕜 E where
coe f | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁵ : SeminormedRing 𝕜
inst✝⁴ : AddGroup E
inst✝³ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
c : 𝕜
x y : E
r✝ : ℝ
inst✝² : SMul R ℝ
inst✝¹ : SMul R ℝ≥0
inst✝ :... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp only [← smul_one_smul ℝ≥0 r (_ : ℝ), NNReal.smul_def, smul_eq_mul] | /-- Any action on `ℝ` which factors through `ℝ≥0` applies to a seminorm. -/
instance instSMul [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] : SMul R (Seminorm 𝕜 E) where
smul r p :=
{ r • p.toAddGroupSeminorm with
toFun := fun x => r • p x
smul' := fun _ _ => by
| Mathlib.Analysis.Seminorm.160_0.ywwMCgoKeIFKDZ3 | /-- Any action on `ℝ` which factors through `ℝ≥0` applies to a seminorm. -/
instance instSMul [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] : SMul R (Seminorm 𝕜 E) where
smul r p | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁵ : SeminormedRing 𝕜
inst✝⁴ : AddGroup E
inst✝³ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
c : 𝕜
x y : E
r✝ : ℝ
inst✝² : SMul R ℝ
inst✝¹ : SMul R ℝ≥0
inst✝ :... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [map_smul_eq_mul, mul_left_comm] | /-- Any action on `ℝ` which factors through `ℝ≥0` applies to a seminorm. -/
instance instSMul [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] : SMul R (Seminorm 𝕜 E) where
smul r p :=
{ r • p.toAddGroupSeminorm with
toFun := fun x => r • p x
smul' := fun _ _ => by
simp only [← smul_one_smul ℝ... | Mathlib.Analysis.Seminorm.160_0.ywwMCgoKeIFKDZ3 | /-- Any action on `ℝ` which factors through `ℝ≥0` applies to a seminorm. -/
instance instSMul [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] : SMul R (Seminorm 𝕜 E) where
smul r p | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddGroup E
inst✝ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
c : 𝕜
x✝ y : E
r : ℝ
p q : Seminorm 𝕜 E
src✝ : AddGroupSeminorm E ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] | instance instAdd : Add (Seminorm 𝕜 E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by | Mathlib.Analysis.Seminorm.185_0.ywwMCgoKeIFKDZ3 | instance instAdd : Add (Seminorm 𝕜 E) where
add p q | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : SeminormedRing 𝕜
inst✝¹ : AddGroup E
inst✝ : SMul 𝕜 E
p : Seminorm 𝕜 E
c : 𝕜
x y : E
r : ℝ
x✝¹ : Seminorm 𝕜 E
x✝ : ℕ
⊢ ⇑(x✝ • x✝¹) = x✝ • ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rfl | instance instAddMonoid : AddMonoid (Seminorm 𝕜 E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by | Mathlib.Analysis.Seminorm.200_0.ywwMCgoKeIFKDZ3 | instance instAddMonoid : AddMonoid (Seminorm 𝕜 E) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddGroup E
inst✝⁴ : SMul 𝕜 E
p : Seminorm 𝕜 E
c : 𝕜
x y : E
r : ℝ
inst✝³ : Monoid R
inst✝² : MulAction R ℝ
inst✝¹... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | intros | instance instMulAction [Monoid R] [MulAction R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] :
MulAction R (Seminorm 𝕜 E) :=
FunLike.coe_injective.mulAction _ (by | Mathlib.Analysis.Seminorm.206_0.ywwMCgoKeIFKDZ3 | instance instMulAction [Monoid R] [MulAction R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] :
MulAction R (Seminorm 𝕜 E) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddGroup E
inst✝⁴ : SMul 𝕜 E
p : Seminorm 𝕜 E
c : 𝕜
x y : E
r : ℝ
inst✝³ : Monoid R
inst✝² : MulAction R ℝ
inst✝¹... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rfl | instance instMulAction [Monoid R] [MulAction R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] :
MulAction R (Seminorm 𝕜 E) :=
FunLike.coe_injective.mulAction _ (by intros; | Mathlib.Analysis.Seminorm.206_0.ywwMCgoKeIFKDZ3 | instance instMulAction [Monoid R] [MulAction R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] :
MulAction R (Seminorm 𝕜 E) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddGroup E
inst✝⁴ : SMul 𝕜 E
p : Seminorm 𝕜 E
c : 𝕜
x y : E
r : ℝ
inst✝³ : Monoid R
inst✝² : DistribMulAction R ℝ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | intros | instance instDistribMulAction [Monoid R] [DistribMulAction R ℝ] [SMul R ℝ≥0]
[IsScalarTower R ℝ≥0 ℝ] : DistribMulAction R (Seminorm 𝕜 E) :=
(coeFnAddMonoidHom_injective 𝕜 E).distribMulAction _ (by | Mathlib.Analysis.Seminorm.226_0.ywwMCgoKeIFKDZ3 | instance instDistribMulAction [Monoid R] [DistribMulAction R ℝ] [SMul R ℝ≥0]
[IsScalarTower R ℝ≥0 ℝ] : DistribMulAction R (Seminorm 𝕜 E) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddGroup E
inst✝⁴ : SMul 𝕜 E
p : Seminorm 𝕜 E
c : 𝕜
x y : E
r : ℝ
inst✝³ : Monoid R
inst✝² : DistribMulAction R ℝ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rfl | instance instDistribMulAction [Monoid R] [DistribMulAction R ℝ] [SMul R ℝ≥0]
[IsScalarTower R ℝ≥0 ℝ] : DistribMulAction R (Seminorm 𝕜 E) :=
(coeFnAddMonoidHom_injective 𝕜 E).distribMulAction _ (by intros; | Mathlib.Analysis.Seminorm.226_0.ywwMCgoKeIFKDZ3 | instance instDistribMulAction [Monoid R] [DistribMulAction R ℝ] [SMul R ℝ≥0]
[IsScalarTower R ℝ≥0 ℝ] : DistribMulAction R (Seminorm 𝕜 E) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddGroup E
inst✝⁴ : SMul 𝕜 E
p : Seminorm 𝕜 E
c : 𝕜
x y : E
r : ℝ
inst✝³ : Semiring R
inst✝² : Module R ℝ
inst✝¹ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | intros | instance instModule [Semiring R] [Module R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] :
Module R (Seminorm 𝕜 E) :=
(coeFnAddMonoidHom_injective 𝕜 E).module R _ (by | Mathlib.Analysis.Seminorm.230_0.ywwMCgoKeIFKDZ3 | instance instModule [Semiring R] [Module R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] :
Module R (Seminorm 𝕜 E) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁶ : SeminormedRing 𝕜
inst✝⁵ : AddGroup E
inst✝⁴ : SMul 𝕜 E
p : Seminorm 𝕜 E
c : 𝕜
x y : E
r : ℝ
inst✝³ : Semiring R
inst✝² : Module R ℝ
inst✝¹ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rfl | instance instModule [Semiring R] [Module R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] :
Module R (Seminorm 𝕜 E) :=
(coeFnAddMonoidHom_injective 𝕜 E).module R _ (by intros; | Mathlib.Analysis.Seminorm.230_0.ywwMCgoKeIFKDZ3 | instance instModule [Semiring R] [Module R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] :
Module R (Seminorm 𝕜 E) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁵ : SeminormedRing 𝕜
inst✝⁴ : AddGroup E
inst✝³ : SMul 𝕜 E
p✝ : Seminorm 𝕜 E
c : 𝕜
x✝ y✝ : E
r✝ : ℝ
inst✝² : SMul R ℝ
inst✝¹ : SMul R ℝ≥0
inst✝... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simpa only [← smul_eq_mul, ← NNReal.smul_def, smul_one_smul ℝ≥0 r (_ : ℝ)] using
mul_max_of_nonneg x y (r • (1 : ℝ≥0) : ℝ≥0).coe_nonneg | theorem smul_sup [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p q : Seminorm 𝕜 E) :
r • (p ⊔ q) = r • p ⊔ r • q :=
have real.smul_max : ∀ x y : ℝ, r • max x y = max (r • x) (r • y) := fun x y => by
| Mathlib.Analysis.Seminorm.251_0.ywwMCgoKeIFKDZ3 | theorem smul_sup [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p q : Seminorm 𝕜 E) :
r • (p ⊔ q) = r • p ⊔ r • q | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp only [map_smulₛₗ] | /-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) : Seminorm 𝕜 E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this c... | Mathlib.Analysis.Seminorm.309_0.ywwMCgoKeIFKDZ3 | /-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) : Seminorm 𝕜 E | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [map_smul_eq_mul, RingHomIsometric.is_iso] | /-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) : Seminorm 𝕜 E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this c... | Mathlib.Analysis.Seminorm.309_0.ywwMCgoKeIFKDZ3 | /-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) : Seminorm 𝕜 E | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | simp_rw [le_def] | theorem smul_le_smul {p q : Seminorm 𝕜 E} {a b : ℝ≥0} (hpq : p ≤ q) (hab : a ≤ b) :
a • p ≤ b • q := by
| Mathlib.Analysis.Seminorm.386_0.ywwMCgoKeIFKDZ3 | theorem smul_le_smul {p q : Seminorm 𝕜 E} {a b : ℝ≥0} (hpq : p ≤ q) (hab : a ≤ b) :
a • p ≤ b • q | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | intro x | theorem smul_le_smul {p q : Seminorm 𝕜 E} {a b : ℝ≥0} (hpq : p ≤ q) (hab : a ≤ b) :
a • p ≤ b • q := by
simp_rw [le_def]
| Mathlib.Analysis.Seminorm.386_0.ywwMCgoKeIFKDZ3 | theorem smul_le_smul {p q : Seminorm 𝕜 E} {a b : ℝ≥0} (hpq : p ≤ q) (hab : a ≤ b) :
a • p ≤ b • q | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b) | theorem smul_le_smul {p q : Seminorm 𝕜 E} {a b : ℝ≥0} (hpq : p ≤ q) (hab : a ≤ b) :
a • p ≤ b • q := by
simp_rw [le_def]
intro x
| Mathlib.Analysis.Seminorm.386_0.ywwMCgoKeIFKDZ3 | theorem smul_le_smul {p q : Seminorm 𝕜 E} {a b : ℝ≥0} (hpq : p ≤ q) (hab : a ≤ b) :
a • p ≤ b • q | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | induction' s using Finset.cons_induction_on with a s ha ih | theorem finset_sup_apply (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) :
s.sup p x = ↑(s.sup fun i => ⟨p i x, map_nonneg (p i) x⟩ : ℝ≥0) := by
| Mathlib.Analysis.Seminorm.393_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_apply (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) :
s.sup p x = ↑(s.sup fun i => ⟨p i x, map_nonneg (p i) x⟩ : ℝ≥0) | Mathlib_Analysis_Seminorm |
case h₁
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply] | theorem finset_sup_apply (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) :
s.sup p x = ↑(s.sup fun i => ⟨p i x, map_nonneg (p i) x⟩ : ℝ≥0) := by
induction' s using Finset.cons_induction_on with a s ha ih
· | Mathlib.Analysis.Seminorm.393_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_apply (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) :
s.sup p x = ↑(s.sup fun i => ⟨p i x, map_nonneg (p i) x⟩ : ℝ≥0) | Mathlib_Analysis_Seminorm |
case h₁
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | norm_cast | theorem finset_sup_apply (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) :
s.sup p x = ↑(s.sup fun i => ⟨p i x, map_nonneg (p i) x⟩ : ℝ≥0) := by
induction' s using Finset.cons_induction_on with a s ha ih
· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
| Mathlib.Analysis.Seminorm.393_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_apply (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) :
s.sup p x = ↑(s.sup fun i => ⟨p i x, map_nonneg (p i) x⟩ : ℝ≥0) | Mathlib_Analysis_Seminorm |
case h₂
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih] | theorem finset_sup_apply (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) :
s.sup p x = ↑(s.sup fun i => ⟨p i x, map_nonneg (p i) x⟩ : ℝ≥0) := by
induction' s using Finset.cons_induction_on with a s ha ih
· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
· | Mathlib.Analysis.Seminorm.393_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_apply (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) :
s.sup p x = ↑(s.sup fun i => ⟨p i x, map_nonneg (p i) x⟩ : ℝ≥0) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rcases Finset.exists_mem_eq_sup s hs (fun i ↦ (⟨p i x, map_nonneg _ _⟩ : ℝ≥0)) with ⟨i, hi, hix⟩ | theorem exists_apply_eq_finset_sup (p : ι → Seminorm 𝕜 E) {s : Finset ι} (hs : s.Nonempty) (x : E) :
∃ i ∈ s, s.sup p x = p i x := by
| Mathlib.Analysis.Seminorm.402_0.ywwMCgoKeIFKDZ3 | theorem exists_apply_eq_finset_sup (p : ι → Seminorm 𝕜 E) {s : Finset ι} (hs : s.Nonempty) (x : E) :
∃ i ∈ s, s.sup p x = p i x | Mathlib_Analysis_Seminorm |
case intro.intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rw [finset_sup_apply] | theorem exists_apply_eq_finset_sup (p : ι → Seminorm 𝕜 E) {s : Finset ι} (hs : s.Nonempty) (x : E) :
∃ i ∈ s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i ↦ (⟨p i x, map_nonneg _ _⟩ : ℝ≥0)) with ⟨i, hi, hix⟩
| Mathlib.Analysis.Seminorm.402_0.ywwMCgoKeIFKDZ3 | theorem exists_apply_eq_finset_sup (p : ι → Seminorm 𝕜 E) {s : Finset ι} (hs : s.Nonempty) (x : E) :
∃ i ∈ s, s.sup p x = p i x | Mathlib_Analysis_Seminorm |
case intro.intro
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | exact ⟨i, hi, congr_arg _ hix⟩ | theorem exists_apply_eq_finset_sup (p : ι → Seminorm 𝕜 E) {s : Finset ι} (hs : s.Nonempty) (x : E) :
∃ i ∈ s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i ↦ (⟨p i x, map_nonneg _ _⟩ : ℝ≥0)) with ⟨i, hi, hix⟩
rw [finset_sup_apply]
| Mathlib.Analysis.Seminorm.402_0.ywwMCgoKeIFKDZ3 | theorem exists_apply_eq_finset_sup (p : ι → Seminorm 𝕜 E) {s : Finset ι} (hs : s.Nonempty) (x : E) :
∃ i ∈ s, s.sup p x = p i x | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝¹⁸ : SeminormedRing 𝕜
inst✝¹⁷ : SeminormedRing 𝕜₂
inst✝¹⁶ : SeminormedRing 𝕜₃
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
... | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Gro... | rcases Finset.eq_empty_or_nonempty s with (rfl|hs) | theorem zero_or_exists_apply_eq_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) :
s.sup p x = 0 ∨ ∃ i ∈ s, s.sup p x = p i x := by
| Mathlib.Analysis.Seminorm.408_0.ywwMCgoKeIFKDZ3 | theorem zero_or_exists_apply_eq_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) :
s.sup p x = 0 ∨ ∃ i ∈ s, s.sup p x = p i x | Mathlib_Analysis_Seminorm |
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