Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 | complexity_score float64 2.72 139,370,958,066,637,970,000,000,000,000,000,000,000,000,000,000,000,000,000B | diff_level int64 0 2 | file_diff_level float64 0 2 | theorem_same_file int64 1 32 | rank_file int64 0 2.51k |
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import Mathlib.Algebra.Quaternion
import Mathlib.Tactic.Ring
#align_import algebra.quaternion_basis from "leanprover-community/mathlib"@"3aa5b8a9ed7a7cabd36e6e1d022c9858ab8a8c2d"
open Quaternion
namespace QuaternionAlgebra
structure Basis {R : Type*} (A : Type*) [CommRing R] [Ring A] [Algebra R A] (c₁ c₂ : R) where
(i j k : A)
i_mul_i : i * i = c₁ • (1 : A)
j_mul_j : j * j = c₂ • (1 : A)
i_mul_j : i * j = k
j_mul_i : j * i = -k
#align quaternion_algebra.basis QuaternionAlgebra.Basis
variable {R : Type*} {A B : Type*} [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B]
variable {c₁ c₂ : R}
namespace Basis
@[ext]
protected theorem ext ⦃q₁ q₂ : Basis A c₁ c₂⦄ (hi : q₁.i = q₂.i) (hj : q₁.j = q₂.j) : q₁ = q₂ := by
cases q₁; rename_i q₁_i_mul_j _
cases q₂; rename_i q₂_i_mul_j _
congr
rw [← q₁_i_mul_j, ← q₂_i_mul_j]
congr
#align quaternion_algebra.basis.ext QuaternionAlgebra.Basis.ext
variable (R)
@[simps i j k]
protected def self : Basis ℍ[R,c₁,c₂] c₁ c₂ where
i := ⟨0, 1, 0, 0⟩
i_mul_i := by ext <;> simp
j := ⟨0, 0, 1, 0⟩
j_mul_j := by ext <;> simp
k := ⟨0, 0, 0, 1⟩
i_mul_j := by ext <;> simp
j_mul_i := by ext <;> simp
#align quaternion_algebra.basis.self QuaternionAlgebra.Basis.self
variable {R}
instance : Inhabited (Basis ℍ[R,c₁,c₂] c₁ c₂) :=
⟨Basis.self R⟩
variable (q : Basis A c₁ c₂)
attribute [simp] i_mul_i j_mul_j i_mul_j j_mul_i
@[simp]
theorem i_mul_k : q.i * q.k = c₁ • q.j := by
rw [← i_mul_j, ← mul_assoc, i_mul_i, smul_mul_assoc, one_mul]
#align quaternion_algebra.basis.i_mul_k QuaternionAlgebra.Basis.i_mul_k
@[simp]
theorem k_mul_i : q.k * q.i = -c₁ • q.j := by
rw [← i_mul_j, mul_assoc, j_mul_i, mul_neg, i_mul_k, neg_smul]
#align quaternion_algebra.basis.k_mul_i QuaternionAlgebra.Basis.k_mul_i
@[simp]
theorem k_mul_j : q.k * q.j = c₂ • q.i := by
rw [← i_mul_j, mul_assoc, j_mul_j, mul_smul_comm, mul_one]
#align quaternion_algebra.basis.k_mul_j QuaternionAlgebra.Basis.k_mul_j
@[simp]
theorem j_mul_k : q.j * q.k = -c₂ • q.i := by
rw [← i_mul_j, ← mul_assoc, j_mul_i, neg_mul, k_mul_j, neg_smul]
#align quaternion_algebra.basis.j_mul_k QuaternionAlgebra.Basis.j_mul_k
@[simp]
theorem k_mul_k : q.k * q.k = -((c₁ * c₂) • (1 : A)) := by
rw [← i_mul_j, mul_assoc, ← mul_assoc q.j _ _, j_mul_i, ← i_mul_j, ← mul_assoc, mul_neg, ←
mul_assoc, i_mul_i, smul_mul_assoc, one_mul, neg_mul, smul_mul_assoc, j_mul_j, smul_smul]
#align quaternion_algebra.basis.k_mul_k QuaternionAlgebra.Basis.k_mul_k
def lift (x : ℍ[R,c₁,c₂]) : A :=
algebraMap R _ x.re + x.imI • q.i + x.imJ • q.j + x.imK • q.k
#align quaternion_algebra.basis.lift QuaternionAlgebra.Basis.lift
theorem lift_zero : q.lift (0 : ℍ[R,c₁,c₂]) = 0 := by simp [lift]
#align quaternion_algebra.basis.lift_zero QuaternionAlgebra.Basis.lift_zero
theorem lift_one : q.lift (1 : ℍ[R,c₁,c₂]) = 1 := by simp [lift]
#align quaternion_algebra.basis.lift_one QuaternionAlgebra.Basis.lift_one
theorem lift_add (x y : ℍ[R,c₁,c₂]) : q.lift (x + y) = q.lift x + q.lift y := by
simp only [lift, add_re, map_add, add_imI, add_smul, add_imJ, add_imK]
abel
#align quaternion_algebra.basis.lift_add QuaternionAlgebra.Basis.lift_add
theorem lift_mul (x y : ℍ[R,c₁,c₂]) : q.lift (x * y) = q.lift x * q.lift y := by
simp only [lift, Algebra.algebraMap_eq_smul_one]
simp_rw [add_mul, mul_add, smul_mul_assoc, mul_smul_comm, one_mul, mul_one, smul_smul]
simp only [i_mul_i, j_mul_j, i_mul_j, j_mul_i, i_mul_k, k_mul_i, k_mul_j, j_mul_k, k_mul_k]
simp only [smul_smul, smul_neg, sub_eq_add_neg, add_smul, ← add_assoc, mul_neg, neg_smul]
simp only [mul_right_comm _ _ (c₁ * c₂), mul_comm _ (c₁ * c₂)]
simp only [mul_comm _ c₁, mul_right_comm _ _ c₁]
simp only [mul_comm _ c₂, mul_right_comm _ _ c₂]
simp only [← mul_comm c₁ c₂, ← mul_assoc]
simp only [mul_re, sub_eq_add_neg, add_smul, neg_smul, mul_imI, ← add_assoc, mul_imJ, mul_imK]
abel
#align quaternion_algebra.basis.lift_mul QuaternionAlgebra.Basis.lift_mul
| Mathlib/Algebra/QuaternionBasis.lean | 138 | 139 | theorem lift_smul (r : R) (x : ℍ[R,c₁,c₂]) : q.lift (r • x) = r • q.lift x := by |
simp [lift, mul_smul, ← Algebra.smul_def]
| 1 | 2.718282 | 0 | 0.4 | 10 | 394 |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.RelIso.Basic
#align_import order.ord_continuous from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w x
variable {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x}
open Function OrderDual Set
def LeftOrdContinuous [Preorder α] [Preorder β] (f : α → β) :=
∀ ⦃s : Set α⦄ ⦃x⦄, IsLUB s x → IsLUB (f '' s) (f x)
#align left_ord_continuous LeftOrdContinuous
def RightOrdContinuous [Preorder α] [Preorder β] (f : α → β) :=
∀ ⦃s : Set α⦄ ⦃x⦄, IsGLB s x → IsGLB (f '' s) (f x)
#align right_ord_continuous RightOrdContinuous
namespace LeftOrdContinuous
section SemilatticeSup
variable [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
theorem map_sup (hf : LeftOrdContinuous f) (x y : α) : f (x ⊔ y) = f x ⊔ f y :=
(hf isLUB_pair).unique <| by simp only [image_pair, isLUB_pair]
#align left_ord_continuous.map_sup LeftOrdContinuous.map_sup
| Mathlib/Order/OrdContinuous.lean | 98 | 99 | theorem le_iff (hf : LeftOrdContinuous f) (h : Injective f) {x y} : f x ≤ f y ↔ x ≤ y := by |
simp only [← sup_eq_right, ← hf.map_sup, h.eq_iff]
| 1 | 2.718282 | 0 | 0.4 | 5 | 395 |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.RelIso.Basic
#align_import order.ord_continuous from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w x
variable {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x}
open Function OrderDual Set
def LeftOrdContinuous [Preorder α] [Preorder β] (f : α → β) :=
∀ ⦃s : Set α⦄ ⦃x⦄, IsLUB s x → IsLUB (f '' s) (f x)
#align left_ord_continuous LeftOrdContinuous
def RightOrdContinuous [Preorder α] [Preorder β] (f : α → β) :=
∀ ⦃s : Set α⦄ ⦃x⦄, IsGLB s x → IsGLB (f '' s) (f x)
#align right_ord_continuous RightOrdContinuous
namespace LeftOrdContinuous
section SemilatticeSup
variable [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
theorem map_sup (hf : LeftOrdContinuous f) (x y : α) : f (x ⊔ y) = f x ⊔ f y :=
(hf isLUB_pair).unique <| by simp only [image_pair, isLUB_pair]
#align left_ord_continuous.map_sup LeftOrdContinuous.map_sup
theorem le_iff (hf : LeftOrdContinuous f) (h : Injective f) {x y} : f x ≤ f y ↔ x ≤ y := by
simp only [← sup_eq_right, ← hf.map_sup, h.eq_iff]
#align left_ord_continuous.le_iff LeftOrdContinuous.le_iff
| Mathlib/Order/OrdContinuous.lean | 102 | 103 | theorem lt_iff (hf : LeftOrdContinuous f) (h : Injective f) {x y} : f x < f y ↔ x < y := by |
simp only [lt_iff_le_not_le, hf.le_iff h]
| 1 | 2.718282 | 0 | 0.4 | 5 | 395 |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.RelIso.Basic
#align_import order.ord_continuous from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w x
variable {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x}
open Function OrderDual Set
def LeftOrdContinuous [Preorder α] [Preorder β] (f : α → β) :=
∀ ⦃s : Set α⦄ ⦃x⦄, IsLUB s x → IsLUB (f '' s) (f x)
#align left_ord_continuous LeftOrdContinuous
def RightOrdContinuous [Preorder α] [Preorder β] (f : α → β) :=
∀ ⦃s : Set α⦄ ⦃x⦄, IsGLB s x → IsGLB (f '' s) (f x)
#align right_ord_continuous RightOrdContinuous
namespace LeftOrdContinuous
section CompleteLattice
variable [CompleteLattice α] [CompleteLattice β] {f : α → β}
theorem map_sSup' (hf : LeftOrdContinuous f) (s : Set α) : f (sSup s) = sSup (f '' s) :=
(hf <| isLUB_sSup s).sSup_eq.symm
#align left_ord_continuous.map_Sup' LeftOrdContinuous.map_sSup'
| Mathlib/Order/OrdContinuous.lean | 131 | 132 | theorem map_sSup (hf : LeftOrdContinuous f) (s : Set α) : f (sSup s) = ⨆ x ∈ s, f x := by |
rw [hf.map_sSup', sSup_image]
| 1 | 2.718282 | 0 | 0.4 | 5 | 395 |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.RelIso.Basic
#align_import order.ord_continuous from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w x
variable {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x}
open Function OrderDual Set
def LeftOrdContinuous [Preorder α] [Preorder β] (f : α → β) :=
∀ ⦃s : Set α⦄ ⦃x⦄, IsLUB s x → IsLUB (f '' s) (f x)
#align left_ord_continuous LeftOrdContinuous
def RightOrdContinuous [Preorder α] [Preorder β] (f : α → β) :=
∀ ⦃s : Set α⦄ ⦃x⦄, IsGLB s x → IsGLB (f '' s) (f x)
#align right_ord_continuous RightOrdContinuous
namespace LeftOrdContinuous
section CompleteLattice
variable [CompleteLattice α] [CompleteLattice β] {f : α → β}
theorem map_sSup' (hf : LeftOrdContinuous f) (s : Set α) : f (sSup s) = sSup (f '' s) :=
(hf <| isLUB_sSup s).sSup_eq.symm
#align left_ord_continuous.map_Sup' LeftOrdContinuous.map_sSup'
theorem map_sSup (hf : LeftOrdContinuous f) (s : Set α) : f (sSup s) = ⨆ x ∈ s, f x := by
rw [hf.map_sSup', sSup_image]
#align left_ord_continuous.map_Sup LeftOrdContinuous.map_sSup
| Mathlib/Order/OrdContinuous.lean | 135 | 137 | theorem map_iSup (hf : LeftOrdContinuous f) (g : ι → α) : f (⨆ i, g i) = ⨆ i, f (g i) := by |
simp only [iSup, hf.map_sSup', ← range_comp]
rfl
| 2 | 7.389056 | 1 | 0.4 | 5 | 395 |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.RelIso.Basic
#align_import order.ord_continuous from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w x
variable {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x}
open Function OrderDual Set
def LeftOrdContinuous [Preorder α] [Preorder β] (f : α → β) :=
∀ ⦃s : Set α⦄ ⦃x⦄, IsLUB s x → IsLUB (f '' s) (f x)
#align left_ord_continuous LeftOrdContinuous
def RightOrdContinuous [Preorder α] [Preorder β] (f : α → β) :=
∀ ⦃s : Set α⦄ ⦃x⦄, IsGLB s x → IsGLB (f '' s) (f x)
#align right_ord_continuous RightOrdContinuous
namespace LeftOrdContinuous
section ConditionallyCompleteLattice
variable [ConditionallyCompleteLattice α] [ConditionallyCompleteLattice β] [Nonempty ι] {f : α → β}
theorem map_csSup (hf : LeftOrdContinuous f) {s : Set α} (sne : s.Nonempty) (sbdd : BddAbove s) :
f (sSup s) = sSup (f '' s) :=
((hf <| isLUB_csSup sne sbdd).csSup_eq <| sne.image f).symm
#align left_ord_continuous.map_cSup LeftOrdContinuous.map_csSup
| Mathlib/Order/OrdContinuous.lean | 151 | 154 | theorem map_ciSup (hf : LeftOrdContinuous f) {g : ι → α} (hg : BddAbove (range g)) :
f (⨆ i, g i) = ⨆ i, f (g i) := by |
simp only [iSup, hf.map_csSup (range_nonempty _) hg, ← range_comp]
rfl
| 2 | 7.389056 | 1 | 0.4 | 5 | 395 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality
import Mathlib.MeasureTheory.Measure.OpenPos
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Order.Filter.IndicatorFunction
#align_import measure_theory.function.lp_space from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
noncomputable section
set_option linter.uppercaseLean3 false
open TopologicalSpace MeasureTheory Filter
open scoped NNReal ENNReal Topology MeasureTheory Uniformity
variable {α E F G : Type*} {m m0 : MeasurableSpace α} {p : ℝ≥0∞} {q : ℝ} {μ ν : Measure α}
[NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup G]
namespace MeasureTheory
@[simp]
theorem snorm_aeeqFun {α E : Type*} [MeasurableSpace α] {μ : Measure α} [NormedAddCommGroup E]
{p : ℝ≥0∞} {f : α → E} (hf : AEStronglyMeasurable f μ) :
snorm (AEEqFun.mk f hf) p μ = snorm f p μ :=
snorm_congr_ae (AEEqFun.coeFn_mk _ _)
#align measure_theory.snorm_ae_eq_fun MeasureTheory.snorm_aeeqFun
| Mathlib/MeasureTheory/Function/LpSpace.lean | 95 | 97 | theorem Memℒp.snorm_mk_lt_top {α E : Type*} [MeasurableSpace α] {μ : Measure α}
[NormedAddCommGroup E] {p : ℝ≥0∞} {f : α → E} (hfp : Memℒp f p μ) :
snorm (AEEqFun.mk f hfp.1) p μ < ∞ := by | simp [hfp.2]
| 1 | 2.718282 | 0 | 0.4 | 5 | 396 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality
import Mathlib.MeasureTheory.Measure.OpenPos
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Order.Filter.IndicatorFunction
#align_import measure_theory.function.lp_space from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
noncomputable section
set_option linter.uppercaseLean3 false
open TopologicalSpace MeasureTheory Filter
open scoped NNReal ENNReal Topology MeasureTheory Uniformity
variable {α E F G : Type*} {m m0 : MeasurableSpace α} {p : ℝ≥0∞} {q : ℝ} {μ ν : Measure α}
[NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup G]
namespace MeasureTheory
@[simp]
theorem snorm_aeeqFun {α E : Type*} [MeasurableSpace α] {μ : Measure α} [NormedAddCommGroup E]
{p : ℝ≥0∞} {f : α → E} (hf : AEStronglyMeasurable f μ) :
snorm (AEEqFun.mk f hf) p μ = snorm f p μ :=
snorm_congr_ae (AEEqFun.coeFn_mk _ _)
#align measure_theory.snorm_ae_eq_fun MeasureTheory.snorm_aeeqFun
theorem Memℒp.snorm_mk_lt_top {α E : Type*} [MeasurableSpace α] {μ : Measure α}
[NormedAddCommGroup E] {p : ℝ≥0∞} {f : α → E} (hfp : Memℒp f p μ) :
snorm (AEEqFun.mk f hfp.1) p μ < ∞ := by simp [hfp.2]
#align measure_theory.mem_ℒp.snorm_mk_lt_top MeasureTheory.Memℒp.snorm_mk_lt_top
def Lp {α} (E : Type*) {m : MeasurableSpace α} [NormedAddCommGroup E] (p : ℝ≥0∞)
(μ : Measure α := by volume_tac) : AddSubgroup (α →ₘ[μ] E) where
carrier := { f | snorm f p μ < ∞ }
zero_mem' := by simp [snorm_congr_ae AEEqFun.coeFn_zero, snorm_zero]
add_mem' {f g} hf hg := by
simp [snorm_congr_ae (AEEqFun.coeFn_add f g),
snorm_add_lt_top ⟨f.aestronglyMeasurable, hf⟩ ⟨g.aestronglyMeasurable, hg⟩]
neg_mem' {f} hf := by rwa [Set.mem_setOf_eq, snorm_congr_ae (AEEqFun.coeFn_neg f), snorm_neg]
#align measure_theory.Lp MeasureTheory.Lp
-- Porting note: calling the first argument `α` breaks the `(α := ·)` notation
scoped notation:25 α' " →₁[" μ "] " E => MeasureTheory.Lp (α := α') E 1 μ
scoped notation:25 α' " →₂[" μ "] " E => MeasureTheory.Lp (α := α') E 2 μ
namespace Memℒp
def toLp (f : α → E) (h_mem_ℒp : Memℒp f p μ) : Lp E p μ :=
⟨AEEqFun.mk f h_mem_ℒp.1, h_mem_ℒp.snorm_mk_lt_top⟩
#align measure_theory.mem_ℒp.to_Lp MeasureTheory.Memℒp.toLp
theorem coeFn_toLp {f : α → E} (hf : Memℒp f p μ) : hf.toLp f =ᵐ[μ] f :=
AEEqFun.coeFn_mk _ _
#align measure_theory.mem_ℒp.coe_fn_to_Lp MeasureTheory.Memℒp.coeFn_toLp
| Mathlib/MeasureTheory/Function/LpSpace.lean | 126 | 127 | theorem toLp_congr {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) (hfg : f =ᵐ[μ] g) :
hf.toLp f = hg.toLp g := by | simp [toLp, hfg]
| 1 | 2.718282 | 0 | 0.4 | 5 | 396 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality
import Mathlib.MeasureTheory.Measure.OpenPos
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Order.Filter.IndicatorFunction
#align_import measure_theory.function.lp_space from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
noncomputable section
set_option linter.uppercaseLean3 false
open TopologicalSpace MeasureTheory Filter
open scoped NNReal ENNReal Topology MeasureTheory Uniformity
variable {α E F G : Type*} {m m0 : MeasurableSpace α} {p : ℝ≥0∞} {q : ℝ} {μ ν : Measure α}
[NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup G]
namespace MeasureTheory
@[simp]
theorem snorm_aeeqFun {α E : Type*} [MeasurableSpace α] {μ : Measure α} [NormedAddCommGroup E]
{p : ℝ≥0∞} {f : α → E} (hf : AEStronglyMeasurable f μ) :
snorm (AEEqFun.mk f hf) p μ = snorm f p μ :=
snorm_congr_ae (AEEqFun.coeFn_mk _ _)
#align measure_theory.snorm_ae_eq_fun MeasureTheory.snorm_aeeqFun
theorem Memℒp.snorm_mk_lt_top {α E : Type*} [MeasurableSpace α] {μ : Measure α}
[NormedAddCommGroup E] {p : ℝ≥0∞} {f : α → E} (hfp : Memℒp f p μ) :
snorm (AEEqFun.mk f hfp.1) p μ < ∞ := by simp [hfp.2]
#align measure_theory.mem_ℒp.snorm_mk_lt_top MeasureTheory.Memℒp.snorm_mk_lt_top
def Lp {α} (E : Type*) {m : MeasurableSpace α} [NormedAddCommGroup E] (p : ℝ≥0∞)
(μ : Measure α := by volume_tac) : AddSubgroup (α →ₘ[μ] E) where
carrier := { f | snorm f p μ < ∞ }
zero_mem' := by simp [snorm_congr_ae AEEqFun.coeFn_zero, snorm_zero]
add_mem' {f g} hf hg := by
simp [snorm_congr_ae (AEEqFun.coeFn_add f g),
snorm_add_lt_top ⟨f.aestronglyMeasurable, hf⟩ ⟨g.aestronglyMeasurable, hg⟩]
neg_mem' {f} hf := by rwa [Set.mem_setOf_eq, snorm_congr_ae (AEEqFun.coeFn_neg f), snorm_neg]
#align measure_theory.Lp MeasureTheory.Lp
-- Porting note: calling the first argument `α` breaks the `(α := ·)` notation
scoped notation:25 α' " →₁[" μ "] " E => MeasureTheory.Lp (α := α') E 1 μ
scoped notation:25 α' " →₂[" μ "] " E => MeasureTheory.Lp (α := α') E 2 μ
namespace Memℒp
def toLp (f : α → E) (h_mem_ℒp : Memℒp f p μ) : Lp E p μ :=
⟨AEEqFun.mk f h_mem_ℒp.1, h_mem_ℒp.snorm_mk_lt_top⟩
#align measure_theory.mem_ℒp.to_Lp MeasureTheory.Memℒp.toLp
theorem coeFn_toLp {f : α → E} (hf : Memℒp f p μ) : hf.toLp f =ᵐ[μ] f :=
AEEqFun.coeFn_mk _ _
#align measure_theory.mem_ℒp.coe_fn_to_Lp MeasureTheory.Memℒp.coeFn_toLp
theorem toLp_congr {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) (hfg : f =ᵐ[μ] g) :
hf.toLp f = hg.toLp g := by simp [toLp, hfg]
#align measure_theory.mem_ℒp.to_Lp_congr MeasureTheory.Memℒp.toLp_congr
@[simp]
| Mathlib/MeasureTheory/Function/LpSpace.lean | 131 | 132 | theorem toLp_eq_toLp_iff {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) :
hf.toLp f = hg.toLp g ↔ f =ᵐ[μ] g := by | simp [toLp]
| 1 | 2.718282 | 0 | 0.4 | 5 | 396 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality
import Mathlib.MeasureTheory.Measure.OpenPos
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Order.Filter.IndicatorFunction
#align_import measure_theory.function.lp_space from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
noncomputable section
set_option linter.uppercaseLean3 false
open TopologicalSpace MeasureTheory Filter
open scoped NNReal ENNReal Topology MeasureTheory Uniformity
variable {α E F G : Type*} {m m0 : MeasurableSpace α} {p : ℝ≥0∞} {q : ℝ} {μ ν : Measure α}
[NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup G]
namespace MeasureTheory
@[simp]
theorem snorm_aeeqFun {α E : Type*} [MeasurableSpace α] {μ : Measure α} [NormedAddCommGroup E]
{p : ℝ≥0∞} {f : α → E} (hf : AEStronglyMeasurable f μ) :
snorm (AEEqFun.mk f hf) p μ = snorm f p μ :=
snorm_congr_ae (AEEqFun.coeFn_mk _ _)
#align measure_theory.snorm_ae_eq_fun MeasureTheory.snorm_aeeqFun
theorem Memℒp.snorm_mk_lt_top {α E : Type*} [MeasurableSpace α] {μ : Measure α}
[NormedAddCommGroup E] {p : ℝ≥0∞} {f : α → E} (hfp : Memℒp f p μ) :
snorm (AEEqFun.mk f hfp.1) p μ < ∞ := by simp [hfp.2]
#align measure_theory.mem_ℒp.snorm_mk_lt_top MeasureTheory.Memℒp.snorm_mk_lt_top
def Lp {α} (E : Type*) {m : MeasurableSpace α} [NormedAddCommGroup E] (p : ℝ≥0∞)
(μ : Measure α := by volume_tac) : AddSubgroup (α →ₘ[μ] E) where
carrier := { f | snorm f p μ < ∞ }
zero_mem' := by simp [snorm_congr_ae AEEqFun.coeFn_zero, snorm_zero]
add_mem' {f g} hf hg := by
simp [snorm_congr_ae (AEEqFun.coeFn_add f g),
snorm_add_lt_top ⟨f.aestronglyMeasurable, hf⟩ ⟨g.aestronglyMeasurable, hg⟩]
neg_mem' {f} hf := by rwa [Set.mem_setOf_eq, snorm_congr_ae (AEEqFun.coeFn_neg f), snorm_neg]
#align measure_theory.Lp MeasureTheory.Lp
-- Porting note: calling the first argument `α` breaks the `(α := ·)` notation
scoped notation:25 α' " →₁[" μ "] " E => MeasureTheory.Lp (α := α') E 1 μ
scoped notation:25 α' " →₂[" μ "] " E => MeasureTheory.Lp (α := α') E 2 μ
namespace Lp
instance instCoeFun : CoeFun (Lp E p μ) (fun _ => α → E) :=
⟨fun f => ((f : α →ₘ[μ] E) : α → E)⟩
#align measure_theory.Lp.has_coe_to_fun MeasureTheory.Lp.instCoeFun
@[ext high]
| Mathlib/MeasureTheory/Function/LpSpace.lean | 163 | 167 | theorem ext {f g : Lp E p μ} (h : f =ᵐ[μ] g) : f = g := by |
cases f
cases g
simp only [Subtype.mk_eq_mk]
exact AEEqFun.ext h
| 4 | 54.59815 | 2 | 0.4 | 5 | 396 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality
import Mathlib.MeasureTheory.Measure.OpenPos
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Order.Filter.IndicatorFunction
#align_import measure_theory.function.lp_space from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
noncomputable section
set_option linter.uppercaseLean3 false
open TopologicalSpace MeasureTheory Filter
open scoped NNReal ENNReal Topology MeasureTheory Uniformity
variable {α E F G : Type*} {m m0 : MeasurableSpace α} {p : ℝ≥0∞} {q : ℝ} {μ ν : Measure α}
[NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup G]
namespace MeasureTheory
@[simp]
theorem snorm_aeeqFun {α E : Type*} [MeasurableSpace α] {μ : Measure α} [NormedAddCommGroup E]
{p : ℝ≥0∞} {f : α → E} (hf : AEStronglyMeasurable f μ) :
snorm (AEEqFun.mk f hf) p μ = snorm f p μ :=
snorm_congr_ae (AEEqFun.coeFn_mk _ _)
#align measure_theory.snorm_ae_eq_fun MeasureTheory.snorm_aeeqFun
theorem Memℒp.snorm_mk_lt_top {α E : Type*} [MeasurableSpace α] {μ : Measure α}
[NormedAddCommGroup E] {p : ℝ≥0∞} {f : α → E} (hfp : Memℒp f p μ) :
snorm (AEEqFun.mk f hfp.1) p μ < ∞ := by simp [hfp.2]
#align measure_theory.mem_ℒp.snorm_mk_lt_top MeasureTheory.Memℒp.snorm_mk_lt_top
def Lp {α} (E : Type*) {m : MeasurableSpace α} [NormedAddCommGroup E] (p : ℝ≥0∞)
(μ : Measure α := by volume_tac) : AddSubgroup (α →ₘ[μ] E) where
carrier := { f | snorm f p μ < ∞ }
zero_mem' := by simp [snorm_congr_ae AEEqFun.coeFn_zero, snorm_zero]
add_mem' {f g} hf hg := by
simp [snorm_congr_ae (AEEqFun.coeFn_add f g),
snorm_add_lt_top ⟨f.aestronglyMeasurable, hf⟩ ⟨g.aestronglyMeasurable, hg⟩]
neg_mem' {f} hf := by rwa [Set.mem_setOf_eq, snorm_congr_ae (AEEqFun.coeFn_neg f), snorm_neg]
#align measure_theory.Lp MeasureTheory.Lp
-- Porting note: calling the first argument `α` breaks the `(α := ·)` notation
scoped notation:25 α' " →₁[" μ "] " E => MeasureTheory.Lp (α := α') E 1 μ
scoped notation:25 α' " →₂[" μ "] " E => MeasureTheory.Lp (α := α') E 2 μ
namespace Lp
instance instCoeFun : CoeFun (Lp E p μ) (fun _ => α → E) :=
⟨fun f => ((f : α →ₘ[μ] E) : α → E)⟩
#align measure_theory.Lp.has_coe_to_fun MeasureTheory.Lp.instCoeFun
@[ext high]
theorem ext {f g : Lp E p μ} (h : f =ᵐ[μ] g) : f = g := by
cases f
cases g
simp only [Subtype.mk_eq_mk]
exact AEEqFun.ext h
#align measure_theory.Lp.ext MeasureTheory.Lp.ext
theorem ext_iff {f g : Lp E p μ} : f = g ↔ f =ᵐ[μ] g :=
⟨fun h => by rw [h], fun h => ext h⟩
#align measure_theory.Lp.ext_iff MeasureTheory.Lp.ext_iff
theorem mem_Lp_iff_snorm_lt_top {f : α →ₘ[μ] E} : f ∈ Lp E p μ ↔ snorm f p μ < ∞ := Iff.rfl
#align measure_theory.Lp.mem_Lp_iff_snorm_lt_top MeasureTheory.Lp.mem_Lp_iff_snorm_lt_top
| Mathlib/MeasureTheory/Function/LpSpace.lean | 177 | 178 | theorem mem_Lp_iff_memℒp {f : α →ₘ[μ] E} : f ∈ Lp E p μ ↔ Memℒp f p μ := by |
simp [mem_Lp_iff_snorm_lt_top, Memℒp, f.stronglyMeasurable.aestronglyMeasurable]
| 1 | 2.718282 | 0 | 0.4 | 5 | 396 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $type → $type))
universe u
variable {R : Type u}
namespace NonUnitalNonAssocSemiring
@[ext] theorem ext ⦃inst₁ inst₂ : NonUnitalNonAssocSemiring R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_mul : local_hMul[R, inst₁] = local_hMul[R, inst₂]) :
inst₁ = inst₂ := by
-- Split into `AddMonoid` instance, `mul` function and properties.
rcases inst₁ with @⟨_, ⟨⟩⟩
rcases inst₂ with @⟨_, ⟨⟩⟩
-- Prove equality of parts using already-proved extensionality lemmas.
congr; ext : 1; assumption
| Mathlib/Algebra/Ring/Ext.lean | 73 | 77 | theorem toDistrib_injective : Function.Injective (@toDistrib R) := by |
intro _ _ h
ext x y
· exact congrArg (·.toAdd.add x y) h
· exact congrArg (·.toMul.mul x y) h
| 4 | 54.59815 | 2 | 0.4 | 10 | 397 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $type → $type))
universe u
variable {R : Type u}
namespace NonUnitalSemiring
| Mathlib/Algebra/Ring/Ext.lean | 90 | 92 | theorem toNonUnitalNonAssocSemiring_injective :
Function.Injective (@toNonUnitalNonAssocSemiring R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
| 1 | 2.718282 | 0 | 0.4 | 10 | 397 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $type → $type))
universe u
variable {R : Type u}
@[ext] theorem AddMonoidWithOne.ext ⦃inst₁ inst₂ : AddMonoidWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one : R)) :
inst₁ = inst₂ := by
have h_monoid : inst₁.toAddMonoid = inst₂.toAddMonoid := by ext : 1; exact h_add
have h_zero' : inst₁.toZero = inst₂.toZero := congrArg (·.toZero) h_monoid
have h_one' : inst₁.toOne = inst₂.toOne :=
congrArg One.mk h_one
have h_natCast : inst₁.toNatCast.natCast = inst₂.toNatCast.natCast := by
funext n; induction n with
| zero => rewrite [inst₁.natCast_zero, inst₂.natCast_zero]
exact congrArg (@Zero.zero R) h_zero'
| succ n h => rw [inst₁.natCast_succ, inst₂.natCast_succ, h_add]
exact congrArg₂ _ h h_one
rcases inst₁ with @⟨⟨⟩⟩; rcases inst₂ with @⟨⟨⟩⟩
congr
| Mathlib/Algebra/Ring/Ext.lean | 133 | 135 | theorem AddCommMonoidWithOne.toAddMonoidWithOne_injective :
Function.Injective (@AddCommMonoidWithOne.toAddMonoidWithOne R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
| 1 | 2.718282 | 0 | 0.4 | 10 | 397 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $type → $type))
universe u
variable {R : Type u}
@[ext] theorem AddMonoidWithOne.ext ⦃inst₁ inst₂ : AddMonoidWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one : R)) :
inst₁ = inst₂ := by
have h_monoid : inst₁.toAddMonoid = inst₂.toAddMonoid := by ext : 1; exact h_add
have h_zero' : inst₁.toZero = inst₂.toZero := congrArg (·.toZero) h_monoid
have h_one' : inst₁.toOne = inst₂.toOne :=
congrArg One.mk h_one
have h_natCast : inst₁.toNatCast.natCast = inst₂.toNatCast.natCast := by
funext n; induction n with
| zero => rewrite [inst₁.natCast_zero, inst₂.natCast_zero]
exact congrArg (@Zero.zero R) h_zero'
| succ n h => rw [inst₁.natCast_succ, inst₂.natCast_succ, h_add]
exact congrArg₂ _ h h_one
rcases inst₁ with @⟨⟨⟩⟩; rcases inst₂ with @⟨⟨⟩⟩
congr
theorem AddCommMonoidWithOne.toAddMonoidWithOne_injective :
Function.Injective (@AddCommMonoidWithOne.toAddMonoidWithOne R) := by
rintro ⟨⟩ ⟨⟩ _; congr
@[ext] theorem AddCommMonoidWithOne.ext ⦃inst₁ inst₂ : AddCommMonoidWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one : R)) :
inst₁ = inst₂ :=
AddCommMonoidWithOne.toAddMonoidWithOne_injective <|
AddMonoidWithOne.ext h_add h_one
namespace NonUnitalNonAssocRing
@[ext] theorem ext ⦃inst₁ inst₂ : NonUnitalNonAssocRing R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_mul : local_hMul[R, inst₁] = local_hMul[R, inst₂]) :
inst₁ = inst₂ := by
-- Split into `AddCommGroup` instance, `mul` function and properties.
rcases inst₁ with @⟨_, ⟨⟩⟩; rcases inst₂ with @⟨_, ⟨⟩⟩
congr; (ext : 1; assumption)
| Mathlib/Algebra/Ring/Ext.lean | 195 | 201 | theorem toNonUnitalNonAssocSemiring_injective :
Function.Injective (@toNonUnitalNonAssocSemiring R) := by |
intro _ _ h
-- Use above extensionality lemma to prove injectivity by showing that `h_add` and `h_mul` hold.
ext x y
· exact congrArg (·.toAdd.add x y) h
· exact congrArg (·.toMul.mul x y) h
| 5 | 148.413159 | 2 | 0.4 | 10 | 397 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $type → $type))
universe u
variable {R : Type u}
@[ext] theorem AddMonoidWithOne.ext ⦃inst₁ inst₂ : AddMonoidWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one : R)) :
inst₁ = inst₂ := by
have h_monoid : inst₁.toAddMonoid = inst₂.toAddMonoid := by ext : 1; exact h_add
have h_zero' : inst₁.toZero = inst₂.toZero := congrArg (·.toZero) h_monoid
have h_one' : inst₁.toOne = inst₂.toOne :=
congrArg One.mk h_one
have h_natCast : inst₁.toNatCast.natCast = inst₂.toNatCast.natCast := by
funext n; induction n with
| zero => rewrite [inst₁.natCast_zero, inst₂.natCast_zero]
exact congrArg (@Zero.zero R) h_zero'
| succ n h => rw [inst₁.natCast_succ, inst₂.natCast_succ, h_add]
exact congrArg₂ _ h h_one
rcases inst₁ with @⟨⟨⟩⟩; rcases inst₂ with @⟨⟨⟩⟩
congr
theorem AddCommMonoidWithOne.toAddMonoidWithOne_injective :
Function.Injective (@AddCommMonoidWithOne.toAddMonoidWithOne R) := by
rintro ⟨⟩ ⟨⟩ _; congr
@[ext] theorem AddCommMonoidWithOne.ext ⦃inst₁ inst₂ : AddCommMonoidWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one : R)) :
inst₁ = inst₂ :=
AddCommMonoidWithOne.toAddMonoidWithOne_injective <|
AddMonoidWithOne.ext h_add h_one
@[ext] theorem AddGroupWithOne.ext ⦃inst₁ inst₂ : AddGroupWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one)) :
inst₁ = inst₂ := by
have : inst₁.toAddMonoidWithOne = inst₂.toAddMonoidWithOne :=
AddMonoidWithOne.ext h_add h_one
have : inst₁.toNatCast = inst₂.toNatCast := congrArg (·.toNatCast) this
have h_group : inst₁.toAddGroup = inst₂.toAddGroup := by ext : 1; exact h_add
-- Extract equality of necessary substructures from h_group
injection h_group with h_group; injection h_group
have : inst₁.toIntCast.intCast = inst₂.toIntCast.intCast := by
funext n; cases n with
| ofNat n => rewrite [Int.ofNat_eq_coe, inst₁.intCast_ofNat, inst₂.intCast_ofNat]; congr
| negSucc n => rewrite [inst₁.intCast_negSucc, inst₂.intCast_negSucc]; congr
rcases inst₁ with @⟨⟨⟩⟩; rcases inst₂ with @⟨⟨⟩⟩
congr
@[ext] theorem AddCommGroupWithOne.ext ⦃inst₁ inst₂ : AddCommGroupWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one)) :
inst₁ = inst₂ := by
have : inst₁.toAddCommGroup = inst₂.toAddCommGroup :=
AddCommGroup.ext h_add
have : inst₁.toAddGroupWithOne = inst₂.toAddGroupWithOne :=
AddGroupWithOne.ext h_add h_one
injection this with _ h_addMonoidWithOne; injection h_addMonoidWithOne
cases inst₁; cases inst₂
congr
namespace NonUnitalNonAssocCommSemiring
| Mathlib/Algebra/Ring/Ext.lean | 405 | 407 | theorem toNonUnitalNonAssocSemiring_injective :
Function.Injective (@toNonUnitalNonAssocSemiring R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
| 1 | 2.718282 | 0 | 0.4 | 10 | 397 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $type → $type))
universe u
variable {R : Type u}
@[ext] theorem AddMonoidWithOne.ext ⦃inst₁ inst₂ : AddMonoidWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one : R)) :
inst₁ = inst₂ := by
have h_monoid : inst₁.toAddMonoid = inst₂.toAddMonoid := by ext : 1; exact h_add
have h_zero' : inst₁.toZero = inst₂.toZero := congrArg (·.toZero) h_monoid
have h_one' : inst₁.toOne = inst₂.toOne :=
congrArg One.mk h_one
have h_natCast : inst₁.toNatCast.natCast = inst₂.toNatCast.natCast := by
funext n; induction n with
| zero => rewrite [inst₁.natCast_zero, inst₂.natCast_zero]
exact congrArg (@Zero.zero R) h_zero'
| succ n h => rw [inst₁.natCast_succ, inst₂.natCast_succ, h_add]
exact congrArg₂ _ h h_one
rcases inst₁ with @⟨⟨⟩⟩; rcases inst₂ with @⟨⟨⟩⟩
congr
theorem AddCommMonoidWithOne.toAddMonoidWithOne_injective :
Function.Injective (@AddCommMonoidWithOne.toAddMonoidWithOne R) := by
rintro ⟨⟩ ⟨⟩ _; congr
@[ext] theorem AddCommMonoidWithOne.ext ⦃inst₁ inst₂ : AddCommMonoidWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one : R)) :
inst₁ = inst₂ :=
AddCommMonoidWithOne.toAddMonoidWithOne_injective <|
AddMonoidWithOne.ext h_add h_one
@[ext] theorem AddGroupWithOne.ext ⦃inst₁ inst₂ : AddGroupWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one)) :
inst₁ = inst₂ := by
have : inst₁.toAddMonoidWithOne = inst₂.toAddMonoidWithOne :=
AddMonoidWithOne.ext h_add h_one
have : inst₁.toNatCast = inst₂.toNatCast := congrArg (·.toNatCast) this
have h_group : inst₁.toAddGroup = inst₂.toAddGroup := by ext : 1; exact h_add
-- Extract equality of necessary substructures from h_group
injection h_group with h_group; injection h_group
have : inst₁.toIntCast.intCast = inst₂.toIntCast.intCast := by
funext n; cases n with
| ofNat n => rewrite [Int.ofNat_eq_coe, inst₁.intCast_ofNat, inst₂.intCast_ofNat]; congr
| negSucc n => rewrite [inst₁.intCast_negSucc, inst₂.intCast_negSucc]; congr
rcases inst₁ with @⟨⟨⟩⟩; rcases inst₂ with @⟨⟨⟩⟩
congr
@[ext] theorem AddCommGroupWithOne.ext ⦃inst₁ inst₂ : AddCommGroupWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one)) :
inst₁ = inst₂ := by
have : inst₁.toAddCommGroup = inst₂.toAddCommGroup :=
AddCommGroup.ext h_add
have : inst₁.toAddGroupWithOne = inst₂.toAddGroupWithOne :=
AddGroupWithOne.ext h_add h_one
injection this with _ h_addMonoidWithOne; injection h_addMonoidWithOne
cases inst₁; cases inst₂
congr
namespace NonUnitalCommSemiring
| Mathlib/Algebra/Ring/Ext.lean | 427 | 429 | theorem toNonUnitalSemiring_injective :
Function.Injective (@toNonUnitalSemiring R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
| 1 | 2.718282 | 0 | 0.4 | 10 | 397 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $type → $type))
universe u
variable {R : Type u}
@[ext] theorem AddMonoidWithOne.ext ⦃inst₁ inst₂ : AddMonoidWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one : R)) :
inst₁ = inst₂ := by
have h_monoid : inst₁.toAddMonoid = inst₂.toAddMonoid := by ext : 1; exact h_add
have h_zero' : inst₁.toZero = inst₂.toZero := congrArg (·.toZero) h_monoid
have h_one' : inst₁.toOne = inst₂.toOne :=
congrArg One.mk h_one
have h_natCast : inst₁.toNatCast.natCast = inst₂.toNatCast.natCast := by
funext n; induction n with
| zero => rewrite [inst₁.natCast_zero, inst₂.natCast_zero]
exact congrArg (@Zero.zero R) h_zero'
| succ n h => rw [inst₁.natCast_succ, inst₂.natCast_succ, h_add]
exact congrArg₂ _ h h_one
rcases inst₁ with @⟨⟨⟩⟩; rcases inst₂ with @⟨⟨⟩⟩
congr
theorem AddCommMonoidWithOne.toAddMonoidWithOne_injective :
Function.Injective (@AddCommMonoidWithOne.toAddMonoidWithOne R) := by
rintro ⟨⟩ ⟨⟩ _; congr
@[ext] theorem AddCommMonoidWithOne.ext ⦃inst₁ inst₂ : AddCommMonoidWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one : R)) :
inst₁ = inst₂ :=
AddCommMonoidWithOne.toAddMonoidWithOne_injective <|
AddMonoidWithOne.ext h_add h_one
@[ext] theorem AddGroupWithOne.ext ⦃inst₁ inst₂ : AddGroupWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one)) :
inst₁ = inst₂ := by
have : inst₁.toAddMonoidWithOne = inst₂.toAddMonoidWithOne :=
AddMonoidWithOne.ext h_add h_one
have : inst₁.toNatCast = inst₂.toNatCast := congrArg (·.toNatCast) this
have h_group : inst₁.toAddGroup = inst₂.toAddGroup := by ext : 1; exact h_add
-- Extract equality of necessary substructures from h_group
injection h_group with h_group; injection h_group
have : inst₁.toIntCast.intCast = inst₂.toIntCast.intCast := by
funext n; cases n with
| ofNat n => rewrite [Int.ofNat_eq_coe, inst₁.intCast_ofNat, inst₂.intCast_ofNat]; congr
| negSucc n => rewrite [inst₁.intCast_negSucc, inst₂.intCast_negSucc]; congr
rcases inst₁ with @⟨⟨⟩⟩; rcases inst₂ with @⟨⟨⟩⟩
congr
@[ext] theorem AddCommGroupWithOne.ext ⦃inst₁ inst₂ : AddCommGroupWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one)) :
inst₁ = inst₂ := by
have : inst₁.toAddCommGroup = inst₂.toAddCommGroup :=
AddCommGroup.ext h_add
have : inst₁.toAddGroupWithOne = inst₂.toAddGroupWithOne :=
AddGroupWithOne.ext h_add h_one
injection this with _ h_addMonoidWithOne; injection h_addMonoidWithOne
cases inst₁; cases inst₂
congr
-- At present, there is no `NonAssocCommSemiring` in Mathlib.
namespace NonUnitalNonAssocCommRing
| Mathlib/Algebra/Ring/Ext.lean | 451 | 453 | theorem toNonUnitalNonAssocRing_injective :
Function.Injective (@toNonUnitalNonAssocRing R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
| 1 | 2.718282 | 0 | 0.4 | 10 | 397 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $type → $type))
universe u
variable {R : Type u}
@[ext] theorem AddMonoidWithOne.ext ⦃inst₁ inst₂ : AddMonoidWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one : R)) :
inst₁ = inst₂ := by
have h_monoid : inst₁.toAddMonoid = inst₂.toAddMonoid := by ext : 1; exact h_add
have h_zero' : inst₁.toZero = inst₂.toZero := congrArg (·.toZero) h_monoid
have h_one' : inst₁.toOne = inst₂.toOne :=
congrArg One.mk h_one
have h_natCast : inst₁.toNatCast.natCast = inst₂.toNatCast.natCast := by
funext n; induction n with
| zero => rewrite [inst₁.natCast_zero, inst₂.natCast_zero]
exact congrArg (@Zero.zero R) h_zero'
| succ n h => rw [inst₁.natCast_succ, inst₂.natCast_succ, h_add]
exact congrArg₂ _ h h_one
rcases inst₁ with @⟨⟨⟩⟩; rcases inst₂ with @⟨⟨⟩⟩
congr
theorem AddCommMonoidWithOne.toAddMonoidWithOne_injective :
Function.Injective (@AddCommMonoidWithOne.toAddMonoidWithOne R) := by
rintro ⟨⟩ ⟨⟩ _; congr
@[ext] theorem AddCommMonoidWithOne.ext ⦃inst₁ inst₂ : AddCommMonoidWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one : R)) :
inst₁ = inst₂ :=
AddCommMonoidWithOne.toAddMonoidWithOne_injective <|
AddMonoidWithOne.ext h_add h_one
@[ext] theorem AddGroupWithOne.ext ⦃inst₁ inst₂ : AddGroupWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one)) :
inst₁ = inst₂ := by
have : inst₁.toAddMonoidWithOne = inst₂.toAddMonoidWithOne :=
AddMonoidWithOne.ext h_add h_one
have : inst₁.toNatCast = inst₂.toNatCast := congrArg (·.toNatCast) this
have h_group : inst₁.toAddGroup = inst₂.toAddGroup := by ext : 1; exact h_add
-- Extract equality of necessary substructures from h_group
injection h_group with h_group; injection h_group
have : inst₁.toIntCast.intCast = inst₂.toIntCast.intCast := by
funext n; cases n with
| ofNat n => rewrite [Int.ofNat_eq_coe, inst₁.intCast_ofNat, inst₂.intCast_ofNat]; congr
| negSucc n => rewrite [inst₁.intCast_negSucc, inst₂.intCast_negSucc]; congr
rcases inst₁ with @⟨⟨⟩⟩; rcases inst₂ with @⟨⟨⟩⟩
congr
@[ext] theorem AddCommGroupWithOne.ext ⦃inst₁ inst₂ : AddCommGroupWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one)) :
inst₁ = inst₂ := by
have : inst₁.toAddCommGroup = inst₂.toAddCommGroup :=
AddCommGroup.ext h_add
have : inst₁.toAddGroupWithOne = inst₂.toAddGroupWithOne :=
AddGroupWithOne.ext h_add h_one
injection this with _ h_addMonoidWithOne; injection h_addMonoidWithOne
cases inst₁; cases inst₂
congr
-- At present, there is no `NonAssocCommSemiring` in Mathlib.
namespace NonUnitalCommRing
| Mathlib/Algebra/Ring/Ext.lean | 473 | 475 | theorem toNonUnitalRing_injective :
Function.Injective (@toNonUnitalRing R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
| 1 | 2.718282 | 0 | 0.4 | 10 | 397 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $type → $type))
universe u
variable {R : Type u}
@[ext] theorem AddMonoidWithOne.ext ⦃inst₁ inst₂ : AddMonoidWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one : R)) :
inst₁ = inst₂ := by
have h_monoid : inst₁.toAddMonoid = inst₂.toAddMonoid := by ext : 1; exact h_add
have h_zero' : inst₁.toZero = inst₂.toZero := congrArg (·.toZero) h_monoid
have h_one' : inst₁.toOne = inst₂.toOne :=
congrArg One.mk h_one
have h_natCast : inst₁.toNatCast.natCast = inst₂.toNatCast.natCast := by
funext n; induction n with
| zero => rewrite [inst₁.natCast_zero, inst₂.natCast_zero]
exact congrArg (@Zero.zero R) h_zero'
| succ n h => rw [inst₁.natCast_succ, inst₂.natCast_succ, h_add]
exact congrArg₂ _ h h_one
rcases inst₁ with @⟨⟨⟩⟩; rcases inst₂ with @⟨⟨⟩⟩
congr
theorem AddCommMonoidWithOne.toAddMonoidWithOne_injective :
Function.Injective (@AddCommMonoidWithOne.toAddMonoidWithOne R) := by
rintro ⟨⟩ ⟨⟩ _; congr
@[ext] theorem AddCommMonoidWithOne.ext ⦃inst₁ inst₂ : AddCommMonoidWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one : R)) :
inst₁ = inst₂ :=
AddCommMonoidWithOne.toAddMonoidWithOne_injective <|
AddMonoidWithOne.ext h_add h_one
@[ext] theorem AddGroupWithOne.ext ⦃inst₁ inst₂ : AddGroupWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one)) :
inst₁ = inst₂ := by
have : inst₁.toAddMonoidWithOne = inst₂.toAddMonoidWithOne :=
AddMonoidWithOne.ext h_add h_one
have : inst₁.toNatCast = inst₂.toNatCast := congrArg (·.toNatCast) this
have h_group : inst₁.toAddGroup = inst₂.toAddGroup := by ext : 1; exact h_add
-- Extract equality of necessary substructures from h_group
injection h_group with h_group; injection h_group
have : inst₁.toIntCast.intCast = inst₂.toIntCast.intCast := by
funext n; cases n with
| ofNat n => rewrite [Int.ofNat_eq_coe, inst₁.intCast_ofNat, inst₂.intCast_ofNat]; congr
| negSucc n => rewrite [inst₁.intCast_negSucc, inst₂.intCast_negSucc]; congr
rcases inst₁ with @⟨⟨⟩⟩; rcases inst₂ with @⟨⟨⟩⟩
congr
@[ext] theorem AddCommGroupWithOne.ext ⦃inst₁ inst₂ : AddCommGroupWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one)) :
inst₁ = inst₂ := by
have : inst₁.toAddCommGroup = inst₂.toAddCommGroup :=
AddCommGroup.ext h_add
have : inst₁.toAddGroupWithOne = inst₂.toAddGroupWithOne :=
AddGroupWithOne.ext h_add h_one
injection this with _ h_addMonoidWithOne; injection h_addMonoidWithOne
cases inst₁; cases inst₂
congr
-- At present, there is no `NonAssocCommSemiring` in Mathlib.
-- At present, there is no `NonAssocCommRing` in Mathlib.
namespace CommSemiring
| Mathlib/Algebra/Ring/Ext.lean | 497 | 499 | theorem toSemiring_injective :
Function.Injective (@toSemiring R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
| 1 | 2.718282 | 0 | 0.4 | 10 | 397 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $type → $type))
universe u
variable {R : Type u}
@[ext] theorem AddMonoidWithOne.ext ⦃inst₁ inst₂ : AddMonoidWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one : R)) :
inst₁ = inst₂ := by
have h_monoid : inst₁.toAddMonoid = inst₂.toAddMonoid := by ext : 1; exact h_add
have h_zero' : inst₁.toZero = inst₂.toZero := congrArg (·.toZero) h_monoid
have h_one' : inst₁.toOne = inst₂.toOne :=
congrArg One.mk h_one
have h_natCast : inst₁.toNatCast.natCast = inst₂.toNatCast.natCast := by
funext n; induction n with
| zero => rewrite [inst₁.natCast_zero, inst₂.natCast_zero]
exact congrArg (@Zero.zero R) h_zero'
| succ n h => rw [inst₁.natCast_succ, inst₂.natCast_succ, h_add]
exact congrArg₂ _ h h_one
rcases inst₁ with @⟨⟨⟩⟩; rcases inst₂ with @⟨⟨⟩⟩
congr
theorem AddCommMonoidWithOne.toAddMonoidWithOne_injective :
Function.Injective (@AddCommMonoidWithOne.toAddMonoidWithOne R) := by
rintro ⟨⟩ ⟨⟩ _; congr
@[ext] theorem AddCommMonoidWithOne.ext ⦃inst₁ inst₂ : AddCommMonoidWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one : R)) :
inst₁ = inst₂ :=
AddCommMonoidWithOne.toAddMonoidWithOne_injective <|
AddMonoidWithOne.ext h_add h_one
@[ext] theorem AddGroupWithOne.ext ⦃inst₁ inst₂ : AddGroupWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one)) :
inst₁ = inst₂ := by
have : inst₁.toAddMonoidWithOne = inst₂.toAddMonoidWithOne :=
AddMonoidWithOne.ext h_add h_one
have : inst₁.toNatCast = inst₂.toNatCast := congrArg (·.toNatCast) this
have h_group : inst₁.toAddGroup = inst₂.toAddGroup := by ext : 1; exact h_add
-- Extract equality of necessary substructures from h_group
injection h_group with h_group; injection h_group
have : inst₁.toIntCast.intCast = inst₂.toIntCast.intCast := by
funext n; cases n with
| ofNat n => rewrite [Int.ofNat_eq_coe, inst₁.intCast_ofNat, inst₂.intCast_ofNat]; congr
| negSucc n => rewrite [inst₁.intCast_negSucc, inst₂.intCast_negSucc]; congr
rcases inst₁ with @⟨⟨⟩⟩; rcases inst₂ with @⟨⟨⟩⟩
congr
@[ext] theorem AddCommGroupWithOne.ext ⦃inst₁ inst₂ : AddCommGroupWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one)) :
inst₁ = inst₂ := by
have : inst₁.toAddCommGroup = inst₂.toAddCommGroup :=
AddCommGroup.ext h_add
have : inst₁.toAddGroupWithOne = inst₂.toAddGroupWithOne :=
AddGroupWithOne.ext h_add h_one
injection this with _ h_addMonoidWithOne; injection h_addMonoidWithOne
cases inst₁; cases inst₂
congr
-- At present, there is no `NonAssocCommSemiring` in Mathlib.
-- At present, there is no `NonAssocCommRing` in Mathlib.
namespace CommRing
| Mathlib/Algebra/Ring/Ext.lean | 519 | 520 | theorem toRing_injective : Function.Injective (@toRing R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
| 1 | 2.718282 | 0 | 0.4 | 10 | 397 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Matrix
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.LinearAlgebra.Matrix.Hermitian
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.Topology.UniformSpace.Matrix
#align_import analysis.normed_space.matrix_exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c4510ad5126f6f9"
open scoped Matrix
open NormedSpace -- For `exp`.
variable (𝕂 : Type*) {m n p : Type*} {n' : m → Type*} {𝔸 : Type*}
namespace Matrix
section Topological
section Ring
variable [Fintype m] [DecidableEq m] [Fintype n] [DecidableEq n] [∀ i, Fintype (n' i)]
[∀ i, DecidableEq (n' i)] [Field 𝕂] [Ring 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸]
[Algebra 𝕂 𝔸] [T2Space 𝔸]
| Mathlib/Analysis/NormedSpace/MatrixExponential.lean | 80 | 81 | theorem exp_diagonal (v : m → 𝔸) : exp 𝕂 (diagonal v) = diagonal (exp 𝕂 v) := by |
simp_rw [exp_eq_tsum, diagonal_pow, ← diagonal_smul, ← diagonal_tsum]
| 1 | 2.718282 | 0 | 0.4 | 5 | 398 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Matrix
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.LinearAlgebra.Matrix.Hermitian
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.Topology.UniformSpace.Matrix
#align_import analysis.normed_space.matrix_exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c4510ad5126f6f9"
open scoped Matrix
open NormedSpace -- For `exp`.
variable (𝕂 : Type*) {m n p : Type*} {n' : m → Type*} {𝔸 : Type*}
namespace Matrix
section Topological
section Ring
variable [Fintype m] [DecidableEq m] [Fintype n] [DecidableEq n] [∀ i, Fintype (n' i)]
[∀ i, DecidableEq (n' i)] [Field 𝕂] [Ring 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸]
[Algebra 𝕂 𝔸] [T2Space 𝔸]
theorem exp_diagonal (v : m → 𝔸) : exp 𝕂 (diagonal v) = diagonal (exp 𝕂 v) := by
simp_rw [exp_eq_tsum, diagonal_pow, ← diagonal_smul, ← diagonal_tsum]
#align matrix.exp_diagonal Matrix.exp_diagonal
| Mathlib/Analysis/NormedSpace/MatrixExponential.lean | 84 | 86 | theorem exp_blockDiagonal (v : m → Matrix n n 𝔸) :
exp 𝕂 (blockDiagonal v) = blockDiagonal (exp 𝕂 v) := by |
simp_rw [exp_eq_tsum, ← blockDiagonal_pow, ← blockDiagonal_smul, ← blockDiagonal_tsum]
| 1 | 2.718282 | 0 | 0.4 | 5 | 398 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Matrix
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.LinearAlgebra.Matrix.Hermitian
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.Topology.UniformSpace.Matrix
#align_import analysis.normed_space.matrix_exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c4510ad5126f6f9"
open scoped Matrix
open NormedSpace -- For `exp`.
variable (𝕂 : Type*) {m n p : Type*} {n' : m → Type*} {𝔸 : Type*}
namespace Matrix
section Topological
section Ring
variable [Fintype m] [DecidableEq m] [Fintype n] [DecidableEq n] [∀ i, Fintype (n' i)]
[∀ i, DecidableEq (n' i)] [Field 𝕂] [Ring 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸]
[Algebra 𝕂 𝔸] [T2Space 𝔸]
theorem exp_diagonal (v : m → 𝔸) : exp 𝕂 (diagonal v) = diagonal (exp 𝕂 v) := by
simp_rw [exp_eq_tsum, diagonal_pow, ← diagonal_smul, ← diagonal_tsum]
#align matrix.exp_diagonal Matrix.exp_diagonal
theorem exp_blockDiagonal (v : m → Matrix n n 𝔸) :
exp 𝕂 (blockDiagonal v) = blockDiagonal (exp 𝕂 v) := by
simp_rw [exp_eq_tsum, ← blockDiagonal_pow, ← blockDiagonal_smul, ← blockDiagonal_tsum]
#align matrix.exp_block_diagonal Matrix.exp_blockDiagonal
| Mathlib/Analysis/NormedSpace/MatrixExponential.lean | 89 | 91 | theorem exp_blockDiagonal' (v : ∀ i, Matrix (n' i) (n' i) 𝔸) :
exp 𝕂 (blockDiagonal' v) = blockDiagonal' (exp 𝕂 v) := by |
simp_rw [exp_eq_tsum, ← blockDiagonal'_pow, ← blockDiagonal'_smul, ← blockDiagonal'_tsum]
| 1 | 2.718282 | 0 | 0.4 | 5 | 398 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Matrix
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.LinearAlgebra.Matrix.Hermitian
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.Topology.UniformSpace.Matrix
#align_import analysis.normed_space.matrix_exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c4510ad5126f6f9"
open scoped Matrix
open NormedSpace -- For `exp`.
variable (𝕂 : Type*) {m n p : Type*} {n' : m → Type*} {𝔸 : Type*}
namespace Matrix
section Topological
section CommRing
variable [Fintype m] [DecidableEq m] [Field 𝕂] [CommRing 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸]
[Algebra 𝕂 𝔸] [T2Space 𝔸]
| Mathlib/Analysis/NormedSpace/MatrixExponential.lean | 111 | 112 | theorem exp_transpose (A : Matrix m m 𝔸) : exp 𝕂 Aᵀ = (exp 𝕂 A)ᵀ := by |
simp_rw [exp_eq_tsum, transpose_tsum, transpose_smul, transpose_pow]
| 1 | 2.718282 | 0 | 0.4 | 5 | 398 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Matrix
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.LinearAlgebra.Matrix.Hermitian
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.Topology.UniformSpace.Matrix
#align_import analysis.normed_space.matrix_exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c4510ad5126f6f9"
open scoped Matrix
open NormedSpace -- For `exp`.
variable (𝕂 : Type*) {m n p : Type*} {n' : m → Type*} {𝔸 : Type*}
namespace Matrix
section Topological
section NormedComm
variable [RCLike 𝕂] [Fintype m] [DecidableEq m] [Fintype n] [DecidableEq n] [∀ i, Fintype (n' i)]
[∀ i, DecidableEq (n' i)] [NormedCommRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸]
| Mathlib/Analysis/NormedSpace/MatrixExponential.lean | 182 | 187 | theorem exp_neg (A : Matrix m m 𝔸) : exp 𝕂 (-A) = (exp 𝕂 A)⁻¹ := by |
rw [nonsing_inv_eq_ring_inverse]
letI : SeminormedRing (Matrix m m 𝔸) := Matrix.linftyOpSemiNormedRing
letI : NormedRing (Matrix m m 𝔸) := Matrix.linftyOpNormedRing
letI : NormedAlgebra 𝕂 (Matrix m m 𝔸) := Matrix.linftyOpNormedAlgebra
exact (Ring.inverse_exp _ A).symm
| 5 | 148.413159 | 2 | 0.4 | 5 | 398 |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.pow.nnreal from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real NNReal ENNReal ComplexConjugate
open Finset Function Set
namespace NNReal
variable {w x y z : ℝ}
noncomputable def rpow (x : ℝ≥0) (y : ℝ) : ℝ≥0 :=
⟨(x : ℝ) ^ y, Real.rpow_nonneg x.2 y⟩
#align nnreal.rpow NNReal.rpow
noncomputable instance : Pow ℝ≥0 ℝ :=
⟨rpow⟩
@[simp]
theorem rpow_eq_pow (x : ℝ≥0) (y : ℝ) : rpow x y = x ^ y :=
rfl
#align nnreal.rpow_eq_pow NNReal.rpow_eq_pow
@[simp, norm_cast]
theorem coe_rpow (x : ℝ≥0) (y : ℝ) : ((x ^ y : ℝ≥0) : ℝ) = (x : ℝ) ^ y :=
rfl
#align nnreal.coe_rpow NNReal.coe_rpow
@[simp]
theorem rpow_zero (x : ℝ≥0) : x ^ (0 : ℝ) = 1 :=
NNReal.eq <| Real.rpow_zero _
#align nnreal.rpow_zero NNReal.rpow_zero
@[simp]
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 57 | 59 | theorem rpow_eq_zero_iff {x : ℝ≥0} {y : ℝ} : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by |
rw [← NNReal.coe_inj, coe_rpow, ← NNReal.coe_eq_zero]
exact Real.rpow_eq_zero_iff_of_nonneg x.2
| 2 | 7.389056 | 1 | 0.4 | 5 | 399 |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.pow.nnreal from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real NNReal ENNReal ComplexConjugate
open Finset Function Set
namespace NNReal
variable {w x y z : ℝ}
noncomputable def rpow (x : ℝ≥0) (y : ℝ) : ℝ≥0 :=
⟨(x : ℝ) ^ y, Real.rpow_nonneg x.2 y⟩
#align nnreal.rpow NNReal.rpow
noncomputable instance : Pow ℝ≥0 ℝ :=
⟨rpow⟩
@[simp]
theorem rpow_eq_pow (x : ℝ≥0) (y : ℝ) : rpow x y = x ^ y :=
rfl
#align nnreal.rpow_eq_pow NNReal.rpow_eq_pow
@[simp, norm_cast]
theorem coe_rpow (x : ℝ≥0) (y : ℝ) : ((x ^ y : ℝ≥0) : ℝ) = (x : ℝ) ^ y :=
rfl
#align nnreal.coe_rpow NNReal.coe_rpow
@[simp]
theorem rpow_zero (x : ℝ≥0) : x ^ (0 : ℝ) = 1 :=
NNReal.eq <| Real.rpow_zero _
#align nnreal.rpow_zero NNReal.rpow_zero
@[simp]
theorem rpow_eq_zero_iff {x : ℝ≥0} {y : ℝ} : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by
rw [← NNReal.coe_inj, coe_rpow, ← NNReal.coe_eq_zero]
exact Real.rpow_eq_zero_iff_of_nonneg x.2
#align nnreal.rpow_eq_zero_iff NNReal.rpow_eq_zero_iff
@[simp]
theorem zero_rpow {x : ℝ} (h : x ≠ 0) : (0 : ℝ≥0) ^ x = 0 :=
NNReal.eq <| Real.zero_rpow h
#align nnreal.zero_rpow NNReal.zero_rpow
@[simp]
theorem rpow_one (x : ℝ≥0) : x ^ (1 : ℝ) = x :=
NNReal.eq <| Real.rpow_one _
#align nnreal.rpow_one NNReal.rpow_one
@[simp]
theorem one_rpow (x : ℝ) : (1 : ℝ≥0) ^ x = 1 :=
NNReal.eq <| Real.one_rpow _
#align nnreal.one_rpow NNReal.one_rpow
theorem rpow_add {x : ℝ≥0} (hx : x ≠ 0) (y z : ℝ) : x ^ (y + z) = x ^ y * x ^ z :=
NNReal.eq <| Real.rpow_add (pos_iff_ne_zero.2 hx) _ _
#align nnreal.rpow_add NNReal.rpow_add
theorem rpow_add' (x : ℝ≥0) {y z : ℝ} (h : y + z ≠ 0) : x ^ (y + z) = x ^ y * x ^ z :=
NNReal.eq <| Real.rpow_add' x.2 h
#align nnreal.rpow_add' NNReal.rpow_add'
lemma rpow_of_add_eq (x : ℝ≥0) (hw : w ≠ 0) (h : y + z = w) : x ^ w = x ^ y * x ^ z := by
rw [← h, rpow_add']; rwa [h]
theorem rpow_mul (x : ℝ≥0) (y z : ℝ) : x ^ (y * z) = (x ^ y) ^ z :=
NNReal.eq <| Real.rpow_mul x.2 y z
#align nnreal.rpow_mul NNReal.rpow_mul
theorem rpow_neg (x : ℝ≥0) (y : ℝ) : x ^ (-y) = (x ^ y)⁻¹ :=
NNReal.eq <| Real.rpow_neg x.2 _
#align nnreal.rpow_neg NNReal.rpow_neg
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 97 | 97 | theorem rpow_neg_one (x : ℝ≥0) : x ^ (-1 : ℝ) = x⁻¹ := by | simp [rpow_neg]
| 1 | 2.718282 | 0 | 0.4 | 5 | 399 |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.pow.nnreal from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real NNReal ENNReal ComplexConjugate
open Finset Function Set
namespace NNReal
variable {w x y z : ℝ}
noncomputable def rpow (x : ℝ≥0) (y : ℝ) : ℝ≥0 :=
⟨(x : ℝ) ^ y, Real.rpow_nonneg x.2 y⟩
#align nnreal.rpow NNReal.rpow
noncomputable instance : Pow ℝ≥0 ℝ :=
⟨rpow⟩
@[simp]
theorem rpow_eq_pow (x : ℝ≥0) (y : ℝ) : rpow x y = x ^ y :=
rfl
#align nnreal.rpow_eq_pow NNReal.rpow_eq_pow
@[simp, norm_cast]
theorem coe_rpow (x : ℝ≥0) (y : ℝ) : ((x ^ y : ℝ≥0) : ℝ) = (x : ℝ) ^ y :=
rfl
#align nnreal.coe_rpow NNReal.coe_rpow
@[simp]
theorem rpow_zero (x : ℝ≥0) : x ^ (0 : ℝ) = 1 :=
NNReal.eq <| Real.rpow_zero _
#align nnreal.rpow_zero NNReal.rpow_zero
@[simp]
theorem rpow_eq_zero_iff {x : ℝ≥0} {y : ℝ} : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by
rw [← NNReal.coe_inj, coe_rpow, ← NNReal.coe_eq_zero]
exact Real.rpow_eq_zero_iff_of_nonneg x.2
#align nnreal.rpow_eq_zero_iff NNReal.rpow_eq_zero_iff
@[simp]
theorem zero_rpow {x : ℝ} (h : x ≠ 0) : (0 : ℝ≥0) ^ x = 0 :=
NNReal.eq <| Real.zero_rpow h
#align nnreal.zero_rpow NNReal.zero_rpow
@[simp]
theorem rpow_one (x : ℝ≥0) : x ^ (1 : ℝ) = x :=
NNReal.eq <| Real.rpow_one _
#align nnreal.rpow_one NNReal.rpow_one
@[simp]
theorem one_rpow (x : ℝ) : (1 : ℝ≥0) ^ x = 1 :=
NNReal.eq <| Real.one_rpow _
#align nnreal.one_rpow NNReal.one_rpow
theorem rpow_add {x : ℝ≥0} (hx : x ≠ 0) (y z : ℝ) : x ^ (y + z) = x ^ y * x ^ z :=
NNReal.eq <| Real.rpow_add (pos_iff_ne_zero.2 hx) _ _
#align nnreal.rpow_add NNReal.rpow_add
theorem rpow_add' (x : ℝ≥0) {y z : ℝ} (h : y + z ≠ 0) : x ^ (y + z) = x ^ y * x ^ z :=
NNReal.eq <| Real.rpow_add' x.2 h
#align nnreal.rpow_add' NNReal.rpow_add'
lemma rpow_of_add_eq (x : ℝ≥0) (hw : w ≠ 0) (h : y + z = w) : x ^ w = x ^ y * x ^ z := by
rw [← h, rpow_add']; rwa [h]
theorem rpow_mul (x : ℝ≥0) (y z : ℝ) : x ^ (y * z) = (x ^ y) ^ z :=
NNReal.eq <| Real.rpow_mul x.2 y z
#align nnreal.rpow_mul NNReal.rpow_mul
theorem rpow_neg (x : ℝ≥0) (y : ℝ) : x ^ (-y) = (x ^ y)⁻¹ :=
NNReal.eq <| Real.rpow_neg x.2 _
#align nnreal.rpow_neg NNReal.rpow_neg
theorem rpow_neg_one (x : ℝ≥0) : x ^ (-1 : ℝ) = x⁻¹ := by simp [rpow_neg]
#align nnreal.rpow_neg_one NNReal.rpow_neg_one
theorem rpow_sub {x : ℝ≥0} (hx : x ≠ 0) (y z : ℝ) : x ^ (y - z) = x ^ y / x ^ z :=
NNReal.eq <| Real.rpow_sub (pos_iff_ne_zero.2 hx) y z
#align nnreal.rpow_sub NNReal.rpow_sub
theorem rpow_sub' (x : ℝ≥0) {y z : ℝ} (h : y - z ≠ 0) : x ^ (y - z) = x ^ y / x ^ z :=
NNReal.eq <| Real.rpow_sub' x.2 h
#align nnreal.rpow_sub' NNReal.rpow_sub'
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 108 | 109 | theorem rpow_inv_rpow_self {y : ℝ} (hy : y ≠ 0) (x : ℝ≥0) : (x ^ y) ^ (1 / y) = x := by |
field_simp [← rpow_mul]
| 1 | 2.718282 | 0 | 0.4 | 5 | 399 |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.pow.nnreal from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real NNReal ENNReal ComplexConjugate
open Finset Function Set
namespace NNReal
variable {w x y z : ℝ}
noncomputable def rpow (x : ℝ≥0) (y : ℝ) : ℝ≥0 :=
⟨(x : ℝ) ^ y, Real.rpow_nonneg x.2 y⟩
#align nnreal.rpow NNReal.rpow
noncomputable instance : Pow ℝ≥0 ℝ :=
⟨rpow⟩
@[simp]
theorem rpow_eq_pow (x : ℝ≥0) (y : ℝ) : rpow x y = x ^ y :=
rfl
#align nnreal.rpow_eq_pow NNReal.rpow_eq_pow
@[simp, norm_cast]
theorem coe_rpow (x : ℝ≥0) (y : ℝ) : ((x ^ y : ℝ≥0) : ℝ) = (x : ℝ) ^ y :=
rfl
#align nnreal.coe_rpow NNReal.coe_rpow
@[simp]
theorem rpow_zero (x : ℝ≥0) : x ^ (0 : ℝ) = 1 :=
NNReal.eq <| Real.rpow_zero _
#align nnreal.rpow_zero NNReal.rpow_zero
@[simp]
theorem rpow_eq_zero_iff {x : ℝ≥0} {y : ℝ} : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by
rw [← NNReal.coe_inj, coe_rpow, ← NNReal.coe_eq_zero]
exact Real.rpow_eq_zero_iff_of_nonneg x.2
#align nnreal.rpow_eq_zero_iff NNReal.rpow_eq_zero_iff
@[simp]
theorem zero_rpow {x : ℝ} (h : x ≠ 0) : (0 : ℝ≥0) ^ x = 0 :=
NNReal.eq <| Real.zero_rpow h
#align nnreal.zero_rpow NNReal.zero_rpow
@[simp]
theorem rpow_one (x : ℝ≥0) : x ^ (1 : ℝ) = x :=
NNReal.eq <| Real.rpow_one _
#align nnreal.rpow_one NNReal.rpow_one
@[simp]
theorem one_rpow (x : ℝ) : (1 : ℝ≥0) ^ x = 1 :=
NNReal.eq <| Real.one_rpow _
#align nnreal.one_rpow NNReal.one_rpow
theorem rpow_add {x : ℝ≥0} (hx : x ≠ 0) (y z : ℝ) : x ^ (y + z) = x ^ y * x ^ z :=
NNReal.eq <| Real.rpow_add (pos_iff_ne_zero.2 hx) _ _
#align nnreal.rpow_add NNReal.rpow_add
theorem rpow_add' (x : ℝ≥0) {y z : ℝ} (h : y + z ≠ 0) : x ^ (y + z) = x ^ y * x ^ z :=
NNReal.eq <| Real.rpow_add' x.2 h
#align nnreal.rpow_add' NNReal.rpow_add'
lemma rpow_of_add_eq (x : ℝ≥0) (hw : w ≠ 0) (h : y + z = w) : x ^ w = x ^ y * x ^ z := by
rw [← h, rpow_add']; rwa [h]
theorem rpow_mul (x : ℝ≥0) (y z : ℝ) : x ^ (y * z) = (x ^ y) ^ z :=
NNReal.eq <| Real.rpow_mul x.2 y z
#align nnreal.rpow_mul NNReal.rpow_mul
theorem rpow_neg (x : ℝ≥0) (y : ℝ) : x ^ (-y) = (x ^ y)⁻¹ :=
NNReal.eq <| Real.rpow_neg x.2 _
#align nnreal.rpow_neg NNReal.rpow_neg
theorem rpow_neg_one (x : ℝ≥0) : x ^ (-1 : ℝ) = x⁻¹ := by simp [rpow_neg]
#align nnreal.rpow_neg_one NNReal.rpow_neg_one
theorem rpow_sub {x : ℝ≥0} (hx : x ≠ 0) (y z : ℝ) : x ^ (y - z) = x ^ y / x ^ z :=
NNReal.eq <| Real.rpow_sub (pos_iff_ne_zero.2 hx) y z
#align nnreal.rpow_sub NNReal.rpow_sub
theorem rpow_sub' (x : ℝ≥0) {y z : ℝ} (h : y - z ≠ 0) : x ^ (y - z) = x ^ y / x ^ z :=
NNReal.eq <| Real.rpow_sub' x.2 h
#align nnreal.rpow_sub' NNReal.rpow_sub'
theorem rpow_inv_rpow_self {y : ℝ} (hy : y ≠ 0) (x : ℝ≥0) : (x ^ y) ^ (1 / y) = x := by
field_simp [← rpow_mul]
#align nnreal.rpow_inv_rpow_self NNReal.rpow_inv_rpow_self
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 112 | 113 | theorem rpow_self_rpow_inv {y : ℝ} (hy : y ≠ 0) (x : ℝ≥0) : (x ^ (1 / y)) ^ y = x := by |
field_simp [← rpow_mul]
| 1 | 2.718282 | 0 | 0.4 | 5 | 399 |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.pow.nnreal from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real NNReal ENNReal ComplexConjugate
open Finset Function Set
namespace NNReal
variable {w x y z : ℝ}
noncomputable def rpow (x : ℝ≥0) (y : ℝ) : ℝ≥0 :=
⟨(x : ℝ) ^ y, Real.rpow_nonneg x.2 y⟩
#align nnreal.rpow NNReal.rpow
noncomputable instance : Pow ℝ≥0 ℝ :=
⟨rpow⟩
@[simp]
theorem rpow_eq_pow (x : ℝ≥0) (y : ℝ) : rpow x y = x ^ y :=
rfl
#align nnreal.rpow_eq_pow NNReal.rpow_eq_pow
@[simp, norm_cast]
theorem coe_rpow (x : ℝ≥0) (y : ℝ) : ((x ^ y : ℝ≥0) : ℝ) = (x : ℝ) ^ y :=
rfl
#align nnreal.coe_rpow NNReal.coe_rpow
@[simp]
theorem rpow_zero (x : ℝ≥0) : x ^ (0 : ℝ) = 1 :=
NNReal.eq <| Real.rpow_zero _
#align nnreal.rpow_zero NNReal.rpow_zero
@[simp]
theorem rpow_eq_zero_iff {x : ℝ≥0} {y : ℝ} : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by
rw [← NNReal.coe_inj, coe_rpow, ← NNReal.coe_eq_zero]
exact Real.rpow_eq_zero_iff_of_nonneg x.2
#align nnreal.rpow_eq_zero_iff NNReal.rpow_eq_zero_iff
@[simp]
theorem zero_rpow {x : ℝ} (h : x ≠ 0) : (0 : ℝ≥0) ^ x = 0 :=
NNReal.eq <| Real.zero_rpow h
#align nnreal.zero_rpow NNReal.zero_rpow
@[simp]
theorem rpow_one (x : ℝ≥0) : x ^ (1 : ℝ) = x :=
NNReal.eq <| Real.rpow_one _
#align nnreal.rpow_one NNReal.rpow_one
@[simp]
theorem one_rpow (x : ℝ) : (1 : ℝ≥0) ^ x = 1 :=
NNReal.eq <| Real.one_rpow _
#align nnreal.one_rpow NNReal.one_rpow
theorem rpow_add {x : ℝ≥0} (hx : x ≠ 0) (y z : ℝ) : x ^ (y + z) = x ^ y * x ^ z :=
NNReal.eq <| Real.rpow_add (pos_iff_ne_zero.2 hx) _ _
#align nnreal.rpow_add NNReal.rpow_add
theorem rpow_add' (x : ℝ≥0) {y z : ℝ} (h : y + z ≠ 0) : x ^ (y + z) = x ^ y * x ^ z :=
NNReal.eq <| Real.rpow_add' x.2 h
#align nnreal.rpow_add' NNReal.rpow_add'
lemma rpow_of_add_eq (x : ℝ≥0) (hw : w ≠ 0) (h : y + z = w) : x ^ w = x ^ y * x ^ z := by
rw [← h, rpow_add']; rwa [h]
theorem rpow_mul (x : ℝ≥0) (y z : ℝ) : x ^ (y * z) = (x ^ y) ^ z :=
NNReal.eq <| Real.rpow_mul x.2 y z
#align nnreal.rpow_mul NNReal.rpow_mul
theorem rpow_neg (x : ℝ≥0) (y : ℝ) : x ^ (-y) = (x ^ y)⁻¹ :=
NNReal.eq <| Real.rpow_neg x.2 _
#align nnreal.rpow_neg NNReal.rpow_neg
theorem rpow_neg_one (x : ℝ≥0) : x ^ (-1 : ℝ) = x⁻¹ := by simp [rpow_neg]
#align nnreal.rpow_neg_one NNReal.rpow_neg_one
theorem rpow_sub {x : ℝ≥0} (hx : x ≠ 0) (y z : ℝ) : x ^ (y - z) = x ^ y / x ^ z :=
NNReal.eq <| Real.rpow_sub (pos_iff_ne_zero.2 hx) y z
#align nnreal.rpow_sub NNReal.rpow_sub
theorem rpow_sub' (x : ℝ≥0) {y z : ℝ} (h : y - z ≠ 0) : x ^ (y - z) = x ^ y / x ^ z :=
NNReal.eq <| Real.rpow_sub' x.2 h
#align nnreal.rpow_sub' NNReal.rpow_sub'
theorem rpow_inv_rpow_self {y : ℝ} (hy : y ≠ 0) (x : ℝ≥0) : (x ^ y) ^ (1 / y) = x := by
field_simp [← rpow_mul]
#align nnreal.rpow_inv_rpow_self NNReal.rpow_inv_rpow_self
theorem rpow_self_rpow_inv {y : ℝ} (hy : y ≠ 0) (x : ℝ≥0) : (x ^ (1 / y)) ^ y = x := by
field_simp [← rpow_mul]
#align nnreal.rpow_self_rpow_inv NNReal.rpow_self_rpow_inv
theorem inv_rpow (x : ℝ≥0) (y : ℝ) : x⁻¹ ^ y = (x ^ y)⁻¹ :=
NNReal.eq <| Real.inv_rpow x.2 y
#align nnreal.inv_rpow NNReal.inv_rpow
theorem div_rpow (x y : ℝ≥0) (z : ℝ) : (x / y) ^ z = x ^ z / y ^ z :=
NNReal.eq <| Real.div_rpow x.2 y.2 z
#align nnreal.div_rpow NNReal.div_rpow
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 124 | 127 | theorem sqrt_eq_rpow (x : ℝ≥0) : sqrt x = x ^ (1 / (2 : ℝ)) := by |
refine NNReal.eq ?_
push_cast
exact Real.sqrt_eq_rpow x.1
| 3 | 20.085537 | 1 | 0.4 | 5 | 399 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
| Mathlib/Algebra/Order/Group/Defs.lean | 71 | 73 | theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by | simpa using mul_le_mul_left' bc a⁻¹
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
| Mathlib/Algebra/Order/Group/Defs.lean | 82 | 84 | theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by | simpa using mul_le_mul_right' bc a⁻¹
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesLeftLE
variable [LE α] [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α}
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
| Mathlib/Algebra/Order/Group/Defs.lean | 98 | 100 | theorem Left.inv_le_one_iff : a⁻¹ ≤ 1 ↔ 1 ≤ a := by |
rw [← mul_le_mul_iff_left a]
simp
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesLeftLE
variable [LE α] [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α}
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
theorem Left.inv_le_one_iff : a⁻¹ ≤ 1 ↔ 1 ≤ a := by
rw [← mul_le_mul_iff_left a]
simp
#align left.inv_le_one_iff Left.inv_le_one_iff
#align left.neg_nonpos_iff Left.neg_nonpos_iff
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
| Mathlib/Algebra/Order/Group/Defs.lean | 106 | 108 | theorem Left.one_le_inv_iff : 1 ≤ a⁻¹ ↔ a ≤ 1 := by |
rw [← mul_le_mul_iff_left a]
simp
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesLeftLE
variable [LE α] [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α}
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
theorem Left.inv_le_one_iff : a⁻¹ ≤ 1 ↔ 1 ≤ a := by
rw [← mul_le_mul_iff_left a]
simp
#align left.inv_le_one_iff Left.inv_le_one_iff
#align left.neg_nonpos_iff Left.neg_nonpos_iff
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
theorem Left.one_le_inv_iff : 1 ≤ a⁻¹ ↔ a ≤ 1 := by
rw [← mul_le_mul_iff_left a]
simp
#align left.one_le_inv_iff Left.one_le_inv_iff
#align left.nonneg_neg_iff Left.nonneg_neg_iff
@[to_additive (attr := simp)]
| Mathlib/Algebra/Order/Group/Defs.lean | 113 | 115 | theorem le_inv_mul_iff_mul_le : b ≤ a⁻¹ * c ↔ a * b ≤ c := by |
rw [← mul_le_mul_iff_left a]
simp
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesLeftLE
variable [LE α] [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α}
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
theorem Left.inv_le_one_iff : a⁻¹ ≤ 1 ↔ 1 ≤ a := by
rw [← mul_le_mul_iff_left a]
simp
#align left.inv_le_one_iff Left.inv_le_one_iff
#align left.neg_nonpos_iff Left.neg_nonpos_iff
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
theorem Left.one_le_inv_iff : 1 ≤ a⁻¹ ↔ a ≤ 1 := by
rw [← mul_le_mul_iff_left a]
simp
#align left.one_le_inv_iff Left.one_le_inv_iff
#align left.nonneg_neg_iff Left.nonneg_neg_iff
@[to_additive (attr := simp)]
theorem le_inv_mul_iff_mul_le : b ≤ a⁻¹ * c ↔ a * b ≤ c := by
rw [← mul_le_mul_iff_left a]
simp
#align le_inv_mul_iff_mul_le le_inv_mul_iff_mul_le
#align le_neg_add_iff_add_le le_neg_add_iff_add_le
@[to_additive (attr := simp)]
| Mathlib/Algebra/Order/Group/Defs.lean | 120 | 121 | theorem inv_mul_le_iff_le_mul : b⁻¹ * a ≤ c ↔ a ≤ b * c := by |
rw [← mul_le_mul_iff_left b, mul_inv_cancel_left]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesLeftLE
variable [LE α] [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α}
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
theorem Left.inv_le_one_iff : a⁻¹ ≤ 1 ↔ 1 ≤ a := by
rw [← mul_le_mul_iff_left a]
simp
#align left.inv_le_one_iff Left.inv_le_one_iff
#align left.neg_nonpos_iff Left.neg_nonpos_iff
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
theorem Left.one_le_inv_iff : 1 ≤ a⁻¹ ↔ a ≤ 1 := by
rw [← mul_le_mul_iff_left a]
simp
#align left.one_le_inv_iff Left.one_le_inv_iff
#align left.nonneg_neg_iff Left.nonneg_neg_iff
@[to_additive (attr := simp)]
theorem le_inv_mul_iff_mul_le : b ≤ a⁻¹ * c ↔ a * b ≤ c := by
rw [← mul_le_mul_iff_left a]
simp
#align le_inv_mul_iff_mul_le le_inv_mul_iff_mul_le
#align le_neg_add_iff_add_le le_neg_add_iff_add_le
@[to_additive (attr := simp)]
theorem inv_mul_le_iff_le_mul : b⁻¹ * a ≤ c ↔ a ≤ b * c := by
rw [← mul_le_mul_iff_left b, mul_inv_cancel_left]
#align inv_mul_le_iff_le_mul inv_mul_le_iff_le_mul
#align neg_add_le_iff_le_add neg_add_le_iff_le_add
@[to_additive neg_le_iff_add_nonneg']
theorem inv_le_iff_one_le_mul' : a⁻¹ ≤ b ↔ 1 ≤ a * b :=
(mul_le_mul_iff_left a).symm.trans <| by rw [mul_inv_self]
#align inv_le_iff_one_le_mul' inv_le_iff_one_le_mul'
#align neg_le_iff_add_nonneg' neg_le_iff_add_nonneg'
@[to_additive]
theorem le_inv_iff_mul_le_one_left : a ≤ b⁻¹ ↔ b * a ≤ 1 :=
(mul_le_mul_iff_left b).symm.trans <| by rw [mul_inv_self]
#align le_inv_iff_mul_le_one_left le_inv_iff_mul_le_one_left
#align le_neg_iff_add_nonpos_left le_neg_iff_add_nonpos_left
@[to_additive]
| Mathlib/Algebra/Order/Group/Defs.lean | 138 | 139 | theorem le_inv_mul_iff_le : 1 ≤ b⁻¹ * a ↔ b ≤ a := by |
rw [← mul_le_mul_iff_left b, mul_one, mul_inv_cancel_left]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesLeftLT
variable [LT α] [CovariantClass α α (· * ·) (· < ·)] {a b c : α}
@[to_additive (attr := simp) Left.neg_pos_iff "Uses `left` co(ntra)variant."]
| Mathlib/Algebra/Order/Group/Defs.lean | 158 | 159 | theorem Left.one_lt_inv_iff : 1 < a⁻¹ ↔ a < 1 := by |
rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesLeftLT
variable [LT α] [CovariantClass α α (· * ·) (· < ·)] {a b c : α}
@[to_additive (attr := simp) Left.neg_pos_iff "Uses `left` co(ntra)variant."]
theorem Left.one_lt_inv_iff : 1 < a⁻¹ ↔ a < 1 := by
rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one]
#align left.one_lt_inv_iff Left.one_lt_inv_iff
#align left.neg_pos_iff Left.neg_pos_iff
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
| Mathlib/Algebra/Order/Group/Defs.lean | 165 | 166 | theorem Left.inv_lt_one_iff : a⁻¹ < 1 ↔ 1 < a := by |
rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesLeftLT
variable [LT α] [CovariantClass α α (· * ·) (· < ·)] {a b c : α}
@[to_additive (attr := simp) Left.neg_pos_iff "Uses `left` co(ntra)variant."]
theorem Left.one_lt_inv_iff : 1 < a⁻¹ ↔ a < 1 := by
rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one]
#align left.one_lt_inv_iff Left.one_lt_inv_iff
#align left.neg_pos_iff Left.neg_pos_iff
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
theorem Left.inv_lt_one_iff : a⁻¹ < 1 ↔ 1 < a := by
rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one]
#align left.inv_lt_one_iff Left.inv_lt_one_iff
#align left.neg_neg_iff Left.neg_neg_iff
@[to_additive (attr := simp)]
| Mathlib/Algebra/Order/Group/Defs.lean | 171 | 173 | theorem lt_inv_mul_iff_mul_lt : b < a⁻¹ * c ↔ a * b < c := by |
rw [← mul_lt_mul_iff_left a]
simp
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesLeftLT
variable [LT α] [CovariantClass α α (· * ·) (· < ·)] {a b c : α}
@[to_additive (attr := simp) Left.neg_pos_iff "Uses `left` co(ntra)variant."]
theorem Left.one_lt_inv_iff : 1 < a⁻¹ ↔ a < 1 := by
rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one]
#align left.one_lt_inv_iff Left.one_lt_inv_iff
#align left.neg_pos_iff Left.neg_pos_iff
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
theorem Left.inv_lt_one_iff : a⁻¹ < 1 ↔ 1 < a := by
rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one]
#align left.inv_lt_one_iff Left.inv_lt_one_iff
#align left.neg_neg_iff Left.neg_neg_iff
@[to_additive (attr := simp)]
theorem lt_inv_mul_iff_mul_lt : b < a⁻¹ * c ↔ a * b < c := by
rw [← mul_lt_mul_iff_left a]
simp
#align lt_inv_mul_iff_mul_lt lt_inv_mul_iff_mul_lt
#align lt_neg_add_iff_add_lt lt_neg_add_iff_add_lt
@[to_additive (attr := simp)]
| Mathlib/Algebra/Order/Group/Defs.lean | 178 | 179 | theorem inv_mul_lt_iff_lt_mul : b⁻¹ * a < c ↔ a < b * c := by |
rw [← mul_lt_mul_iff_left b, mul_inv_cancel_left]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesLeftLT
variable [LT α] [CovariantClass α α (· * ·) (· < ·)] {a b c : α}
@[to_additive (attr := simp) Left.neg_pos_iff "Uses `left` co(ntra)variant."]
theorem Left.one_lt_inv_iff : 1 < a⁻¹ ↔ a < 1 := by
rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one]
#align left.one_lt_inv_iff Left.one_lt_inv_iff
#align left.neg_pos_iff Left.neg_pos_iff
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
theorem Left.inv_lt_one_iff : a⁻¹ < 1 ↔ 1 < a := by
rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one]
#align left.inv_lt_one_iff Left.inv_lt_one_iff
#align left.neg_neg_iff Left.neg_neg_iff
@[to_additive (attr := simp)]
theorem lt_inv_mul_iff_mul_lt : b < a⁻¹ * c ↔ a * b < c := by
rw [← mul_lt_mul_iff_left a]
simp
#align lt_inv_mul_iff_mul_lt lt_inv_mul_iff_mul_lt
#align lt_neg_add_iff_add_lt lt_neg_add_iff_add_lt
@[to_additive (attr := simp)]
theorem inv_mul_lt_iff_lt_mul : b⁻¹ * a < c ↔ a < b * c := by
rw [← mul_lt_mul_iff_left b, mul_inv_cancel_left]
#align inv_mul_lt_iff_lt_mul inv_mul_lt_iff_lt_mul
#align neg_add_lt_iff_lt_add neg_add_lt_iff_lt_add
@[to_additive]
theorem inv_lt_iff_one_lt_mul' : a⁻¹ < b ↔ 1 < a * b :=
(mul_lt_mul_iff_left a).symm.trans <| by rw [mul_inv_self]
#align inv_lt_iff_one_lt_mul' inv_lt_iff_one_lt_mul'
#align neg_lt_iff_pos_add' neg_lt_iff_pos_add'
@[to_additive]
theorem lt_inv_iff_mul_lt_one' : a < b⁻¹ ↔ b * a < 1 :=
(mul_lt_mul_iff_left b).symm.trans <| by rw [mul_inv_self]
#align lt_inv_iff_mul_lt_one' lt_inv_iff_mul_lt_one'
#align lt_neg_iff_add_neg' lt_neg_iff_add_neg'
@[to_additive]
| Mathlib/Algebra/Order/Group/Defs.lean | 196 | 197 | theorem lt_inv_mul_iff_lt : 1 < b⁻¹ * a ↔ b < a := by |
rw [← mul_lt_mul_iff_left b, mul_one, mul_inv_cancel_left]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesRightLE
variable [LE α] [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α}
@[to_additive (attr := simp) "Uses `right` co(ntra)variant."]
| Mathlib/Algebra/Order/Group/Defs.lean | 215 | 217 | theorem Right.inv_le_one_iff : a⁻¹ ≤ 1 ↔ 1 ≤ a := by |
rw [← mul_le_mul_iff_right a]
simp
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesRightLE
variable [LE α] [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α}
@[to_additive (attr := simp) "Uses `right` co(ntra)variant."]
theorem Right.inv_le_one_iff : a⁻¹ ≤ 1 ↔ 1 ≤ a := by
rw [← mul_le_mul_iff_right a]
simp
#align right.inv_le_one_iff Right.inv_le_one_iff
#align right.neg_nonpos_iff Right.neg_nonpos_iff
@[to_additive (attr := simp) "Uses `right` co(ntra)variant."]
| Mathlib/Algebra/Order/Group/Defs.lean | 223 | 225 | theorem Right.one_le_inv_iff : 1 ≤ a⁻¹ ↔ a ≤ 1 := by |
rw [← mul_le_mul_iff_right a]
simp
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesRightLT
variable [LT α] [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c : α}
@[to_additive (attr := simp) "Uses `right` co(ntra)variant."]
| Mathlib/Algebra/Order/Group/Defs.lean | 280 | 281 | theorem Right.inv_lt_one_iff : a⁻¹ < 1 ↔ 1 < a := by |
rw [← mul_lt_mul_iff_right a, inv_mul_self, one_mul]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesRightLT
variable [LT α] [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c : α}
@[to_additive (attr := simp) "Uses `right` co(ntra)variant."]
theorem Right.inv_lt_one_iff : a⁻¹ < 1 ↔ 1 < a := by
rw [← mul_lt_mul_iff_right a, inv_mul_self, one_mul]
#align right.inv_lt_one_iff Right.inv_lt_one_iff
#align right.neg_neg_iff Right.neg_neg_iff
@[to_additive (attr := simp) Right.neg_pos_iff "Uses `right` co(ntra)variant."]
| Mathlib/Algebra/Order/Group/Defs.lean | 287 | 288 | theorem Right.one_lt_inv_iff : 1 < a⁻¹ ↔ a < 1 := by |
rw [← mul_lt_mul_iff_right a, inv_mul_self, one_mul]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesRightLT
variable [LT α] [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c : α}
@[to_additive (attr := simp) "Uses `right` co(ntra)variant."]
theorem Right.inv_lt_one_iff : a⁻¹ < 1 ↔ 1 < a := by
rw [← mul_lt_mul_iff_right a, inv_mul_self, one_mul]
#align right.inv_lt_one_iff Right.inv_lt_one_iff
#align right.neg_neg_iff Right.neg_neg_iff
@[to_additive (attr := simp) Right.neg_pos_iff "Uses `right` co(ntra)variant."]
theorem Right.one_lt_inv_iff : 1 < a⁻¹ ↔ a < 1 := by
rw [← mul_lt_mul_iff_right a, inv_mul_self, one_mul]
#align right.one_lt_inv_iff Right.one_lt_inv_iff
#align right.neg_pos_iff Right.neg_pos_iff
@[to_additive]
theorem inv_lt_iff_one_lt_mul : a⁻¹ < b ↔ 1 < b * a :=
(mul_lt_mul_iff_right a).symm.trans <| by rw [inv_mul_self]
#align inv_lt_iff_one_lt_mul inv_lt_iff_one_lt_mul
#align neg_lt_iff_pos_add neg_lt_iff_pos_add
@[to_additive]
theorem lt_inv_iff_mul_lt_one : a < b⁻¹ ↔ a * b < 1 :=
(mul_lt_mul_iff_right b).symm.trans <| by rw [inv_mul_self]
#align lt_inv_iff_mul_lt_one lt_inv_iff_mul_lt_one
#align lt_neg_iff_add_neg lt_neg_iff_add_neg
@[to_additive (attr := simp)]
| Mathlib/Algebra/Order/Group/Defs.lean | 305 | 306 | theorem mul_inv_lt_iff_lt_mul : a * b⁻¹ < c ↔ a < c * b := by |
rw [← mul_lt_mul_iff_right b, inv_mul_cancel_right]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesRightLT
variable [LT α] [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c : α}
@[to_additive (attr := simp) "Uses `right` co(ntra)variant."]
theorem Right.inv_lt_one_iff : a⁻¹ < 1 ↔ 1 < a := by
rw [← mul_lt_mul_iff_right a, inv_mul_self, one_mul]
#align right.inv_lt_one_iff Right.inv_lt_one_iff
#align right.neg_neg_iff Right.neg_neg_iff
@[to_additive (attr := simp) Right.neg_pos_iff "Uses `right` co(ntra)variant."]
theorem Right.one_lt_inv_iff : 1 < a⁻¹ ↔ a < 1 := by
rw [← mul_lt_mul_iff_right a, inv_mul_self, one_mul]
#align right.one_lt_inv_iff Right.one_lt_inv_iff
#align right.neg_pos_iff Right.neg_pos_iff
@[to_additive]
theorem inv_lt_iff_one_lt_mul : a⁻¹ < b ↔ 1 < b * a :=
(mul_lt_mul_iff_right a).symm.trans <| by rw [inv_mul_self]
#align inv_lt_iff_one_lt_mul inv_lt_iff_one_lt_mul
#align neg_lt_iff_pos_add neg_lt_iff_pos_add
@[to_additive]
theorem lt_inv_iff_mul_lt_one : a < b⁻¹ ↔ a * b < 1 :=
(mul_lt_mul_iff_right b).symm.trans <| by rw [inv_mul_self]
#align lt_inv_iff_mul_lt_one lt_inv_iff_mul_lt_one
#align lt_neg_iff_add_neg lt_neg_iff_add_neg
@[to_additive (attr := simp)]
theorem mul_inv_lt_iff_lt_mul : a * b⁻¹ < c ↔ a < c * b := by
rw [← mul_lt_mul_iff_right b, inv_mul_cancel_right]
#align mul_inv_lt_iff_lt_mul mul_inv_lt_iff_lt_mul
#align add_neg_lt_iff_lt_add add_neg_lt_iff_lt_add
@[to_additive (attr := simp)]
theorem lt_mul_inv_iff_mul_lt : c < a * b⁻¹ ↔ c * b < a :=
(mul_lt_mul_iff_right b).symm.trans <| by rw [inv_mul_cancel_right]
#align lt_mul_inv_iff_mul_lt lt_mul_inv_iff_mul_lt
#align lt_add_neg_iff_add_lt lt_add_neg_iff_add_lt
-- Porting note (#10618): `simp` can prove this
@[to_additive]
| Mathlib/Algebra/Order/Group/Defs.lean | 318 | 319 | theorem inv_mul_lt_one_iff_lt : a * b⁻¹ < 1 ↔ a < b := by |
rw [← mul_lt_mul_iff_right b, inv_mul_cancel_right, one_mul]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesLeftRightLE
variable [LE α] [CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{a b c d : α}
@[to_additive (attr := simp)]
| Mathlib/Algebra/Order/Group/Defs.lean | 343 | 345 | theorem inv_le_inv_iff : a⁻¹ ≤ b⁻¹ ↔ b ≤ a := by |
rw [← mul_le_mul_iff_left a, ← mul_le_mul_iff_right b]
simp
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesLeftRightLE
variable [LE α] [CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{a b c d : α}
@[to_additive (attr := simp)]
theorem inv_le_inv_iff : a⁻¹ ≤ b⁻¹ ↔ b ≤ a := by
rw [← mul_le_mul_iff_left a, ← mul_le_mul_iff_right b]
simp
#align inv_le_inv_iff inv_le_inv_iff
#align neg_le_neg_iff neg_le_neg_iff
alias ⟨le_of_neg_le_neg, _⟩ := neg_le_neg_iff
#align le_of_neg_le_neg le_of_neg_le_neg
@[to_additive]
| Mathlib/Algebra/Order/Group/Defs.lean | 353 | 355 | theorem mul_inv_le_inv_mul_iff : a * b⁻¹ ≤ d⁻¹ * c ↔ d * a ≤ c * b := by |
rw [← mul_le_mul_iff_left d, ← mul_le_mul_iff_right b, mul_inv_cancel_left, mul_assoc,
inv_mul_cancel_right]
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesLeftRightLT
variable [LT α] [CovariantClass α α (· * ·) (· < ·)] [CovariantClass α α (swap (· * ·)) (· < ·)]
{a b c d : α}
@[to_additive (attr := simp)]
| Mathlib/Algebra/Order/Group/Defs.lean | 382 | 384 | theorem inv_lt_inv_iff : a⁻¹ < b⁻¹ ↔ b < a := by |
rw [← mul_lt_mul_iff_left a, ← mul_lt_mul_iff_right b]
simp
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesLeftRightLT
variable [LT α] [CovariantClass α α (· * ·) (· < ·)] [CovariantClass α α (swap (· * ·)) (· < ·)]
{a b c d : α}
@[to_additive (attr := simp)]
theorem inv_lt_inv_iff : a⁻¹ < b⁻¹ ↔ b < a := by
rw [← mul_lt_mul_iff_left a, ← mul_lt_mul_iff_right b]
simp
#align inv_lt_inv_iff inv_lt_inv_iff
#align neg_lt_neg_iff neg_lt_neg_iff
@[to_additive neg_lt]
| Mathlib/Algebra/Order/Group/Defs.lean | 389 | 389 | theorem inv_lt' : a⁻¹ < b ↔ b⁻¹ < a := by | rw [← inv_lt_inv_iff, inv_inv]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesLeftRightLT
variable [LT α] [CovariantClass α α (· * ·) (· < ·)] [CovariantClass α α (swap (· * ·)) (· < ·)]
{a b c d : α}
@[to_additive (attr := simp)]
theorem inv_lt_inv_iff : a⁻¹ < b⁻¹ ↔ b < a := by
rw [← mul_lt_mul_iff_left a, ← mul_lt_mul_iff_right b]
simp
#align inv_lt_inv_iff inv_lt_inv_iff
#align neg_lt_neg_iff neg_lt_neg_iff
@[to_additive neg_lt]
theorem inv_lt' : a⁻¹ < b ↔ b⁻¹ < a := by rw [← inv_lt_inv_iff, inv_inv]
#align inv_lt' inv_lt'
#align neg_lt neg_lt
@[to_additive lt_neg]
| Mathlib/Algebra/Order/Group/Defs.lean | 394 | 394 | theorem lt_inv' : a < b⁻¹ ↔ b < a⁻¹ := by | rw [← inv_lt_inv_iff, inv_inv]
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section TypeclassesLeftRightLT
variable [LT α] [CovariantClass α α (· * ·) (· < ·)] [CovariantClass α α (swap (· * ·)) (· < ·)]
{a b c d : α}
@[to_additive (attr := simp)]
theorem inv_lt_inv_iff : a⁻¹ < b⁻¹ ↔ b < a := by
rw [← mul_lt_mul_iff_left a, ← mul_lt_mul_iff_right b]
simp
#align inv_lt_inv_iff inv_lt_inv_iff
#align neg_lt_neg_iff neg_lt_neg_iff
@[to_additive neg_lt]
theorem inv_lt' : a⁻¹ < b ↔ b⁻¹ < a := by rw [← inv_lt_inv_iff, inv_inv]
#align inv_lt' inv_lt'
#align neg_lt neg_lt
@[to_additive lt_neg]
theorem lt_inv' : a < b⁻¹ ↔ b < a⁻¹ := by rw [← inv_lt_inv_iff, inv_inv]
#align lt_inv' lt_inv'
#align lt_neg lt_neg
alias ⟨lt_inv_of_lt_inv, _⟩ := lt_inv'
#align lt_inv_of_lt_inv lt_inv_of_lt_inv
attribute [to_additive] lt_inv_of_lt_inv
#align lt_neg_of_lt_neg lt_neg_of_lt_neg
alias ⟨inv_lt_of_inv_lt', _⟩ := inv_lt'
#align inv_lt_of_inv_lt' inv_lt_of_inv_lt'
attribute [to_additive neg_lt_of_neg_lt] inv_lt_of_inv_lt'
#align neg_lt_of_neg_lt neg_lt_of_neg_lt
@[to_additive]
| Mathlib/Algebra/Order/Group/Defs.lean | 411 | 413 | theorem mul_inv_lt_inv_mul_iff : a * b⁻¹ < d⁻¹ * c ↔ d * a < c * b := by |
rw [← mul_lt_mul_iff_left d, ← mul_lt_mul_iff_right b, mul_inv_cancel_left, mul_assoc,
inv_mul_cancel_right]
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Type u) extends AddCommGroup α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
#align ordered_add_comm_group OrderedAddCommGroup
class OrderedCommGroup (α : Type u) extends CommGroup α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c : α, c * a ≤ c * b
#align ordered_comm_group OrderedCommGroup
attribute [to_additive] OrderedCommGroup
@[to_additive]
instance OrderedCommGroup.to_covariantClass_left_le (α : Type u) [OrderedCommGroup α] :
CovariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := OrderedCommGroup.mul_le_mul_left b c bc a
#align ordered_comm_group.to_covariant_class_left_le OrderedCommGroup.to_covariantClass_left_le
#align ordered_add_comm_group.to_covariant_class_left_le OrderedAddCommGroup.to_covariantClass_left_le
-- See note [lower instance priority]
@[to_additive OrderedAddCommGroup.toOrderedCancelAddCommMonoid]
instance (priority := 100) OrderedCommGroup.toOrderedCancelCommMonoid [OrderedCommGroup α] :
OrderedCancelCommMonoid α :=
{ ‹OrderedCommGroup α› with le_of_mul_le_mul_left := fun a b c ↦ le_of_mul_le_mul_left' }
#align ordered_comm_group.to_ordered_cancel_comm_monoid OrderedCommGroup.toOrderedCancelCommMonoid
#align ordered_add_comm_group.to_ordered_cancel_add_comm_monoid OrderedAddCommGroup.toOrderedCancelAddCommMonoid
example (α : Type u) [OrderedAddCommGroup α] : CovariantClass α α (swap (· + ·)) (· < ·) :=
IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le α
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- It was introduced in https://github.com/leanprover-community/mathlib/pull/17564
-- but without the motivation clearly explained.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_left' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_left_le OrderedCommGroup.to_contravariantClass_left_le
#align ordered_add_comm_group.to_contravariant_class_left_le OrderedAddCommGroup.to_contravariantClass_left_le
-- Porting note: this instance is not used,
-- and causes timeouts after lean4#2210.
-- See further explanation on `OrderedCommGroup.to_contravariantClass_left_le`.
@[to_additive "A choice-free shortcut instance."]
theorem OrderedCommGroup.to_contravariantClass_right_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (swap (· * ·)) (· ≤ ·) where
elim a b c bc := by simpa using mul_le_mul_right' bc a⁻¹
#align ordered_comm_group.to_contravariant_class_right_le OrderedCommGroup.to_contravariantClass_right_le
#align ordered_add_comm_group.to_contravariant_class_right_le OrderedAddCommGroup.to_contravariantClass_right_le
section Group
variable [Group α]
section Preorder
variable [Preorder α]
section CommGroup
variable [CommGroup α]
-- Most of the lemmas that are primed in this section appear in ordered_field.
-- I (DT) did not try to minimise the assumptions.
section Group
variable [Group α] [LT α]
section Right
variable [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c d : α}
@[to_additive (attr := simp)]
| Mathlib/Algebra/Order/Group/Defs.lean | 875 | 876 | theorem div_lt_div_iff_right (c : α) : a / c < b / c ↔ a < b := by |
simpa only [div_eq_mul_inv] using mul_lt_mul_iff_right _
| 1 | 2.718282 | 0 | 0.4 | 25 | 400 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} = (∅ : Set α) :=
eq_empty_of_subset_empty fun _ => coe_ne_top
#align with_top.preimage_coe_top WithTop.preimage_coe_top
variable [Preorder α] {a b : α}
| Mathlib/Order/Interval/Set/WithBotTop.lean | 33 | 35 | theorem range_coe : range (some : α → WithTop α) = Iio ⊤ := by |
ext x
rw [mem_Iio, WithTop.lt_top_iff_ne_top, mem_range, ne_top_iff_exists]
| 2 | 7.389056 | 1 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} = (∅ : Set α) :=
eq_empty_of_subset_empty fun _ => coe_ne_top
#align with_top.preimage_coe_top WithTop.preimage_coe_top
variable [Preorder α] {a b : α}
theorem range_coe : range (some : α → WithTop α) = Iio ⊤ := by
ext x
rw [mem_Iio, WithTop.lt_top_iff_ne_top, mem_range, ne_top_iff_exists]
#align with_top.range_coe WithTop.range_coe
@[simp]
theorem preimage_coe_Ioi : (some : α → WithTop α) ⁻¹' Ioi a = Ioi a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Ioi WithTop.preimage_coe_Ioi
@[simp]
theorem preimage_coe_Ici : (some : α → WithTop α) ⁻¹' Ici a = Ici a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Ici WithTop.preimage_coe_Ici
@[simp]
theorem preimage_coe_Iio : (some : α → WithTop α) ⁻¹' Iio a = Iio a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Iio WithTop.preimage_coe_Iio
@[simp]
theorem preimage_coe_Iic : (some : α → WithTop α) ⁻¹' Iic a = Iic a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Iic WithTop.preimage_coe_Iic
@[simp]
| Mathlib/Order/Interval/Set/WithBotTop.lean | 59 | 59 | theorem preimage_coe_Icc : (some : α → WithTop α) ⁻¹' Icc a b = Icc a b := by | simp [← Ici_inter_Iic]
| 1 | 2.718282 | 0 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} = (∅ : Set α) :=
eq_empty_of_subset_empty fun _ => coe_ne_top
#align with_top.preimage_coe_top WithTop.preimage_coe_top
variable [Preorder α] {a b : α}
theorem range_coe : range (some : α → WithTop α) = Iio ⊤ := by
ext x
rw [mem_Iio, WithTop.lt_top_iff_ne_top, mem_range, ne_top_iff_exists]
#align with_top.range_coe WithTop.range_coe
@[simp]
theorem preimage_coe_Ioi : (some : α → WithTop α) ⁻¹' Ioi a = Ioi a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Ioi WithTop.preimage_coe_Ioi
@[simp]
theorem preimage_coe_Ici : (some : α → WithTop α) ⁻¹' Ici a = Ici a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Ici WithTop.preimage_coe_Ici
@[simp]
theorem preimage_coe_Iio : (some : α → WithTop α) ⁻¹' Iio a = Iio a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Iio WithTop.preimage_coe_Iio
@[simp]
theorem preimage_coe_Iic : (some : α → WithTop α) ⁻¹' Iic a = Iic a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Iic WithTop.preimage_coe_Iic
@[simp]
theorem preimage_coe_Icc : (some : α → WithTop α) ⁻¹' Icc a b = Icc a b := by simp [← Ici_inter_Iic]
#align with_top.preimage_coe_Icc WithTop.preimage_coe_Icc
@[simp]
| Mathlib/Order/Interval/Set/WithBotTop.lean | 63 | 63 | theorem preimage_coe_Ico : (some : α → WithTop α) ⁻¹' Ico a b = Ico a b := by | simp [← Ici_inter_Iio]
| 1 | 2.718282 | 0 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} = (∅ : Set α) :=
eq_empty_of_subset_empty fun _ => coe_ne_top
#align with_top.preimage_coe_top WithTop.preimage_coe_top
variable [Preorder α] {a b : α}
theorem range_coe : range (some : α → WithTop α) = Iio ⊤ := by
ext x
rw [mem_Iio, WithTop.lt_top_iff_ne_top, mem_range, ne_top_iff_exists]
#align with_top.range_coe WithTop.range_coe
@[simp]
theorem preimage_coe_Ioi : (some : α → WithTop α) ⁻¹' Ioi a = Ioi a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Ioi WithTop.preimage_coe_Ioi
@[simp]
theorem preimage_coe_Ici : (some : α → WithTop α) ⁻¹' Ici a = Ici a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Ici WithTop.preimage_coe_Ici
@[simp]
theorem preimage_coe_Iio : (some : α → WithTop α) ⁻¹' Iio a = Iio a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Iio WithTop.preimage_coe_Iio
@[simp]
theorem preimage_coe_Iic : (some : α → WithTop α) ⁻¹' Iic a = Iic a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Iic WithTop.preimage_coe_Iic
@[simp]
theorem preimage_coe_Icc : (some : α → WithTop α) ⁻¹' Icc a b = Icc a b := by simp [← Ici_inter_Iic]
#align with_top.preimage_coe_Icc WithTop.preimage_coe_Icc
@[simp]
theorem preimage_coe_Ico : (some : α → WithTop α) ⁻¹' Ico a b = Ico a b := by simp [← Ici_inter_Iio]
#align with_top.preimage_coe_Ico WithTop.preimage_coe_Ico
@[simp]
| Mathlib/Order/Interval/Set/WithBotTop.lean | 67 | 67 | theorem preimage_coe_Ioc : (some : α → WithTop α) ⁻¹' Ioc a b = Ioc a b := by | simp [← Ioi_inter_Iic]
| 1 | 2.718282 | 0 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} = (∅ : Set α) :=
eq_empty_of_subset_empty fun _ => coe_ne_top
#align with_top.preimage_coe_top WithTop.preimage_coe_top
variable [Preorder α] {a b : α}
theorem range_coe : range (some : α → WithTop α) = Iio ⊤ := by
ext x
rw [mem_Iio, WithTop.lt_top_iff_ne_top, mem_range, ne_top_iff_exists]
#align with_top.range_coe WithTop.range_coe
@[simp]
theorem preimage_coe_Ioi : (some : α → WithTop α) ⁻¹' Ioi a = Ioi a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Ioi WithTop.preimage_coe_Ioi
@[simp]
theorem preimage_coe_Ici : (some : α → WithTop α) ⁻¹' Ici a = Ici a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Ici WithTop.preimage_coe_Ici
@[simp]
theorem preimage_coe_Iio : (some : α → WithTop α) ⁻¹' Iio a = Iio a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Iio WithTop.preimage_coe_Iio
@[simp]
theorem preimage_coe_Iic : (some : α → WithTop α) ⁻¹' Iic a = Iic a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Iic WithTop.preimage_coe_Iic
@[simp]
theorem preimage_coe_Icc : (some : α → WithTop α) ⁻¹' Icc a b = Icc a b := by simp [← Ici_inter_Iic]
#align with_top.preimage_coe_Icc WithTop.preimage_coe_Icc
@[simp]
theorem preimage_coe_Ico : (some : α → WithTop α) ⁻¹' Ico a b = Ico a b := by simp [← Ici_inter_Iio]
#align with_top.preimage_coe_Ico WithTop.preimage_coe_Ico
@[simp]
theorem preimage_coe_Ioc : (some : α → WithTop α) ⁻¹' Ioc a b = Ioc a b := by simp [← Ioi_inter_Iic]
#align with_top.preimage_coe_Ioc WithTop.preimage_coe_Ioc
@[simp]
| Mathlib/Order/Interval/Set/WithBotTop.lean | 71 | 71 | theorem preimage_coe_Ioo : (some : α → WithTop α) ⁻¹' Ioo a b = Ioo a b := by | simp [← Ioi_inter_Iio]
| 1 | 2.718282 | 0 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} = (∅ : Set α) :=
eq_empty_of_subset_empty fun _ => coe_ne_top
#align with_top.preimage_coe_top WithTop.preimage_coe_top
variable [Preorder α] {a b : α}
theorem range_coe : range (some : α → WithTop α) = Iio ⊤ := by
ext x
rw [mem_Iio, WithTop.lt_top_iff_ne_top, mem_range, ne_top_iff_exists]
#align with_top.range_coe WithTop.range_coe
@[simp]
theorem preimage_coe_Ioi : (some : α → WithTop α) ⁻¹' Ioi a = Ioi a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Ioi WithTop.preimage_coe_Ioi
@[simp]
theorem preimage_coe_Ici : (some : α → WithTop α) ⁻¹' Ici a = Ici a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Ici WithTop.preimage_coe_Ici
@[simp]
theorem preimage_coe_Iio : (some : α → WithTop α) ⁻¹' Iio a = Iio a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Iio WithTop.preimage_coe_Iio
@[simp]
theorem preimage_coe_Iic : (some : α → WithTop α) ⁻¹' Iic a = Iic a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Iic WithTop.preimage_coe_Iic
@[simp]
theorem preimage_coe_Icc : (some : α → WithTop α) ⁻¹' Icc a b = Icc a b := by simp [← Ici_inter_Iic]
#align with_top.preimage_coe_Icc WithTop.preimage_coe_Icc
@[simp]
theorem preimage_coe_Ico : (some : α → WithTop α) ⁻¹' Ico a b = Ico a b := by simp [← Ici_inter_Iio]
#align with_top.preimage_coe_Ico WithTop.preimage_coe_Ico
@[simp]
theorem preimage_coe_Ioc : (some : α → WithTop α) ⁻¹' Ioc a b = Ioc a b := by simp [← Ioi_inter_Iic]
#align with_top.preimage_coe_Ioc WithTop.preimage_coe_Ioc
@[simp]
theorem preimage_coe_Ioo : (some : α → WithTop α) ⁻¹' Ioo a b = Ioo a b := by simp [← Ioi_inter_Iio]
#align with_top.preimage_coe_Ioo WithTop.preimage_coe_Ioo
@[simp]
| Mathlib/Order/Interval/Set/WithBotTop.lean | 75 | 76 | theorem preimage_coe_Iio_top : (some : α → WithTop α) ⁻¹' Iio ⊤ = univ := by |
rw [← range_coe, preimage_range]
| 1 | 2.718282 | 0 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} = (∅ : Set α) :=
eq_empty_of_subset_empty fun _ => coe_ne_top
#align with_top.preimage_coe_top WithTop.preimage_coe_top
variable [Preorder α] {a b : α}
theorem range_coe : range (some : α → WithTop α) = Iio ⊤ := by
ext x
rw [mem_Iio, WithTop.lt_top_iff_ne_top, mem_range, ne_top_iff_exists]
#align with_top.range_coe WithTop.range_coe
@[simp]
theorem preimage_coe_Ioi : (some : α → WithTop α) ⁻¹' Ioi a = Ioi a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Ioi WithTop.preimage_coe_Ioi
@[simp]
theorem preimage_coe_Ici : (some : α → WithTop α) ⁻¹' Ici a = Ici a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Ici WithTop.preimage_coe_Ici
@[simp]
theorem preimage_coe_Iio : (some : α → WithTop α) ⁻¹' Iio a = Iio a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Iio WithTop.preimage_coe_Iio
@[simp]
theorem preimage_coe_Iic : (some : α → WithTop α) ⁻¹' Iic a = Iic a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Iic WithTop.preimage_coe_Iic
@[simp]
theorem preimage_coe_Icc : (some : α → WithTop α) ⁻¹' Icc a b = Icc a b := by simp [← Ici_inter_Iic]
#align with_top.preimage_coe_Icc WithTop.preimage_coe_Icc
@[simp]
theorem preimage_coe_Ico : (some : α → WithTop α) ⁻¹' Ico a b = Ico a b := by simp [← Ici_inter_Iio]
#align with_top.preimage_coe_Ico WithTop.preimage_coe_Ico
@[simp]
theorem preimage_coe_Ioc : (some : α → WithTop α) ⁻¹' Ioc a b = Ioc a b := by simp [← Ioi_inter_Iic]
#align with_top.preimage_coe_Ioc WithTop.preimage_coe_Ioc
@[simp]
theorem preimage_coe_Ioo : (some : α → WithTop α) ⁻¹' Ioo a b = Ioo a b := by simp [← Ioi_inter_Iio]
#align with_top.preimage_coe_Ioo WithTop.preimage_coe_Ioo
@[simp]
theorem preimage_coe_Iio_top : (some : α → WithTop α) ⁻¹' Iio ⊤ = univ := by
rw [← range_coe, preimage_range]
#align with_top.preimage_coe_Iio_top WithTop.preimage_coe_Iio_top
@[simp]
| Mathlib/Order/Interval/Set/WithBotTop.lean | 80 | 81 | theorem preimage_coe_Ico_top : (some : α → WithTop α) ⁻¹' Ico a ⊤ = Ici a := by |
simp [← Ici_inter_Iio]
| 1 | 2.718282 | 0 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} = (∅ : Set α) :=
eq_empty_of_subset_empty fun _ => coe_ne_top
#align with_top.preimage_coe_top WithTop.preimage_coe_top
variable [Preorder α] {a b : α}
theorem range_coe : range (some : α → WithTop α) = Iio ⊤ := by
ext x
rw [mem_Iio, WithTop.lt_top_iff_ne_top, mem_range, ne_top_iff_exists]
#align with_top.range_coe WithTop.range_coe
@[simp]
theorem preimage_coe_Ioi : (some : α → WithTop α) ⁻¹' Ioi a = Ioi a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Ioi WithTop.preimage_coe_Ioi
@[simp]
theorem preimage_coe_Ici : (some : α → WithTop α) ⁻¹' Ici a = Ici a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Ici WithTop.preimage_coe_Ici
@[simp]
theorem preimage_coe_Iio : (some : α → WithTop α) ⁻¹' Iio a = Iio a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Iio WithTop.preimage_coe_Iio
@[simp]
theorem preimage_coe_Iic : (some : α → WithTop α) ⁻¹' Iic a = Iic a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Iic WithTop.preimage_coe_Iic
@[simp]
theorem preimage_coe_Icc : (some : α → WithTop α) ⁻¹' Icc a b = Icc a b := by simp [← Ici_inter_Iic]
#align with_top.preimage_coe_Icc WithTop.preimage_coe_Icc
@[simp]
theorem preimage_coe_Ico : (some : α → WithTop α) ⁻¹' Ico a b = Ico a b := by simp [← Ici_inter_Iio]
#align with_top.preimage_coe_Ico WithTop.preimage_coe_Ico
@[simp]
theorem preimage_coe_Ioc : (some : α → WithTop α) ⁻¹' Ioc a b = Ioc a b := by simp [← Ioi_inter_Iic]
#align with_top.preimage_coe_Ioc WithTop.preimage_coe_Ioc
@[simp]
theorem preimage_coe_Ioo : (some : α → WithTop α) ⁻¹' Ioo a b = Ioo a b := by simp [← Ioi_inter_Iio]
#align with_top.preimage_coe_Ioo WithTop.preimage_coe_Ioo
@[simp]
theorem preimage_coe_Iio_top : (some : α → WithTop α) ⁻¹' Iio ⊤ = univ := by
rw [← range_coe, preimage_range]
#align with_top.preimage_coe_Iio_top WithTop.preimage_coe_Iio_top
@[simp]
theorem preimage_coe_Ico_top : (some : α → WithTop α) ⁻¹' Ico a ⊤ = Ici a := by
simp [← Ici_inter_Iio]
#align with_top.preimage_coe_Ico_top WithTop.preimage_coe_Ico_top
@[simp]
| Mathlib/Order/Interval/Set/WithBotTop.lean | 85 | 86 | theorem preimage_coe_Ioo_top : (some : α → WithTop α) ⁻¹' Ioo a ⊤ = Ioi a := by |
simp [← Ioi_inter_Iio]
| 1 | 2.718282 | 0 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} = (∅ : Set α) :=
eq_empty_of_subset_empty fun _ => coe_ne_top
#align with_top.preimage_coe_top WithTop.preimage_coe_top
variable [Preorder α] {a b : α}
theorem range_coe : range (some : α → WithTop α) = Iio ⊤ := by
ext x
rw [mem_Iio, WithTop.lt_top_iff_ne_top, mem_range, ne_top_iff_exists]
#align with_top.range_coe WithTop.range_coe
@[simp]
theorem preimage_coe_Ioi : (some : α → WithTop α) ⁻¹' Ioi a = Ioi a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Ioi WithTop.preimage_coe_Ioi
@[simp]
theorem preimage_coe_Ici : (some : α → WithTop α) ⁻¹' Ici a = Ici a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Ici WithTop.preimage_coe_Ici
@[simp]
theorem preimage_coe_Iio : (some : α → WithTop α) ⁻¹' Iio a = Iio a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Iio WithTop.preimage_coe_Iio
@[simp]
theorem preimage_coe_Iic : (some : α → WithTop α) ⁻¹' Iic a = Iic a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Iic WithTop.preimage_coe_Iic
@[simp]
theorem preimage_coe_Icc : (some : α → WithTop α) ⁻¹' Icc a b = Icc a b := by simp [← Ici_inter_Iic]
#align with_top.preimage_coe_Icc WithTop.preimage_coe_Icc
@[simp]
theorem preimage_coe_Ico : (some : α → WithTop α) ⁻¹' Ico a b = Ico a b := by simp [← Ici_inter_Iio]
#align with_top.preimage_coe_Ico WithTop.preimage_coe_Ico
@[simp]
theorem preimage_coe_Ioc : (some : α → WithTop α) ⁻¹' Ioc a b = Ioc a b := by simp [← Ioi_inter_Iic]
#align with_top.preimage_coe_Ioc WithTop.preimage_coe_Ioc
@[simp]
theorem preimage_coe_Ioo : (some : α → WithTop α) ⁻¹' Ioo a b = Ioo a b := by simp [← Ioi_inter_Iio]
#align with_top.preimage_coe_Ioo WithTop.preimage_coe_Ioo
@[simp]
theorem preimage_coe_Iio_top : (some : α → WithTop α) ⁻¹' Iio ⊤ = univ := by
rw [← range_coe, preimage_range]
#align with_top.preimage_coe_Iio_top WithTop.preimage_coe_Iio_top
@[simp]
theorem preimage_coe_Ico_top : (some : α → WithTop α) ⁻¹' Ico a ⊤ = Ici a := by
simp [← Ici_inter_Iio]
#align with_top.preimage_coe_Ico_top WithTop.preimage_coe_Ico_top
@[simp]
theorem preimage_coe_Ioo_top : (some : α → WithTop α) ⁻¹' Ioo a ⊤ = Ioi a := by
simp [← Ioi_inter_Iio]
#align with_top.preimage_coe_Ioo_top WithTop.preimage_coe_Ioo_top
| Mathlib/Order/Interval/Set/WithBotTop.lean | 89 | 90 | theorem image_coe_Ioi : (some : α → WithTop α) '' Ioi a = Ioo (a : WithTop α) ⊤ := by |
rw [← preimage_coe_Ioi, image_preimage_eq_inter_range, range_coe, Ioi_inter_Iio]
| 1 | 2.718282 | 0 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} = (∅ : Set α) :=
eq_empty_of_subset_empty fun _ => coe_ne_top
#align with_top.preimage_coe_top WithTop.preimage_coe_top
variable [Preorder α] {a b : α}
theorem range_coe : range (some : α → WithTop α) = Iio ⊤ := by
ext x
rw [mem_Iio, WithTop.lt_top_iff_ne_top, mem_range, ne_top_iff_exists]
#align with_top.range_coe WithTop.range_coe
@[simp]
theorem preimage_coe_Ioi : (some : α → WithTop α) ⁻¹' Ioi a = Ioi a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Ioi WithTop.preimage_coe_Ioi
@[simp]
theorem preimage_coe_Ici : (some : α → WithTop α) ⁻¹' Ici a = Ici a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Ici WithTop.preimage_coe_Ici
@[simp]
theorem preimage_coe_Iio : (some : α → WithTop α) ⁻¹' Iio a = Iio a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Iio WithTop.preimage_coe_Iio
@[simp]
theorem preimage_coe_Iic : (some : α → WithTop α) ⁻¹' Iic a = Iic a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Iic WithTop.preimage_coe_Iic
@[simp]
theorem preimage_coe_Icc : (some : α → WithTop α) ⁻¹' Icc a b = Icc a b := by simp [← Ici_inter_Iic]
#align with_top.preimage_coe_Icc WithTop.preimage_coe_Icc
@[simp]
theorem preimage_coe_Ico : (some : α → WithTop α) ⁻¹' Ico a b = Ico a b := by simp [← Ici_inter_Iio]
#align with_top.preimage_coe_Ico WithTop.preimage_coe_Ico
@[simp]
theorem preimage_coe_Ioc : (some : α → WithTop α) ⁻¹' Ioc a b = Ioc a b := by simp [← Ioi_inter_Iic]
#align with_top.preimage_coe_Ioc WithTop.preimage_coe_Ioc
@[simp]
theorem preimage_coe_Ioo : (some : α → WithTop α) ⁻¹' Ioo a b = Ioo a b := by simp [← Ioi_inter_Iio]
#align with_top.preimage_coe_Ioo WithTop.preimage_coe_Ioo
@[simp]
theorem preimage_coe_Iio_top : (some : α → WithTop α) ⁻¹' Iio ⊤ = univ := by
rw [← range_coe, preimage_range]
#align with_top.preimage_coe_Iio_top WithTop.preimage_coe_Iio_top
@[simp]
theorem preimage_coe_Ico_top : (some : α → WithTop α) ⁻¹' Ico a ⊤ = Ici a := by
simp [← Ici_inter_Iio]
#align with_top.preimage_coe_Ico_top WithTop.preimage_coe_Ico_top
@[simp]
theorem preimage_coe_Ioo_top : (some : α → WithTop α) ⁻¹' Ioo a ⊤ = Ioi a := by
simp [← Ioi_inter_Iio]
#align with_top.preimage_coe_Ioo_top WithTop.preimage_coe_Ioo_top
theorem image_coe_Ioi : (some : α → WithTop α) '' Ioi a = Ioo (a : WithTop α) ⊤ := by
rw [← preimage_coe_Ioi, image_preimage_eq_inter_range, range_coe, Ioi_inter_Iio]
#align with_top.image_coe_Ioi WithTop.image_coe_Ioi
| Mathlib/Order/Interval/Set/WithBotTop.lean | 93 | 94 | theorem image_coe_Ici : (some : α → WithTop α) '' Ici a = Ico (a : WithTop α) ⊤ := by |
rw [← preimage_coe_Ici, image_preimage_eq_inter_range, range_coe, Ici_inter_Iio]
| 1 | 2.718282 | 0 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} = (∅ : Set α) :=
eq_empty_of_subset_empty fun _ => coe_ne_top
#align with_top.preimage_coe_top WithTop.preimage_coe_top
variable [Preorder α] {a b : α}
theorem range_coe : range (some : α → WithTop α) = Iio ⊤ := by
ext x
rw [mem_Iio, WithTop.lt_top_iff_ne_top, mem_range, ne_top_iff_exists]
#align with_top.range_coe WithTop.range_coe
@[simp]
theorem preimage_coe_Ioi : (some : α → WithTop α) ⁻¹' Ioi a = Ioi a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Ioi WithTop.preimage_coe_Ioi
@[simp]
theorem preimage_coe_Ici : (some : α → WithTop α) ⁻¹' Ici a = Ici a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Ici WithTop.preimage_coe_Ici
@[simp]
theorem preimage_coe_Iio : (some : α → WithTop α) ⁻¹' Iio a = Iio a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Iio WithTop.preimage_coe_Iio
@[simp]
theorem preimage_coe_Iic : (some : α → WithTop α) ⁻¹' Iic a = Iic a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Iic WithTop.preimage_coe_Iic
@[simp]
theorem preimage_coe_Icc : (some : α → WithTop α) ⁻¹' Icc a b = Icc a b := by simp [← Ici_inter_Iic]
#align with_top.preimage_coe_Icc WithTop.preimage_coe_Icc
@[simp]
theorem preimage_coe_Ico : (some : α → WithTop α) ⁻¹' Ico a b = Ico a b := by simp [← Ici_inter_Iio]
#align with_top.preimage_coe_Ico WithTop.preimage_coe_Ico
@[simp]
theorem preimage_coe_Ioc : (some : α → WithTop α) ⁻¹' Ioc a b = Ioc a b := by simp [← Ioi_inter_Iic]
#align with_top.preimage_coe_Ioc WithTop.preimage_coe_Ioc
@[simp]
theorem preimage_coe_Ioo : (some : α → WithTop α) ⁻¹' Ioo a b = Ioo a b := by simp [← Ioi_inter_Iio]
#align with_top.preimage_coe_Ioo WithTop.preimage_coe_Ioo
@[simp]
theorem preimage_coe_Iio_top : (some : α → WithTop α) ⁻¹' Iio ⊤ = univ := by
rw [← range_coe, preimage_range]
#align with_top.preimage_coe_Iio_top WithTop.preimage_coe_Iio_top
@[simp]
theorem preimage_coe_Ico_top : (some : α → WithTop α) ⁻¹' Ico a ⊤ = Ici a := by
simp [← Ici_inter_Iio]
#align with_top.preimage_coe_Ico_top WithTop.preimage_coe_Ico_top
@[simp]
theorem preimage_coe_Ioo_top : (some : α → WithTop α) ⁻¹' Ioo a ⊤ = Ioi a := by
simp [← Ioi_inter_Iio]
#align with_top.preimage_coe_Ioo_top WithTop.preimage_coe_Ioo_top
theorem image_coe_Ioi : (some : α → WithTop α) '' Ioi a = Ioo (a : WithTop α) ⊤ := by
rw [← preimage_coe_Ioi, image_preimage_eq_inter_range, range_coe, Ioi_inter_Iio]
#align with_top.image_coe_Ioi WithTop.image_coe_Ioi
theorem image_coe_Ici : (some : α → WithTop α) '' Ici a = Ico (a : WithTop α) ⊤ := by
rw [← preimage_coe_Ici, image_preimage_eq_inter_range, range_coe, Ici_inter_Iio]
#align with_top.image_coe_Ici WithTop.image_coe_Ici
| Mathlib/Order/Interval/Set/WithBotTop.lean | 97 | 99 | theorem image_coe_Iio : (some : α → WithTop α) '' Iio a = Iio (a : WithTop α) := by |
rw [← preimage_coe_Iio, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left (Iio_subset_Iio le_top)]
| 2 | 7.389056 | 1 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} = (∅ : Set α) :=
eq_empty_of_subset_empty fun _ => coe_ne_top
#align with_top.preimage_coe_top WithTop.preimage_coe_top
variable [Preorder α] {a b : α}
theorem range_coe : range (some : α → WithTop α) = Iio ⊤ := by
ext x
rw [mem_Iio, WithTop.lt_top_iff_ne_top, mem_range, ne_top_iff_exists]
#align with_top.range_coe WithTop.range_coe
@[simp]
theorem preimage_coe_Ioi : (some : α → WithTop α) ⁻¹' Ioi a = Ioi a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Ioi WithTop.preimage_coe_Ioi
@[simp]
theorem preimage_coe_Ici : (some : α → WithTop α) ⁻¹' Ici a = Ici a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Ici WithTop.preimage_coe_Ici
@[simp]
theorem preimage_coe_Iio : (some : α → WithTop α) ⁻¹' Iio a = Iio a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Iio WithTop.preimage_coe_Iio
@[simp]
theorem preimage_coe_Iic : (some : α → WithTop α) ⁻¹' Iic a = Iic a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Iic WithTop.preimage_coe_Iic
@[simp]
theorem preimage_coe_Icc : (some : α → WithTop α) ⁻¹' Icc a b = Icc a b := by simp [← Ici_inter_Iic]
#align with_top.preimage_coe_Icc WithTop.preimage_coe_Icc
@[simp]
theorem preimage_coe_Ico : (some : α → WithTop α) ⁻¹' Ico a b = Ico a b := by simp [← Ici_inter_Iio]
#align with_top.preimage_coe_Ico WithTop.preimage_coe_Ico
@[simp]
theorem preimage_coe_Ioc : (some : α → WithTop α) ⁻¹' Ioc a b = Ioc a b := by simp [← Ioi_inter_Iic]
#align with_top.preimage_coe_Ioc WithTop.preimage_coe_Ioc
@[simp]
theorem preimage_coe_Ioo : (some : α → WithTop α) ⁻¹' Ioo a b = Ioo a b := by simp [← Ioi_inter_Iio]
#align with_top.preimage_coe_Ioo WithTop.preimage_coe_Ioo
@[simp]
theorem preimage_coe_Iio_top : (some : α → WithTop α) ⁻¹' Iio ⊤ = univ := by
rw [← range_coe, preimage_range]
#align with_top.preimage_coe_Iio_top WithTop.preimage_coe_Iio_top
@[simp]
theorem preimage_coe_Ico_top : (some : α → WithTop α) ⁻¹' Ico a ⊤ = Ici a := by
simp [← Ici_inter_Iio]
#align with_top.preimage_coe_Ico_top WithTop.preimage_coe_Ico_top
@[simp]
theorem preimage_coe_Ioo_top : (some : α → WithTop α) ⁻¹' Ioo a ⊤ = Ioi a := by
simp [← Ioi_inter_Iio]
#align with_top.preimage_coe_Ioo_top WithTop.preimage_coe_Ioo_top
theorem image_coe_Ioi : (some : α → WithTop α) '' Ioi a = Ioo (a : WithTop α) ⊤ := by
rw [← preimage_coe_Ioi, image_preimage_eq_inter_range, range_coe, Ioi_inter_Iio]
#align with_top.image_coe_Ioi WithTop.image_coe_Ioi
theorem image_coe_Ici : (some : α → WithTop α) '' Ici a = Ico (a : WithTop α) ⊤ := by
rw [← preimage_coe_Ici, image_preimage_eq_inter_range, range_coe, Ici_inter_Iio]
#align with_top.image_coe_Ici WithTop.image_coe_Ici
theorem image_coe_Iio : (some : α → WithTop α) '' Iio a = Iio (a : WithTop α) := by
rw [← preimage_coe_Iio, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left (Iio_subset_Iio le_top)]
#align with_top.image_coe_Iio WithTop.image_coe_Iio
| Mathlib/Order/Interval/Set/WithBotTop.lean | 102 | 104 | theorem image_coe_Iic : (some : α → WithTop α) '' Iic a = Iic (a : WithTop α) := by |
rw [← preimage_coe_Iic, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left (Iic_subset_Iio.2 <| coe_lt_top a)]
| 2 | 7.389056 | 1 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} = (∅ : Set α) :=
eq_empty_of_subset_empty fun _ => coe_ne_top
#align with_top.preimage_coe_top WithTop.preimage_coe_top
variable [Preorder α] {a b : α}
theorem range_coe : range (some : α → WithTop α) = Iio ⊤ := by
ext x
rw [mem_Iio, WithTop.lt_top_iff_ne_top, mem_range, ne_top_iff_exists]
#align with_top.range_coe WithTop.range_coe
@[simp]
theorem preimage_coe_Ioi : (some : α → WithTop α) ⁻¹' Ioi a = Ioi a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Ioi WithTop.preimage_coe_Ioi
@[simp]
theorem preimage_coe_Ici : (some : α → WithTop α) ⁻¹' Ici a = Ici a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Ici WithTop.preimage_coe_Ici
@[simp]
theorem preimage_coe_Iio : (some : α → WithTop α) ⁻¹' Iio a = Iio a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Iio WithTop.preimage_coe_Iio
@[simp]
theorem preimage_coe_Iic : (some : α → WithTop α) ⁻¹' Iic a = Iic a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Iic WithTop.preimage_coe_Iic
@[simp]
theorem preimage_coe_Icc : (some : α → WithTop α) ⁻¹' Icc a b = Icc a b := by simp [← Ici_inter_Iic]
#align with_top.preimage_coe_Icc WithTop.preimage_coe_Icc
@[simp]
theorem preimage_coe_Ico : (some : α → WithTop α) ⁻¹' Ico a b = Ico a b := by simp [← Ici_inter_Iio]
#align with_top.preimage_coe_Ico WithTop.preimage_coe_Ico
@[simp]
theorem preimage_coe_Ioc : (some : α → WithTop α) ⁻¹' Ioc a b = Ioc a b := by simp [← Ioi_inter_Iic]
#align with_top.preimage_coe_Ioc WithTop.preimage_coe_Ioc
@[simp]
theorem preimage_coe_Ioo : (some : α → WithTop α) ⁻¹' Ioo a b = Ioo a b := by simp [← Ioi_inter_Iio]
#align with_top.preimage_coe_Ioo WithTop.preimage_coe_Ioo
@[simp]
theorem preimage_coe_Iio_top : (some : α → WithTop α) ⁻¹' Iio ⊤ = univ := by
rw [← range_coe, preimage_range]
#align with_top.preimage_coe_Iio_top WithTop.preimage_coe_Iio_top
@[simp]
theorem preimage_coe_Ico_top : (some : α → WithTop α) ⁻¹' Ico a ⊤ = Ici a := by
simp [← Ici_inter_Iio]
#align with_top.preimage_coe_Ico_top WithTop.preimage_coe_Ico_top
@[simp]
theorem preimage_coe_Ioo_top : (some : α → WithTop α) ⁻¹' Ioo a ⊤ = Ioi a := by
simp [← Ioi_inter_Iio]
#align with_top.preimage_coe_Ioo_top WithTop.preimage_coe_Ioo_top
theorem image_coe_Ioi : (some : α → WithTop α) '' Ioi a = Ioo (a : WithTop α) ⊤ := by
rw [← preimage_coe_Ioi, image_preimage_eq_inter_range, range_coe, Ioi_inter_Iio]
#align with_top.image_coe_Ioi WithTop.image_coe_Ioi
theorem image_coe_Ici : (some : α → WithTop α) '' Ici a = Ico (a : WithTop α) ⊤ := by
rw [← preimage_coe_Ici, image_preimage_eq_inter_range, range_coe, Ici_inter_Iio]
#align with_top.image_coe_Ici WithTop.image_coe_Ici
theorem image_coe_Iio : (some : α → WithTop α) '' Iio a = Iio (a : WithTop α) := by
rw [← preimage_coe_Iio, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left (Iio_subset_Iio le_top)]
#align with_top.image_coe_Iio WithTop.image_coe_Iio
theorem image_coe_Iic : (some : α → WithTop α) '' Iic a = Iic (a : WithTop α) := by
rw [← preimage_coe_Iic, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left (Iic_subset_Iio.2 <| coe_lt_top a)]
#align with_top.image_coe_Iic WithTop.image_coe_Iic
| Mathlib/Order/Interval/Set/WithBotTop.lean | 107 | 110 | theorem image_coe_Icc : (some : α → WithTop α) '' Icc a b = Icc (a : WithTop α) b := by |
rw [← preimage_coe_Icc, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left
(Subset.trans Icc_subset_Iic_self <| Iic_subset_Iio.2 <| coe_lt_top b)]
| 3 | 20.085537 | 1 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} = (∅ : Set α) :=
eq_empty_of_subset_empty fun _ => coe_ne_top
#align with_top.preimage_coe_top WithTop.preimage_coe_top
variable [Preorder α] {a b : α}
theorem range_coe : range (some : α → WithTop α) = Iio ⊤ := by
ext x
rw [mem_Iio, WithTop.lt_top_iff_ne_top, mem_range, ne_top_iff_exists]
#align with_top.range_coe WithTop.range_coe
@[simp]
theorem preimage_coe_Ioi : (some : α → WithTop α) ⁻¹' Ioi a = Ioi a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Ioi WithTop.preimage_coe_Ioi
@[simp]
theorem preimage_coe_Ici : (some : α → WithTop α) ⁻¹' Ici a = Ici a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Ici WithTop.preimage_coe_Ici
@[simp]
theorem preimage_coe_Iio : (some : α → WithTop α) ⁻¹' Iio a = Iio a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Iio WithTop.preimage_coe_Iio
@[simp]
theorem preimage_coe_Iic : (some : α → WithTop α) ⁻¹' Iic a = Iic a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Iic WithTop.preimage_coe_Iic
@[simp]
theorem preimage_coe_Icc : (some : α → WithTop α) ⁻¹' Icc a b = Icc a b := by simp [← Ici_inter_Iic]
#align with_top.preimage_coe_Icc WithTop.preimage_coe_Icc
@[simp]
theorem preimage_coe_Ico : (some : α → WithTop α) ⁻¹' Ico a b = Ico a b := by simp [← Ici_inter_Iio]
#align with_top.preimage_coe_Ico WithTop.preimage_coe_Ico
@[simp]
theorem preimage_coe_Ioc : (some : α → WithTop α) ⁻¹' Ioc a b = Ioc a b := by simp [← Ioi_inter_Iic]
#align with_top.preimage_coe_Ioc WithTop.preimage_coe_Ioc
@[simp]
theorem preimage_coe_Ioo : (some : α → WithTop α) ⁻¹' Ioo a b = Ioo a b := by simp [← Ioi_inter_Iio]
#align with_top.preimage_coe_Ioo WithTop.preimage_coe_Ioo
@[simp]
theorem preimage_coe_Iio_top : (some : α → WithTop α) ⁻¹' Iio ⊤ = univ := by
rw [← range_coe, preimage_range]
#align with_top.preimage_coe_Iio_top WithTop.preimage_coe_Iio_top
@[simp]
theorem preimage_coe_Ico_top : (some : α → WithTop α) ⁻¹' Ico a ⊤ = Ici a := by
simp [← Ici_inter_Iio]
#align with_top.preimage_coe_Ico_top WithTop.preimage_coe_Ico_top
@[simp]
theorem preimage_coe_Ioo_top : (some : α → WithTop α) ⁻¹' Ioo a ⊤ = Ioi a := by
simp [← Ioi_inter_Iio]
#align with_top.preimage_coe_Ioo_top WithTop.preimage_coe_Ioo_top
theorem image_coe_Ioi : (some : α → WithTop α) '' Ioi a = Ioo (a : WithTop α) ⊤ := by
rw [← preimage_coe_Ioi, image_preimage_eq_inter_range, range_coe, Ioi_inter_Iio]
#align with_top.image_coe_Ioi WithTop.image_coe_Ioi
theorem image_coe_Ici : (some : α → WithTop α) '' Ici a = Ico (a : WithTop α) ⊤ := by
rw [← preimage_coe_Ici, image_preimage_eq_inter_range, range_coe, Ici_inter_Iio]
#align with_top.image_coe_Ici WithTop.image_coe_Ici
theorem image_coe_Iio : (some : α → WithTop α) '' Iio a = Iio (a : WithTop α) := by
rw [← preimage_coe_Iio, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left (Iio_subset_Iio le_top)]
#align with_top.image_coe_Iio WithTop.image_coe_Iio
theorem image_coe_Iic : (some : α → WithTop α) '' Iic a = Iic (a : WithTop α) := by
rw [← preimage_coe_Iic, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left (Iic_subset_Iio.2 <| coe_lt_top a)]
#align with_top.image_coe_Iic WithTop.image_coe_Iic
theorem image_coe_Icc : (some : α → WithTop α) '' Icc a b = Icc (a : WithTop α) b := by
rw [← preimage_coe_Icc, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left
(Subset.trans Icc_subset_Iic_self <| Iic_subset_Iio.2 <| coe_lt_top b)]
#align with_top.image_coe_Icc WithTop.image_coe_Icc
| Mathlib/Order/Interval/Set/WithBotTop.lean | 113 | 115 | theorem image_coe_Ico : (some : α → WithTop α) '' Ico a b = Ico (a : WithTop α) b := by |
rw [← preimage_coe_Ico, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left (Subset.trans Ico_subset_Iio_self <| Iio_subset_Iio le_top)]
| 2 | 7.389056 | 1 | 0.4 | 15 | 401 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} = (∅ : Set α) :=
eq_empty_of_subset_empty fun _ => coe_ne_top
#align with_top.preimage_coe_top WithTop.preimage_coe_top
variable [Preorder α] {a b : α}
theorem range_coe : range (some : α → WithTop α) = Iio ⊤ := by
ext x
rw [mem_Iio, WithTop.lt_top_iff_ne_top, mem_range, ne_top_iff_exists]
#align with_top.range_coe WithTop.range_coe
@[simp]
theorem preimage_coe_Ioi : (some : α → WithTop α) ⁻¹' Ioi a = Ioi a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Ioi WithTop.preimage_coe_Ioi
@[simp]
theorem preimage_coe_Ici : (some : α → WithTop α) ⁻¹' Ici a = Ici a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Ici WithTop.preimage_coe_Ici
@[simp]
theorem preimage_coe_Iio : (some : α → WithTop α) ⁻¹' Iio a = Iio a :=
ext fun _ => coe_lt_coe
#align with_top.preimage_coe_Iio WithTop.preimage_coe_Iio
@[simp]
theorem preimage_coe_Iic : (some : α → WithTop α) ⁻¹' Iic a = Iic a :=
ext fun _ => coe_le_coe
#align with_top.preimage_coe_Iic WithTop.preimage_coe_Iic
@[simp]
theorem preimage_coe_Icc : (some : α → WithTop α) ⁻¹' Icc a b = Icc a b := by simp [← Ici_inter_Iic]
#align with_top.preimage_coe_Icc WithTop.preimage_coe_Icc
@[simp]
theorem preimage_coe_Ico : (some : α → WithTop α) ⁻¹' Ico a b = Ico a b := by simp [← Ici_inter_Iio]
#align with_top.preimage_coe_Ico WithTop.preimage_coe_Ico
@[simp]
theorem preimage_coe_Ioc : (some : α → WithTop α) ⁻¹' Ioc a b = Ioc a b := by simp [← Ioi_inter_Iic]
#align with_top.preimage_coe_Ioc WithTop.preimage_coe_Ioc
@[simp]
theorem preimage_coe_Ioo : (some : α → WithTop α) ⁻¹' Ioo a b = Ioo a b := by simp [← Ioi_inter_Iio]
#align with_top.preimage_coe_Ioo WithTop.preimage_coe_Ioo
@[simp]
theorem preimage_coe_Iio_top : (some : α → WithTop α) ⁻¹' Iio ⊤ = univ := by
rw [← range_coe, preimage_range]
#align with_top.preimage_coe_Iio_top WithTop.preimage_coe_Iio_top
@[simp]
theorem preimage_coe_Ico_top : (some : α → WithTop α) ⁻¹' Ico a ⊤ = Ici a := by
simp [← Ici_inter_Iio]
#align with_top.preimage_coe_Ico_top WithTop.preimage_coe_Ico_top
@[simp]
theorem preimage_coe_Ioo_top : (some : α → WithTop α) ⁻¹' Ioo a ⊤ = Ioi a := by
simp [← Ioi_inter_Iio]
#align with_top.preimage_coe_Ioo_top WithTop.preimage_coe_Ioo_top
theorem image_coe_Ioi : (some : α → WithTop α) '' Ioi a = Ioo (a : WithTop α) ⊤ := by
rw [← preimage_coe_Ioi, image_preimage_eq_inter_range, range_coe, Ioi_inter_Iio]
#align with_top.image_coe_Ioi WithTop.image_coe_Ioi
theorem image_coe_Ici : (some : α → WithTop α) '' Ici a = Ico (a : WithTop α) ⊤ := by
rw [← preimage_coe_Ici, image_preimage_eq_inter_range, range_coe, Ici_inter_Iio]
#align with_top.image_coe_Ici WithTop.image_coe_Ici
theorem image_coe_Iio : (some : α → WithTop α) '' Iio a = Iio (a : WithTop α) := by
rw [← preimage_coe_Iio, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left (Iio_subset_Iio le_top)]
#align with_top.image_coe_Iio WithTop.image_coe_Iio
theorem image_coe_Iic : (some : α → WithTop α) '' Iic a = Iic (a : WithTop α) := by
rw [← preimage_coe_Iic, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left (Iic_subset_Iio.2 <| coe_lt_top a)]
#align with_top.image_coe_Iic WithTop.image_coe_Iic
theorem image_coe_Icc : (some : α → WithTop α) '' Icc a b = Icc (a : WithTop α) b := by
rw [← preimage_coe_Icc, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left
(Subset.trans Icc_subset_Iic_self <| Iic_subset_Iio.2 <| coe_lt_top b)]
#align with_top.image_coe_Icc WithTop.image_coe_Icc
theorem image_coe_Ico : (some : α → WithTop α) '' Ico a b = Ico (a : WithTop α) b := by
rw [← preimage_coe_Ico, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left (Subset.trans Ico_subset_Iio_self <| Iio_subset_Iio le_top)]
#align with_top.image_coe_Ico WithTop.image_coe_Ico
| Mathlib/Order/Interval/Set/WithBotTop.lean | 118 | 121 | theorem image_coe_Ioc : (some : α → WithTop α) '' Ioc a b = Ioc (a : WithTop α) b := by |
rw [← preimage_coe_Ioc, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left
(Subset.trans Ioc_subset_Iic_self <| Iic_subset_Iio.2 <| coe_lt_top b)]
| 3 | 20.085537 | 1 | 0.4 | 15 | 401 |
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
noncomputable section
set_option linter.uppercaseLean3 false
open TopologicalSpace MeasureTheory Filter
open scoped NNReal ENNReal Topology
variable {α E F G : Type*} {m m0 : MeasurableSpace α} {p : ℝ≥0∞} {q : ℝ} {μ ν : Measure α}
[NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup G]
namespace MeasureTheory
section ℒp
section ℒpSpaceDefinition
def snorm' {_ : MeasurableSpace α} (f : α → F) (q : ℝ) (μ : Measure α) : ℝ≥0∞ :=
(∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ q ∂μ) ^ (1 / q)
#align measure_theory.snorm' MeasureTheory.snorm'
def snormEssSup {_ : MeasurableSpace α} (f : α → F) (μ : Measure α) :=
essSup (fun x => (‖f x‖₊ : ℝ≥0∞)) μ
#align measure_theory.snorm_ess_sup MeasureTheory.snormEssSup
def snorm {_ : MeasurableSpace α} (f : α → F) (p : ℝ≥0∞) (μ : Measure α) : ℝ≥0∞ :=
if p = 0 then 0 else if p = ∞ then snormEssSup f μ else snorm' f (ENNReal.toReal p) μ
#align measure_theory.snorm MeasureTheory.snorm
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 87 | 88 | theorem snorm_eq_snorm' (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {f : α → F} :
snorm f p μ = snorm' f (ENNReal.toReal p) μ := by | simp [snorm, hp_ne_zero, hp_ne_top]
| 1 | 2.718282 | 0 | 0.4 | 5 | 402 |
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
noncomputable section
set_option linter.uppercaseLean3 false
open TopologicalSpace MeasureTheory Filter
open scoped NNReal ENNReal Topology
variable {α E F G : Type*} {m m0 : MeasurableSpace α} {p : ℝ≥0∞} {q : ℝ} {μ ν : Measure α}
[NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup G]
namespace MeasureTheory
section ℒp
section ℒpSpaceDefinition
def snorm' {_ : MeasurableSpace α} (f : α → F) (q : ℝ) (μ : Measure α) : ℝ≥0∞ :=
(∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ q ∂μ) ^ (1 / q)
#align measure_theory.snorm' MeasureTheory.snorm'
def snormEssSup {_ : MeasurableSpace α} (f : α → F) (μ : Measure α) :=
essSup (fun x => (‖f x‖₊ : ℝ≥0∞)) μ
#align measure_theory.snorm_ess_sup MeasureTheory.snormEssSup
def snorm {_ : MeasurableSpace α} (f : α → F) (p : ℝ≥0∞) (μ : Measure α) : ℝ≥0∞ :=
if p = 0 then 0 else if p = ∞ then snormEssSup f μ else snorm' f (ENNReal.toReal p) μ
#align measure_theory.snorm MeasureTheory.snorm
theorem snorm_eq_snorm' (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {f : α → F} :
snorm f p μ = snorm' f (ENNReal.toReal p) μ := by simp [snorm, hp_ne_zero, hp_ne_top]
#align measure_theory.snorm_eq_snorm' MeasureTheory.snorm_eq_snorm'
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 91 | 93 | theorem snorm_eq_lintegral_rpow_nnnorm (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {f : α → F} :
snorm f p μ = (∫⁻ x, (‖f x‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) ^ (1 / p.toReal) := by |
rw [snorm_eq_snorm' hp_ne_zero hp_ne_top, snorm']
| 1 | 2.718282 | 0 | 0.4 | 5 | 402 |
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
noncomputable section
set_option linter.uppercaseLean3 false
open TopologicalSpace MeasureTheory Filter
open scoped NNReal ENNReal Topology
variable {α E F G : Type*} {m m0 : MeasurableSpace α} {p : ℝ≥0∞} {q : ℝ} {μ ν : Measure α}
[NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup G]
namespace MeasureTheory
section ℒp
section ℒpSpaceDefinition
def snorm' {_ : MeasurableSpace α} (f : α → F) (q : ℝ) (μ : Measure α) : ℝ≥0∞ :=
(∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ q ∂μ) ^ (1 / q)
#align measure_theory.snorm' MeasureTheory.snorm'
def snormEssSup {_ : MeasurableSpace α} (f : α → F) (μ : Measure α) :=
essSup (fun x => (‖f x‖₊ : ℝ≥0∞)) μ
#align measure_theory.snorm_ess_sup MeasureTheory.snormEssSup
def snorm {_ : MeasurableSpace α} (f : α → F) (p : ℝ≥0∞) (μ : Measure α) : ℝ≥0∞ :=
if p = 0 then 0 else if p = ∞ then snormEssSup f μ else snorm' f (ENNReal.toReal p) μ
#align measure_theory.snorm MeasureTheory.snorm
theorem snorm_eq_snorm' (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {f : α → F} :
snorm f p μ = snorm' f (ENNReal.toReal p) μ := by simp [snorm, hp_ne_zero, hp_ne_top]
#align measure_theory.snorm_eq_snorm' MeasureTheory.snorm_eq_snorm'
theorem snorm_eq_lintegral_rpow_nnnorm (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {f : α → F} :
snorm f p μ = (∫⁻ x, (‖f x‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) ^ (1 / p.toReal) := by
rw [snorm_eq_snorm' hp_ne_zero hp_ne_top, snorm']
#align measure_theory.snorm_eq_lintegral_rpow_nnnorm MeasureTheory.snorm_eq_lintegral_rpow_nnnorm
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 96 | 98 | theorem snorm_one_eq_lintegral_nnnorm {f : α → F} : snorm f 1 μ = ∫⁻ x, ‖f x‖₊ ∂μ := by |
simp_rw [snorm_eq_lintegral_rpow_nnnorm one_ne_zero ENNReal.coe_ne_top, ENNReal.one_toReal,
one_div_one, ENNReal.rpow_one]
| 2 | 7.389056 | 1 | 0.4 | 5 | 402 |
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
noncomputable section
set_option linter.uppercaseLean3 false
open TopologicalSpace MeasureTheory Filter
open scoped NNReal ENNReal Topology
variable {α E F G : Type*} {m m0 : MeasurableSpace α} {p : ℝ≥0∞} {q : ℝ} {μ ν : Measure α}
[NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup G]
namespace MeasureTheory
section ℒp
section ℒpSpaceDefinition
def snorm' {_ : MeasurableSpace α} (f : α → F) (q : ℝ) (μ : Measure α) : ℝ≥0∞ :=
(∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ q ∂μ) ^ (1 / q)
#align measure_theory.snorm' MeasureTheory.snorm'
def snormEssSup {_ : MeasurableSpace α} (f : α → F) (μ : Measure α) :=
essSup (fun x => (‖f x‖₊ : ℝ≥0∞)) μ
#align measure_theory.snorm_ess_sup MeasureTheory.snormEssSup
def snorm {_ : MeasurableSpace α} (f : α → F) (p : ℝ≥0∞) (μ : Measure α) : ℝ≥0∞ :=
if p = 0 then 0 else if p = ∞ then snormEssSup f μ else snorm' f (ENNReal.toReal p) μ
#align measure_theory.snorm MeasureTheory.snorm
theorem snorm_eq_snorm' (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {f : α → F} :
snorm f p μ = snorm' f (ENNReal.toReal p) μ := by simp [snorm, hp_ne_zero, hp_ne_top]
#align measure_theory.snorm_eq_snorm' MeasureTheory.snorm_eq_snorm'
theorem snorm_eq_lintegral_rpow_nnnorm (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {f : α → F} :
snorm f p μ = (∫⁻ x, (‖f x‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) ^ (1 / p.toReal) := by
rw [snorm_eq_snorm' hp_ne_zero hp_ne_top, snorm']
#align measure_theory.snorm_eq_lintegral_rpow_nnnorm MeasureTheory.snorm_eq_lintegral_rpow_nnnorm
theorem snorm_one_eq_lintegral_nnnorm {f : α → F} : snorm f 1 μ = ∫⁻ x, ‖f x‖₊ ∂μ := by
simp_rw [snorm_eq_lintegral_rpow_nnnorm one_ne_zero ENNReal.coe_ne_top, ENNReal.one_toReal,
one_div_one, ENNReal.rpow_one]
#align measure_theory.snorm_one_eq_lintegral_nnnorm MeasureTheory.snorm_one_eq_lintegral_nnnorm
@[simp]
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 102 | 102 | theorem snorm_exponent_top {f : α → F} : snorm f ∞ μ = snormEssSup f μ := by | simp [snorm]
| 1 | 2.718282 | 0 | 0.4 | 5 | 402 |
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
noncomputable section
set_option linter.uppercaseLean3 false
open TopologicalSpace MeasureTheory Filter
open scoped NNReal ENNReal Topology
variable {α E F G : Type*} {m m0 : MeasurableSpace α} {p : ℝ≥0∞} {q : ℝ} {μ ν : Measure α}
[NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup G]
namespace MeasureTheory
section ℒp
section ℒpSpaceDefinition
def snorm' {_ : MeasurableSpace α} (f : α → F) (q : ℝ) (μ : Measure α) : ℝ≥0∞ :=
(∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ q ∂μ) ^ (1 / q)
#align measure_theory.snorm' MeasureTheory.snorm'
def snormEssSup {_ : MeasurableSpace α} (f : α → F) (μ : Measure α) :=
essSup (fun x => (‖f x‖₊ : ℝ≥0∞)) μ
#align measure_theory.snorm_ess_sup MeasureTheory.snormEssSup
def snorm {_ : MeasurableSpace α} (f : α → F) (p : ℝ≥0∞) (μ : Measure α) : ℝ≥0∞ :=
if p = 0 then 0 else if p = ∞ then snormEssSup f μ else snorm' f (ENNReal.toReal p) μ
#align measure_theory.snorm MeasureTheory.snorm
theorem snorm_eq_snorm' (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {f : α → F} :
snorm f p μ = snorm' f (ENNReal.toReal p) μ := by simp [snorm, hp_ne_zero, hp_ne_top]
#align measure_theory.snorm_eq_snorm' MeasureTheory.snorm_eq_snorm'
theorem snorm_eq_lintegral_rpow_nnnorm (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {f : α → F} :
snorm f p μ = (∫⁻ x, (‖f x‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) ^ (1 / p.toReal) := by
rw [snorm_eq_snorm' hp_ne_zero hp_ne_top, snorm']
#align measure_theory.snorm_eq_lintegral_rpow_nnnorm MeasureTheory.snorm_eq_lintegral_rpow_nnnorm
theorem snorm_one_eq_lintegral_nnnorm {f : α → F} : snorm f 1 μ = ∫⁻ x, ‖f x‖₊ ∂μ := by
simp_rw [snorm_eq_lintegral_rpow_nnnorm one_ne_zero ENNReal.coe_ne_top, ENNReal.one_toReal,
one_div_one, ENNReal.rpow_one]
#align measure_theory.snorm_one_eq_lintegral_nnnorm MeasureTheory.snorm_one_eq_lintegral_nnnorm
@[simp]
theorem snorm_exponent_top {f : α → F} : snorm f ∞ μ = snormEssSup f μ := by simp [snorm]
#align measure_theory.snorm_exponent_top MeasureTheory.snorm_exponent_top
def Memℒp {α} {_ : MeasurableSpace α} (f : α → E) (p : ℝ≥0∞)
(μ : Measure α := by volume_tac) : Prop :=
AEStronglyMeasurable f μ ∧ snorm f p μ < ∞
#align measure_theory.mem_ℒp MeasureTheory.Memℒp
theorem Memℒp.aestronglyMeasurable {f : α → E} {p : ℝ≥0∞} (h : Memℒp f p μ) :
AEStronglyMeasurable f μ :=
h.1
#align measure_theory.mem_ℒp.ae_strongly_measurable MeasureTheory.Memℒp.aestronglyMeasurable
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 117 | 120 | theorem lintegral_rpow_nnnorm_eq_rpow_snorm' {f : α → F} (hq0_lt : 0 < q) :
(∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ q ∂μ) = snorm' f q μ ^ q := by |
rw [snorm', ← ENNReal.rpow_mul, one_div, inv_mul_cancel, ENNReal.rpow_one]
exact (ne_of_lt hq0_lt).symm
| 2 | 7.389056 | 1 | 0.4 | 5 | 402 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import geometry.euclidean.angle.unoriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open Real RealInnerProductSpace
namespace EuclideanGeometry
open InnerProductGeometry
variable {V P : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
[NormedAddTorsor V P] {p p₀ p₁ p₂ : P}
nonrec def angle (p1 p2 p3 : P) : ℝ :=
angle (p1 -ᵥ p2 : V) (p3 -ᵥ p2)
#align euclidean_geometry.angle EuclideanGeometry.angle
@[inherit_doc] scoped notation "∠" => EuclideanGeometry.angle
| Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 50 | 57 | theorem continuousAt_angle {x : P × P × P} (hx12 : x.1 ≠ x.2.1) (hx32 : x.2.2 ≠ x.2.1) :
ContinuousAt (fun y : P × P × P => ∠ y.1 y.2.1 y.2.2) x := by |
let f : P × P × P → V × V := fun y => (y.1 -ᵥ y.2.1, y.2.2 -ᵥ y.2.1)
have hf1 : (f x).1 ≠ 0 := by simp [hx12]
have hf2 : (f x).2 ≠ 0 := by simp [hx32]
exact (InnerProductGeometry.continuousAt_angle hf1 hf2).comp
((continuous_fst.vsub continuous_snd.fst).prod_mk
(continuous_snd.snd.vsub continuous_snd.fst)).continuousAt
| 6 | 403.428793 | 2 | 0.4 | 5 | 403 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import geometry.euclidean.angle.unoriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open Real RealInnerProductSpace
namespace EuclideanGeometry
open InnerProductGeometry
variable {V P : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
[NormedAddTorsor V P] {p p₀ p₁ p₂ : P}
nonrec def angle (p1 p2 p3 : P) : ℝ :=
angle (p1 -ᵥ p2 : V) (p3 -ᵥ p2)
#align euclidean_geometry.angle EuclideanGeometry.angle
@[inherit_doc] scoped notation "∠" => EuclideanGeometry.angle
theorem continuousAt_angle {x : P × P × P} (hx12 : x.1 ≠ x.2.1) (hx32 : x.2.2 ≠ x.2.1) :
ContinuousAt (fun y : P × P × P => ∠ y.1 y.2.1 y.2.2) x := by
let f : P × P × P → V × V := fun y => (y.1 -ᵥ y.2.1, y.2.2 -ᵥ y.2.1)
have hf1 : (f x).1 ≠ 0 := by simp [hx12]
have hf2 : (f x).2 ≠ 0 := by simp [hx32]
exact (InnerProductGeometry.continuousAt_angle hf1 hf2).comp
((continuous_fst.vsub continuous_snd.fst).prod_mk
(continuous_snd.snd.vsub continuous_snd.fst)).continuousAt
#align euclidean_geometry.continuous_at_angle EuclideanGeometry.continuousAt_angle
@[simp]
| Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 61 | 64 | theorem _root_.AffineIsometry.angle_map {V₂ P₂ : Type*} [NormedAddCommGroup V₂]
[InnerProductSpace ℝ V₂] [MetricSpace P₂] [NormedAddTorsor V₂ P₂]
(f : P →ᵃⁱ[ℝ] P₂) (p₁ p₂ p₃ : P) : ∠ (f p₁) (f p₂) (f p₃) = ∠ p₁ p₂ p₃ := by |
simp_rw [angle, ← AffineIsometry.map_vsub, LinearIsometry.angle_map]
| 1 | 2.718282 | 0 | 0.4 | 5 | 403 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import geometry.euclidean.angle.unoriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open Real RealInnerProductSpace
namespace EuclideanGeometry
open InnerProductGeometry
variable {V P : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
[NormedAddTorsor V P] {p p₀ p₁ p₂ : P}
nonrec def angle (p1 p2 p3 : P) : ℝ :=
angle (p1 -ᵥ p2 : V) (p3 -ᵥ p2)
#align euclidean_geometry.angle EuclideanGeometry.angle
@[inherit_doc] scoped notation "∠" => EuclideanGeometry.angle
theorem continuousAt_angle {x : P × P × P} (hx12 : x.1 ≠ x.2.1) (hx32 : x.2.2 ≠ x.2.1) :
ContinuousAt (fun y : P × P × P => ∠ y.1 y.2.1 y.2.2) x := by
let f : P × P × P → V × V := fun y => (y.1 -ᵥ y.2.1, y.2.2 -ᵥ y.2.1)
have hf1 : (f x).1 ≠ 0 := by simp [hx12]
have hf2 : (f x).2 ≠ 0 := by simp [hx32]
exact (InnerProductGeometry.continuousAt_angle hf1 hf2).comp
((continuous_fst.vsub continuous_snd.fst).prod_mk
(continuous_snd.snd.vsub continuous_snd.fst)).continuousAt
#align euclidean_geometry.continuous_at_angle EuclideanGeometry.continuousAt_angle
@[simp]
theorem _root_.AffineIsometry.angle_map {V₂ P₂ : Type*} [NormedAddCommGroup V₂]
[InnerProductSpace ℝ V₂] [MetricSpace P₂] [NormedAddTorsor V₂ P₂]
(f : P →ᵃⁱ[ℝ] P₂) (p₁ p₂ p₃ : P) : ∠ (f p₁) (f p₂) (f p₃) = ∠ p₁ p₂ p₃ := by
simp_rw [angle, ← AffineIsometry.map_vsub, LinearIsometry.angle_map]
#align affine_isometry.angle_map AffineIsometry.angle_map
@[simp, norm_cast]
theorem _root_.AffineSubspace.angle_coe {s : AffineSubspace ℝ P} (p₁ p₂ p₃ : s) :
haveI : Nonempty s := ⟨p₁⟩
∠ (p₁ : P) (p₂ : P) (p₃ : P) = ∠ p₁ p₂ p₃ :=
haveI : Nonempty s := ⟨p₁⟩
s.subtypeₐᵢ.angle_map p₁ p₂ p₃
#align affine_subspace.angle_coe AffineSubspace.angle_coe
@[simp]
theorem angle_const_vadd (v : V) (p₁ p₂ p₃ : P) : ∠ (v +ᵥ p₁) (v +ᵥ p₂) (v +ᵥ p₃) = ∠ p₁ p₂ p₃ :=
(AffineIsometryEquiv.constVAdd ℝ P v).toAffineIsometry.angle_map _ _ _
#align euclidean_geometry.angle_const_vadd EuclideanGeometry.angle_const_vadd
@[simp]
theorem angle_vadd_const (v₁ v₂ v₃ : V) (p : P) : ∠ (v₁ +ᵥ p) (v₂ +ᵥ p) (v₃ +ᵥ p) = ∠ v₁ v₂ v₃ :=
(AffineIsometryEquiv.vaddConst ℝ p).toAffineIsometry.angle_map _ _ _
#align euclidean_geometry.angle_vadd_const EuclideanGeometry.angle_vadd_const
@[simp]
theorem angle_const_vsub (p p₁ p₂ p₃ : P) : ∠ (p -ᵥ p₁) (p -ᵥ p₂) (p -ᵥ p₃) = ∠ p₁ p₂ p₃ :=
(AffineIsometryEquiv.constVSub ℝ p).toAffineIsometry.angle_map _ _ _
#align euclidean_geometry.angle_const_vsub EuclideanGeometry.angle_const_vsub
@[simp]
theorem angle_vsub_const (p₁ p₂ p₃ p : P) : ∠ (p₁ -ᵥ p) (p₂ -ᵥ p) (p₃ -ᵥ p) = ∠ p₁ p₂ p₃ :=
(AffineIsometryEquiv.vaddConst ℝ p).symm.toAffineIsometry.angle_map _ _ _
#align euclidean_geometry.angle_vsub_const EuclideanGeometry.angle_vsub_const
@[simp]
theorem angle_add_const (v₁ v₂ v₃ : V) (v : V) : ∠ (v₁ + v) (v₂ + v) (v₃ + v) = ∠ v₁ v₂ v₃ :=
angle_vadd_const _ _ _ _
#align euclidean_geometry.angle_add_const EuclideanGeometry.angle_add_const
@[simp]
theorem angle_const_add (v : V) (v₁ v₂ v₃ : V) : ∠ (v + v₁) (v + v₂) (v + v₃) = ∠ v₁ v₂ v₃ :=
angle_const_vadd _ _ _ _
#align euclidean_geometry.angle_const_add EuclideanGeometry.angle_const_add
@[simp]
| Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 113 | 114 | theorem angle_sub_const (v₁ v₂ v₃ : V) (v : V) : ∠ (v₁ - v) (v₂ - v) (v₃ - v) = ∠ v₁ v₂ v₃ := by |
simpa only [vsub_eq_sub] using angle_vsub_const v₁ v₂ v₃ v
| 1 | 2.718282 | 0 | 0.4 | 5 | 403 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import geometry.euclidean.angle.unoriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open Real RealInnerProductSpace
namespace EuclideanGeometry
open InnerProductGeometry
variable {V P : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
[NormedAddTorsor V P] {p p₀ p₁ p₂ : P}
nonrec def angle (p1 p2 p3 : P) : ℝ :=
angle (p1 -ᵥ p2 : V) (p3 -ᵥ p2)
#align euclidean_geometry.angle EuclideanGeometry.angle
@[inherit_doc] scoped notation "∠" => EuclideanGeometry.angle
theorem continuousAt_angle {x : P × P × P} (hx12 : x.1 ≠ x.2.1) (hx32 : x.2.2 ≠ x.2.1) :
ContinuousAt (fun y : P × P × P => ∠ y.1 y.2.1 y.2.2) x := by
let f : P × P × P → V × V := fun y => (y.1 -ᵥ y.2.1, y.2.2 -ᵥ y.2.1)
have hf1 : (f x).1 ≠ 0 := by simp [hx12]
have hf2 : (f x).2 ≠ 0 := by simp [hx32]
exact (InnerProductGeometry.continuousAt_angle hf1 hf2).comp
((continuous_fst.vsub continuous_snd.fst).prod_mk
(continuous_snd.snd.vsub continuous_snd.fst)).continuousAt
#align euclidean_geometry.continuous_at_angle EuclideanGeometry.continuousAt_angle
@[simp]
theorem _root_.AffineIsometry.angle_map {V₂ P₂ : Type*} [NormedAddCommGroup V₂]
[InnerProductSpace ℝ V₂] [MetricSpace P₂] [NormedAddTorsor V₂ P₂]
(f : P →ᵃⁱ[ℝ] P₂) (p₁ p₂ p₃ : P) : ∠ (f p₁) (f p₂) (f p₃) = ∠ p₁ p₂ p₃ := by
simp_rw [angle, ← AffineIsometry.map_vsub, LinearIsometry.angle_map]
#align affine_isometry.angle_map AffineIsometry.angle_map
@[simp, norm_cast]
theorem _root_.AffineSubspace.angle_coe {s : AffineSubspace ℝ P} (p₁ p₂ p₃ : s) :
haveI : Nonempty s := ⟨p₁⟩
∠ (p₁ : P) (p₂ : P) (p₃ : P) = ∠ p₁ p₂ p₃ :=
haveI : Nonempty s := ⟨p₁⟩
s.subtypeₐᵢ.angle_map p₁ p₂ p₃
#align affine_subspace.angle_coe AffineSubspace.angle_coe
@[simp]
theorem angle_const_vadd (v : V) (p₁ p₂ p₃ : P) : ∠ (v +ᵥ p₁) (v +ᵥ p₂) (v +ᵥ p₃) = ∠ p₁ p₂ p₃ :=
(AffineIsometryEquiv.constVAdd ℝ P v).toAffineIsometry.angle_map _ _ _
#align euclidean_geometry.angle_const_vadd EuclideanGeometry.angle_const_vadd
@[simp]
theorem angle_vadd_const (v₁ v₂ v₃ : V) (p : P) : ∠ (v₁ +ᵥ p) (v₂ +ᵥ p) (v₃ +ᵥ p) = ∠ v₁ v₂ v₃ :=
(AffineIsometryEquiv.vaddConst ℝ p).toAffineIsometry.angle_map _ _ _
#align euclidean_geometry.angle_vadd_const EuclideanGeometry.angle_vadd_const
@[simp]
theorem angle_const_vsub (p p₁ p₂ p₃ : P) : ∠ (p -ᵥ p₁) (p -ᵥ p₂) (p -ᵥ p₃) = ∠ p₁ p₂ p₃ :=
(AffineIsometryEquiv.constVSub ℝ p).toAffineIsometry.angle_map _ _ _
#align euclidean_geometry.angle_const_vsub EuclideanGeometry.angle_const_vsub
@[simp]
theorem angle_vsub_const (p₁ p₂ p₃ p : P) : ∠ (p₁ -ᵥ p) (p₂ -ᵥ p) (p₃ -ᵥ p) = ∠ p₁ p₂ p₃ :=
(AffineIsometryEquiv.vaddConst ℝ p).symm.toAffineIsometry.angle_map _ _ _
#align euclidean_geometry.angle_vsub_const EuclideanGeometry.angle_vsub_const
@[simp]
theorem angle_add_const (v₁ v₂ v₃ : V) (v : V) : ∠ (v₁ + v) (v₂ + v) (v₃ + v) = ∠ v₁ v₂ v₃ :=
angle_vadd_const _ _ _ _
#align euclidean_geometry.angle_add_const EuclideanGeometry.angle_add_const
@[simp]
theorem angle_const_add (v : V) (v₁ v₂ v₃ : V) : ∠ (v + v₁) (v + v₂) (v + v₃) = ∠ v₁ v₂ v₃ :=
angle_const_vadd _ _ _ _
#align euclidean_geometry.angle_const_add EuclideanGeometry.angle_const_add
@[simp]
theorem angle_sub_const (v₁ v₂ v₃ : V) (v : V) : ∠ (v₁ - v) (v₂ - v) (v₃ - v) = ∠ v₁ v₂ v₃ := by
simpa only [vsub_eq_sub] using angle_vsub_const v₁ v₂ v₃ v
#align euclidean_geometry.angle_sub_const EuclideanGeometry.angle_sub_const
@[simp]
| Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 119 | 120 | theorem angle_const_sub (v : V) (v₁ v₂ v₃ : V) : ∠ (v - v₁) (v - v₂) (v - v₃) = ∠ v₁ v₂ v₃ := by |
simpa only [vsub_eq_sub] using angle_const_vsub v v₁ v₂ v₃
| 1 | 2.718282 | 0 | 0.4 | 5 | 403 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import geometry.euclidean.angle.unoriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open Real RealInnerProductSpace
namespace EuclideanGeometry
open InnerProductGeometry
variable {V P : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
[NormedAddTorsor V P] {p p₀ p₁ p₂ : P}
nonrec def angle (p1 p2 p3 : P) : ℝ :=
angle (p1 -ᵥ p2 : V) (p3 -ᵥ p2)
#align euclidean_geometry.angle EuclideanGeometry.angle
@[inherit_doc] scoped notation "∠" => EuclideanGeometry.angle
theorem continuousAt_angle {x : P × P × P} (hx12 : x.1 ≠ x.2.1) (hx32 : x.2.2 ≠ x.2.1) :
ContinuousAt (fun y : P × P × P => ∠ y.1 y.2.1 y.2.2) x := by
let f : P × P × P → V × V := fun y => (y.1 -ᵥ y.2.1, y.2.2 -ᵥ y.2.1)
have hf1 : (f x).1 ≠ 0 := by simp [hx12]
have hf2 : (f x).2 ≠ 0 := by simp [hx32]
exact (InnerProductGeometry.continuousAt_angle hf1 hf2).comp
((continuous_fst.vsub continuous_snd.fst).prod_mk
(continuous_snd.snd.vsub continuous_snd.fst)).continuousAt
#align euclidean_geometry.continuous_at_angle EuclideanGeometry.continuousAt_angle
@[simp]
theorem _root_.AffineIsometry.angle_map {V₂ P₂ : Type*} [NormedAddCommGroup V₂]
[InnerProductSpace ℝ V₂] [MetricSpace P₂] [NormedAddTorsor V₂ P₂]
(f : P →ᵃⁱ[ℝ] P₂) (p₁ p₂ p₃ : P) : ∠ (f p₁) (f p₂) (f p₃) = ∠ p₁ p₂ p₃ := by
simp_rw [angle, ← AffineIsometry.map_vsub, LinearIsometry.angle_map]
#align affine_isometry.angle_map AffineIsometry.angle_map
@[simp, norm_cast]
theorem _root_.AffineSubspace.angle_coe {s : AffineSubspace ℝ P} (p₁ p₂ p₃ : s) :
haveI : Nonempty s := ⟨p₁⟩
∠ (p₁ : P) (p₂ : P) (p₃ : P) = ∠ p₁ p₂ p₃ :=
haveI : Nonempty s := ⟨p₁⟩
s.subtypeₐᵢ.angle_map p₁ p₂ p₃
#align affine_subspace.angle_coe AffineSubspace.angle_coe
@[simp]
theorem angle_const_vadd (v : V) (p₁ p₂ p₃ : P) : ∠ (v +ᵥ p₁) (v +ᵥ p₂) (v +ᵥ p₃) = ∠ p₁ p₂ p₃ :=
(AffineIsometryEquiv.constVAdd ℝ P v).toAffineIsometry.angle_map _ _ _
#align euclidean_geometry.angle_const_vadd EuclideanGeometry.angle_const_vadd
@[simp]
theorem angle_vadd_const (v₁ v₂ v₃ : V) (p : P) : ∠ (v₁ +ᵥ p) (v₂ +ᵥ p) (v₃ +ᵥ p) = ∠ v₁ v₂ v₃ :=
(AffineIsometryEquiv.vaddConst ℝ p).toAffineIsometry.angle_map _ _ _
#align euclidean_geometry.angle_vadd_const EuclideanGeometry.angle_vadd_const
@[simp]
theorem angle_const_vsub (p p₁ p₂ p₃ : P) : ∠ (p -ᵥ p₁) (p -ᵥ p₂) (p -ᵥ p₃) = ∠ p₁ p₂ p₃ :=
(AffineIsometryEquiv.constVSub ℝ p).toAffineIsometry.angle_map _ _ _
#align euclidean_geometry.angle_const_vsub EuclideanGeometry.angle_const_vsub
@[simp]
theorem angle_vsub_const (p₁ p₂ p₃ p : P) : ∠ (p₁ -ᵥ p) (p₂ -ᵥ p) (p₃ -ᵥ p) = ∠ p₁ p₂ p₃ :=
(AffineIsometryEquiv.vaddConst ℝ p).symm.toAffineIsometry.angle_map _ _ _
#align euclidean_geometry.angle_vsub_const EuclideanGeometry.angle_vsub_const
@[simp]
theorem angle_add_const (v₁ v₂ v₃ : V) (v : V) : ∠ (v₁ + v) (v₂ + v) (v₃ + v) = ∠ v₁ v₂ v₃ :=
angle_vadd_const _ _ _ _
#align euclidean_geometry.angle_add_const EuclideanGeometry.angle_add_const
@[simp]
theorem angle_const_add (v : V) (v₁ v₂ v₃ : V) : ∠ (v + v₁) (v + v₂) (v + v₃) = ∠ v₁ v₂ v₃ :=
angle_const_vadd _ _ _ _
#align euclidean_geometry.angle_const_add EuclideanGeometry.angle_const_add
@[simp]
theorem angle_sub_const (v₁ v₂ v₃ : V) (v : V) : ∠ (v₁ - v) (v₂ - v) (v₃ - v) = ∠ v₁ v₂ v₃ := by
simpa only [vsub_eq_sub] using angle_vsub_const v₁ v₂ v₃ v
#align euclidean_geometry.angle_sub_const EuclideanGeometry.angle_sub_const
@[simp]
theorem angle_const_sub (v : V) (v₁ v₂ v₃ : V) : ∠ (v - v₁) (v - v₂) (v - v₃) = ∠ v₁ v₂ v₃ := by
simpa only [vsub_eq_sub] using angle_const_vsub v v₁ v₂ v₃
#align euclidean_geometry.angle_const_sub EuclideanGeometry.angle_const_sub
@[simp]
| Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 125 | 126 | theorem angle_neg (v₁ v₂ v₃ : V) : ∠ (-v₁) (-v₂) (-v₃) = ∠ v₁ v₂ v₃ := by |
simpa only [zero_sub] using angle_const_sub 0 v₁ v₂ v₃
| 1 | 2.718282 | 0 | 0.4 | 5 | 403 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
variable {K : Type u}
namespace RatFunc
section Field
variable [CommRing K]
protected irreducible_def zero : RatFunc K :=
⟨0⟩
#align ratfunc.zero RatFunc.zero
instance : Zero (RatFunc K) :=
⟨RatFunc.zero⟩
-- Porting note: added `OfNat.ofNat`. using `simp?` produces `simp only [zero_def]`
-- that does not close the goal
| Mathlib/FieldTheory/RatFunc/Basic.lean | 75 | 76 | theorem ofFractionRing_zero : (ofFractionRing 0 : RatFunc K) = 0 := by |
simp only [Zero.zero, OfNat.ofNat, RatFunc.zero]
| 1 | 2.718282 | 0 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
variable {K : Type u}
namespace RatFunc
section Field
variable [CommRing K]
protected irreducible_def zero : RatFunc K :=
⟨0⟩
#align ratfunc.zero RatFunc.zero
instance : Zero (RatFunc K) :=
⟨RatFunc.zero⟩
-- Porting note: added `OfNat.ofNat`. using `simp?` produces `simp only [zero_def]`
-- that does not close the goal
theorem ofFractionRing_zero : (ofFractionRing 0 : RatFunc K) = 0 := by
simp only [Zero.zero, OfNat.ofNat, RatFunc.zero]
#align ratfunc.of_fraction_ring_zero RatFunc.ofFractionRing_zero
protected irreducible_def add : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p + q⟩
#align ratfunc.add RatFunc.add
instance : Add (RatFunc K) :=
⟨RatFunc.add⟩
-- Porting note: added `HAdd.hAdd`. using `simp?` produces `simp only [add_def]`
-- that does not close the goal
| Mathlib/FieldTheory/RatFunc/Basic.lean | 89 | 91 | theorem ofFractionRing_add (p q : FractionRing K[X]) :
ofFractionRing (p + q) = ofFractionRing p + ofFractionRing q := by |
simp only [HAdd.hAdd, Add.add, RatFunc.add]
| 1 | 2.718282 | 0 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
variable {K : Type u}
namespace RatFunc
section Field
variable [CommRing K]
protected irreducible_def zero : RatFunc K :=
⟨0⟩
#align ratfunc.zero RatFunc.zero
instance : Zero (RatFunc K) :=
⟨RatFunc.zero⟩
-- Porting note: added `OfNat.ofNat`. using `simp?` produces `simp only [zero_def]`
-- that does not close the goal
theorem ofFractionRing_zero : (ofFractionRing 0 : RatFunc K) = 0 := by
simp only [Zero.zero, OfNat.ofNat, RatFunc.zero]
#align ratfunc.of_fraction_ring_zero RatFunc.ofFractionRing_zero
protected irreducible_def add : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p + q⟩
#align ratfunc.add RatFunc.add
instance : Add (RatFunc K) :=
⟨RatFunc.add⟩
-- Porting note: added `HAdd.hAdd`. using `simp?` produces `simp only [add_def]`
-- that does not close the goal
theorem ofFractionRing_add (p q : FractionRing K[X]) :
ofFractionRing (p + q) = ofFractionRing p + ofFractionRing q := by
simp only [HAdd.hAdd, Add.add, RatFunc.add]
#align ratfunc.of_fraction_ring_add RatFunc.ofFractionRing_add
protected irreducible_def sub : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p - q⟩
#align ratfunc.sub RatFunc.sub
instance : Sub (RatFunc K) :=
⟨RatFunc.sub⟩
-- Porting note: added `HSub.hSub`. using `simp?` produces `simp only [sub_def]`
-- that does not close the goal
| Mathlib/FieldTheory/RatFunc/Basic.lean | 104 | 106 | theorem ofFractionRing_sub (p q : FractionRing K[X]) :
ofFractionRing (p - q) = ofFractionRing p - ofFractionRing q := by |
simp only [Sub.sub, HSub.hSub, RatFunc.sub]
| 1 | 2.718282 | 0 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
variable {K : Type u}
namespace RatFunc
section Field
variable [CommRing K]
protected irreducible_def zero : RatFunc K :=
⟨0⟩
#align ratfunc.zero RatFunc.zero
instance : Zero (RatFunc K) :=
⟨RatFunc.zero⟩
-- Porting note: added `OfNat.ofNat`. using `simp?` produces `simp only [zero_def]`
-- that does not close the goal
theorem ofFractionRing_zero : (ofFractionRing 0 : RatFunc K) = 0 := by
simp only [Zero.zero, OfNat.ofNat, RatFunc.zero]
#align ratfunc.of_fraction_ring_zero RatFunc.ofFractionRing_zero
protected irreducible_def add : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p + q⟩
#align ratfunc.add RatFunc.add
instance : Add (RatFunc K) :=
⟨RatFunc.add⟩
-- Porting note: added `HAdd.hAdd`. using `simp?` produces `simp only [add_def]`
-- that does not close the goal
theorem ofFractionRing_add (p q : FractionRing K[X]) :
ofFractionRing (p + q) = ofFractionRing p + ofFractionRing q := by
simp only [HAdd.hAdd, Add.add, RatFunc.add]
#align ratfunc.of_fraction_ring_add RatFunc.ofFractionRing_add
protected irreducible_def sub : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p - q⟩
#align ratfunc.sub RatFunc.sub
instance : Sub (RatFunc K) :=
⟨RatFunc.sub⟩
-- Porting note: added `HSub.hSub`. using `simp?` produces `simp only [sub_def]`
-- that does not close the goal
theorem ofFractionRing_sub (p q : FractionRing K[X]) :
ofFractionRing (p - q) = ofFractionRing p - ofFractionRing q := by
simp only [Sub.sub, HSub.hSub, RatFunc.sub]
#align ratfunc.of_fraction_ring_sub RatFunc.ofFractionRing_sub
protected irreducible_def neg : RatFunc K → RatFunc K
| ⟨p⟩ => ⟨-p⟩
#align ratfunc.neg RatFunc.neg
instance : Neg (RatFunc K) :=
⟨RatFunc.neg⟩
| Mathlib/FieldTheory/RatFunc/Basic.lean | 117 | 118 | theorem ofFractionRing_neg (p : FractionRing K[X]) :
ofFractionRing (-p) = -ofFractionRing p := by | simp only [Neg.neg, RatFunc.neg]
| 1 | 2.718282 | 0 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
variable {K : Type u}
namespace RatFunc
section Field
variable [CommRing K]
protected irreducible_def zero : RatFunc K :=
⟨0⟩
#align ratfunc.zero RatFunc.zero
instance : Zero (RatFunc K) :=
⟨RatFunc.zero⟩
-- Porting note: added `OfNat.ofNat`. using `simp?` produces `simp only [zero_def]`
-- that does not close the goal
theorem ofFractionRing_zero : (ofFractionRing 0 : RatFunc K) = 0 := by
simp only [Zero.zero, OfNat.ofNat, RatFunc.zero]
#align ratfunc.of_fraction_ring_zero RatFunc.ofFractionRing_zero
protected irreducible_def add : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p + q⟩
#align ratfunc.add RatFunc.add
instance : Add (RatFunc K) :=
⟨RatFunc.add⟩
-- Porting note: added `HAdd.hAdd`. using `simp?` produces `simp only [add_def]`
-- that does not close the goal
theorem ofFractionRing_add (p q : FractionRing K[X]) :
ofFractionRing (p + q) = ofFractionRing p + ofFractionRing q := by
simp only [HAdd.hAdd, Add.add, RatFunc.add]
#align ratfunc.of_fraction_ring_add RatFunc.ofFractionRing_add
protected irreducible_def sub : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p - q⟩
#align ratfunc.sub RatFunc.sub
instance : Sub (RatFunc K) :=
⟨RatFunc.sub⟩
-- Porting note: added `HSub.hSub`. using `simp?` produces `simp only [sub_def]`
-- that does not close the goal
theorem ofFractionRing_sub (p q : FractionRing K[X]) :
ofFractionRing (p - q) = ofFractionRing p - ofFractionRing q := by
simp only [Sub.sub, HSub.hSub, RatFunc.sub]
#align ratfunc.of_fraction_ring_sub RatFunc.ofFractionRing_sub
protected irreducible_def neg : RatFunc K → RatFunc K
| ⟨p⟩ => ⟨-p⟩
#align ratfunc.neg RatFunc.neg
instance : Neg (RatFunc K) :=
⟨RatFunc.neg⟩
theorem ofFractionRing_neg (p : FractionRing K[X]) :
ofFractionRing (-p) = -ofFractionRing p := by simp only [Neg.neg, RatFunc.neg]
#align ratfunc.of_fraction_ring_neg RatFunc.ofFractionRing_neg
protected irreducible_def one : RatFunc K :=
⟨1⟩
#align ratfunc.one RatFunc.one
instance : One (RatFunc K) :=
⟨RatFunc.one⟩
-- Porting note: added `OfNat.ofNat`. using `simp?` produces `simp only [one_def]`
-- that does not close the goal
| Mathlib/FieldTheory/RatFunc/Basic.lean | 131 | 132 | theorem ofFractionRing_one : (ofFractionRing 1 : RatFunc K) = 1 := by |
simp only [One.one, OfNat.ofNat, RatFunc.one]
| 1 | 2.718282 | 0 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
variable {K : Type u}
namespace RatFunc
section Field
variable [CommRing K]
protected irreducible_def zero : RatFunc K :=
⟨0⟩
#align ratfunc.zero RatFunc.zero
instance : Zero (RatFunc K) :=
⟨RatFunc.zero⟩
-- Porting note: added `OfNat.ofNat`. using `simp?` produces `simp only [zero_def]`
-- that does not close the goal
theorem ofFractionRing_zero : (ofFractionRing 0 : RatFunc K) = 0 := by
simp only [Zero.zero, OfNat.ofNat, RatFunc.zero]
#align ratfunc.of_fraction_ring_zero RatFunc.ofFractionRing_zero
protected irreducible_def add : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p + q⟩
#align ratfunc.add RatFunc.add
instance : Add (RatFunc K) :=
⟨RatFunc.add⟩
-- Porting note: added `HAdd.hAdd`. using `simp?` produces `simp only [add_def]`
-- that does not close the goal
theorem ofFractionRing_add (p q : FractionRing K[X]) :
ofFractionRing (p + q) = ofFractionRing p + ofFractionRing q := by
simp only [HAdd.hAdd, Add.add, RatFunc.add]
#align ratfunc.of_fraction_ring_add RatFunc.ofFractionRing_add
protected irreducible_def sub : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p - q⟩
#align ratfunc.sub RatFunc.sub
instance : Sub (RatFunc K) :=
⟨RatFunc.sub⟩
-- Porting note: added `HSub.hSub`. using `simp?` produces `simp only [sub_def]`
-- that does not close the goal
theorem ofFractionRing_sub (p q : FractionRing K[X]) :
ofFractionRing (p - q) = ofFractionRing p - ofFractionRing q := by
simp only [Sub.sub, HSub.hSub, RatFunc.sub]
#align ratfunc.of_fraction_ring_sub RatFunc.ofFractionRing_sub
protected irreducible_def neg : RatFunc K → RatFunc K
| ⟨p⟩ => ⟨-p⟩
#align ratfunc.neg RatFunc.neg
instance : Neg (RatFunc K) :=
⟨RatFunc.neg⟩
theorem ofFractionRing_neg (p : FractionRing K[X]) :
ofFractionRing (-p) = -ofFractionRing p := by simp only [Neg.neg, RatFunc.neg]
#align ratfunc.of_fraction_ring_neg RatFunc.ofFractionRing_neg
protected irreducible_def one : RatFunc K :=
⟨1⟩
#align ratfunc.one RatFunc.one
instance : One (RatFunc K) :=
⟨RatFunc.one⟩
-- Porting note: added `OfNat.ofNat`. using `simp?` produces `simp only [one_def]`
-- that does not close the goal
theorem ofFractionRing_one : (ofFractionRing 1 : RatFunc K) = 1 := by
simp only [One.one, OfNat.ofNat, RatFunc.one]
#align ratfunc.of_fraction_ring_one RatFunc.ofFractionRing_one
protected irreducible_def mul : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p * q⟩
#align ratfunc.mul RatFunc.mul
instance : Mul (RatFunc K) :=
⟨RatFunc.mul⟩
-- Porting note: added `HMul.hMul`. using `simp?` produces `simp only [mul_def]`
-- that does not close the goal
| Mathlib/FieldTheory/RatFunc/Basic.lean | 145 | 147 | theorem ofFractionRing_mul (p q : FractionRing K[X]) :
ofFractionRing (p * q) = ofFractionRing p * ofFractionRing q := by |
simp only [Mul.mul, HMul.hMul, RatFunc.mul]
| 1 | 2.718282 | 0 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
variable {K : Type u}
namespace RatFunc
section Field
variable [CommRing K]
protected irreducible_def zero : RatFunc K :=
⟨0⟩
#align ratfunc.zero RatFunc.zero
instance : Zero (RatFunc K) :=
⟨RatFunc.zero⟩
-- Porting note: added `OfNat.ofNat`. using `simp?` produces `simp only [zero_def]`
-- that does not close the goal
theorem ofFractionRing_zero : (ofFractionRing 0 : RatFunc K) = 0 := by
simp only [Zero.zero, OfNat.ofNat, RatFunc.zero]
#align ratfunc.of_fraction_ring_zero RatFunc.ofFractionRing_zero
protected irreducible_def add : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p + q⟩
#align ratfunc.add RatFunc.add
instance : Add (RatFunc K) :=
⟨RatFunc.add⟩
-- Porting note: added `HAdd.hAdd`. using `simp?` produces `simp only [add_def]`
-- that does not close the goal
theorem ofFractionRing_add (p q : FractionRing K[X]) :
ofFractionRing (p + q) = ofFractionRing p + ofFractionRing q := by
simp only [HAdd.hAdd, Add.add, RatFunc.add]
#align ratfunc.of_fraction_ring_add RatFunc.ofFractionRing_add
protected irreducible_def sub : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p - q⟩
#align ratfunc.sub RatFunc.sub
instance : Sub (RatFunc K) :=
⟨RatFunc.sub⟩
-- Porting note: added `HSub.hSub`. using `simp?` produces `simp only [sub_def]`
-- that does not close the goal
theorem ofFractionRing_sub (p q : FractionRing K[X]) :
ofFractionRing (p - q) = ofFractionRing p - ofFractionRing q := by
simp only [Sub.sub, HSub.hSub, RatFunc.sub]
#align ratfunc.of_fraction_ring_sub RatFunc.ofFractionRing_sub
protected irreducible_def neg : RatFunc K → RatFunc K
| ⟨p⟩ => ⟨-p⟩
#align ratfunc.neg RatFunc.neg
instance : Neg (RatFunc K) :=
⟨RatFunc.neg⟩
theorem ofFractionRing_neg (p : FractionRing K[X]) :
ofFractionRing (-p) = -ofFractionRing p := by simp only [Neg.neg, RatFunc.neg]
#align ratfunc.of_fraction_ring_neg RatFunc.ofFractionRing_neg
protected irreducible_def one : RatFunc K :=
⟨1⟩
#align ratfunc.one RatFunc.one
instance : One (RatFunc K) :=
⟨RatFunc.one⟩
-- Porting note: added `OfNat.ofNat`. using `simp?` produces `simp only [one_def]`
-- that does not close the goal
theorem ofFractionRing_one : (ofFractionRing 1 : RatFunc K) = 1 := by
simp only [One.one, OfNat.ofNat, RatFunc.one]
#align ratfunc.of_fraction_ring_one RatFunc.ofFractionRing_one
protected irreducible_def mul : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p * q⟩
#align ratfunc.mul RatFunc.mul
instance : Mul (RatFunc K) :=
⟨RatFunc.mul⟩
-- Porting note: added `HMul.hMul`. using `simp?` produces `simp only [mul_def]`
-- that does not close the goal
theorem ofFractionRing_mul (p q : FractionRing K[X]) :
ofFractionRing (p * q) = ofFractionRing p * ofFractionRing q := by
simp only [Mul.mul, HMul.hMul, RatFunc.mul]
#align ratfunc.of_fraction_ring_mul RatFunc.ofFractionRing_mul
section IsDomain
variable [IsDomain K]
protected irreducible_def div : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p / q⟩
#align ratfunc.div RatFunc.div
instance : Div (RatFunc K) :=
⟨RatFunc.div⟩
-- Porting note: added `HDiv.hDiv`. using `simp?` produces `simp only [div_def]`
-- that does not close the goal
| Mathlib/FieldTheory/RatFunc/Basic.lean | 164 | 166 | theorem ofFractionRing_div (p q : FractionRing K[X]) :
ofFractionRing (p / q) = ofFractionRing p / ofFractionRing q := by |
simp only [Div.div, HDiv.hDiv, RatFunc.div]
| 1 | 2.718282 | 0 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
variable {K : Type u}
namespace RatFunc
section Field
variable [CommRing K]
protected irreducible_def zero : RatFunc K :=
⟨0⟩
#align ratfunc.zero RatFunc.zero
instance : Zero (RatFunc K) :=
⟨RatFunc.zero⟩
-- Porting note: added `OfNat.ofNat`. using `simp?` produces `simp only [zero_def]`
-- that does not close the goal
theorem ofFractionRing_zero : (ofFractionRing 0 : RatFunc K) = 0 := by
simp only [Zero.zero, OfNat.ofNat, RatFunc.zero]
#align ratfunc.of_fraction_ring_zero RatFunc.ofFractionRing_zero
protected irreducible_def add : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p + q⟩
#align ratfunc.add RatFunc.add
instance : Add (RatFunc K) :=
⟨RatFunc.add⟩
-- Porting note: added `HAdd.hAdd`. using `simp?` produces `simp only [add_def]`
-- that does not close the goal
theorem ofFractionRing_add (p q : FractionRing K[X]) :
ofFractionRing (p + q) = ofFractionRing p + ofFractionRing q := by
simp only [HAdd.hAdd, Add.add, RatFunc.add]
#align ratfunc.of_fraction_ring_add RatFunc.ofFractionRing_add
protected irreducible_def sub : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p - q⟩
#align ratfunc.sub RatFunc.sub
instance : Sub (RatFunc K) :=
⟨RatFunc.sub⟩
-- Porting note: added `HSub.hSub`. using `simp?` produces `simp only [sub_def]`
-- that does not close the goal
theorem ofFractionRing_sub (p q : FractionRing K[X]) :
ofFractionRing (p - q) = ofFractionRing p - ofFractionRing q := by
simp only [Sub.sub, HSub.hSub, RatFunc.sub]
#align ratfunc.of_fraction_ring_sub RatFunc.ofFractionRing_sub
protected irreducible_def neg : RatFunc K → RatFunc K
| ⟨p⟩ => ⟨-p⟩
#align ratfunc.neg RatFunc.neg
instance : Neg (RatFunc K) :=
⟨RatFunc.neg⟩
theorem ofFractionRing_neg (p : FractionRing K[X]) :
ofFractionRing (-p) = -ofFractionRing p := by simp only [Neg.neg, RatFunc.neg]
#align ratfunc.of_fraction_ring_neg RatFunc.ofFractionRing_neg
protected irreducible_def one : RatFunc K :=
⟨1⟩
#align ratfunc.one RatFunc.one
instance : One (RatFunc K) :=
⟨RatFunc.one⟩
-- Porting note: added `OfNat.ofNat`. using `simp?` produces `simp only [one_def]`
-- that does not close the goal
theorem ofFractionRing_one : (ofFractionRing 1 : RatFunc K) = 1 := by
simp only [One.one, OfNat.ofNat, RatFunc.one]
#align ratfunc.of_fraction_ring_one RatFunc.ofFractionRing_one
protected irreducible_def mul : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p * q⟩
#align ratfunc.mul RatFunc.mul
instance : Mul (RatFunc K) :=
⟨RatFunc.mul⟩
-- Porting note: added `HMul.hMul`. using `simp?` produces `simp only [mul_def]`
-- that does not close the goal
theorem ofFractionRing_mul (p q : FractionRing K[X]) :
ofFractionRing (p * q) = ofFractionRing p * ofFractionRing q := by
simp only [Mul.mul, HMul.hMul, RatFunc.mul]
#align ratfunc.of_fraction_ring_mul RatFunc.ofFractionRing_mul
section IsDomain
variable [IsDomain K]
protected irreducible_def div : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p / q⟩
#align ratfunc.div RatFunc.div
instance : Div (RatFunc K) :=
⟨RatFunc.div⟩
-- Porting note: added `HDiv.hDiv`. using `simp?` produces `simp only [div_def]`
-- that does not close the goal
theorem ofFractionRing_div (p q : FractionRing K[X]) :
ofFractionRing (p / q) = ofFractionRing p / ofFractionRing q := by
simp only [Div.div, HDiv.hDiv, RatFunc.div]
#align ratfunc.of_fraction_ring_div RatFunc.ofFractionRing_div
protected irreducible_def inv : RatFunc K → RatFunc K
| ⟨p⟩ => ⟨p⁻¹⟩
#align ratfunc.inv RatFunc.inv
instance : Inv (RatFunc K) :=
⟨RatFunc.inv⟩
| Mathlib/FieldTheory/RatFunc/Basic.lean | 177 | 179 | theorem ofFractionRing_inv (p : FractionRing K[X]) :
ofFractionRing p⁻¹ = (ofFractionRing p)⁻¹ := by |
simp only [Inv.inv, RatFunc.inv]
| 1 | 2.718282 | 0 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
variable {K : Type u}
namespace RatFunc
section Field
variable [CommRing K]
protected irreducible_def zero : RatFunc K :=
⟨0⟩
#align ratfunc.zero RatFunc.zero
instance : Zero (RatFunc K) :=
⟨RatFunc.zero⟩
-- Porting note: added `OfNat.ofNat`. using `simp?` produces `simp only [zero_def]`
-- that does not close the goal
theorem ofFractionRing_zero : (ofFractionRing 0 : RatFunc K) = 0 := by
simp only [Zero.zero, OfNat.ofNat, RatFunc.zero]
#align ratfunc.of_fraction_ring_zero RatFunc.ofFractionRing_zero
protected irreducible_def add : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p + q⟩
#align ratfunc.add RatFunc.add
instance : Add (RatFunc K) :=
⟨RatFunc.add⟩
-- Porting note: added `HAdd.hAdd`. using `simp?` produces `simp only [add_def]`
-- that does not close the goal
theorem ofFractionRing_add (p q : FractionRing K[X]) :
ofFractionRing (p + q) = ofFractionRing p + ofFractionRing q := by
simp only [HAdd.hAdd, Add.add, RatFunc.add]
#align ratfunc.of_fraction_ring_add RatFunc.ofFractionRing_add
protected irreducible_def sub : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p - q⟩
#align ratfunc.sub RatFunc.sub
instance : Sub (RatFunc K) :=
⟨RatFunc.sub⟩
-- Porting note: added `HSub.hSub`. using `simp?` produces `simp only [sub_def]`
-- that does not close the goal
theorem ofFractionRing_sub (p q : FractionRing K[X]) :
ofFractionRing (p - q) = ofFractionRing p - ofFractionRing q := by
simp only [Sub.sub, HSub.hSub, RatFunc.sub]
#align ratfunc.of_fraction_ring_sub RatFunc.ofFractionRing_sub
protected irreducible_def neg : RatFunc K → RatFunc K
| ⟨p⟩ => ⟨-p⟩
#align ratfunc.neg RatFunc.neg
instance : Neg (RatFunc K) :=
⟨RatFunc.neg⟩
theorem ofFractionRing_neg (p : FractionRing K[X]) :
ofFractionRing (-p) = -ofFractionRing p := by simp only [Neg.neg, RatFunc.neg]
#align ratfunc.of_fraction_ring_neg RatFunc.ofFractionRing_neg
protected irreducible_def one : RatFunc K :=
⟨1⟩
#align ratfunc.one RatFunc.one
instance : One (RatFunc K) :=
⟨RatFunc.one⟩
-- Porting note: added `OfNat.ofNat`. using `simp?` produces `simp only [one_def]`
-- that does not close the goal
theorem ofFractionRing_one : (ofFractionRing 1 : RatFunc K) = 1 := by
simp only [One.one, OfNat.ofNat, RatFunc.one]
#align ratfunc.of_fraction_ring_one RatFunc.ofFractionRing_one
protected irreducible_def mul : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p * q⟩
#align ratfunc.mul RatFunc.mul
instance : Mul (RatFunc K) :=
⟨RatFunc.mul⟩
-- Porting note: added `HMul.hMul`. using `simp?` produces `simp only [mul_def]`
-- that does not close the goal
theorem ofFractionRing_mul (p q : FractionRing K[X]) :
ofFractionRing (p * q) = ofFractionRing p * ofFractionRing q := by
simp only [Mul.mul, HMul.hMul, RatFunc.mul]
#align ratfunc.of_fraction_ring_mul RatFunc.ofFractionRing_mul
section SMul
variable {R : Type*}
protected irreducible_def smul [SMul R (FractionRing K[X])] : R → RatFunc K → RatFunc K
| r, ⟨p⟩ => ⟨r • p⟩
#align ratfunc.smul RatFunc.smul
-- cannot reproduce
--@[nolint fails_quickly] -- Porting note: `linter 'fails_quickly' not found`
instance [SMul R (FractionRing K[X])] : SMul R (RatFunc K) :=
⟨RatFunc.smul⟩
-- Porting note: added `SMul.hSMul`. using `simp?` produces `simp only [smul_def]`
-- that does not close the goal
| Mathlib/FieldTheory/RatFunc/Basic.lean | 209 | 211 | theorem ofFractionRing_smul [SMul R (FractionRing K[X])] (c : R) (p : FractionRing K[X]) :
ofFractionRing (c • p) = c • ofFractionRing p := by |
simp only [SMul.smul, HSMul.hSMul, RatFunc.smul]
| 1 | 2.718282 | 0 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
variable {K : Type u}
namespace RatFunc
section Field
variable [CommRing K]
protected irreducible_def zero : RatFunc K :=
⟨0⟩
#align ratfunc.zero RatFunc.zero
instance : Zero (RatFunc K) :=
⟨RatFunc.zero⟩
-- Porting note: added `OfNat.ofNat`. using `simp?` produces `simp only [zero_def]`
-- that does not close the goal
theorem ofFractionRing_zero : (ofFractionRing 0 : RatFunc K) = 0 := by
simp only [Zero.zero, OfNat.ofNat, RatFunc.zero]
#align ratfunc.of_fraction_ring_zero RatFunc.ofFractionRing_zero
protected irreducible_def add : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p + q⟩
#align ratfunc.add RatFunc.add
instance : Add (RatFunc K) :=
⟨RatFunc.add⟩
-- Porting note: added `HAdd.hAdd`. using `simp?` produces `simp only [add_def]`
-- that does not close the goal
theorem ofFractionRing_add (p q : FractionRing K[X]) :
ofFractionRing (p + q) = ofFractionRing p + ofFractionRing q := by
simp only [HAdd.hAdd, Add.add, RatFunc.add]
#align ratfunc.of_fraction_ring_add RatFunc.ofFractionRing_add
protected irreducible_def sub : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p - q⟩
#align ratfunc.sub RatFunc.sub
instance : Sub (RatFunc K) :=
⟨RatFunc.sub⟩
-- Porting note: added `HSub.hSub`. using `simp?` produces `simp only [sub_def]`
-- that does not close the goal
theorem ofFractionRing_sub (p q : FractionRing K[X]) :
ofFractionRing (p - q) = ofFractionRing p - ofFractionRing q := by
simp only [Sub.sub, HSub.hSub, RatFunc.sub]
#align ratfunc.of_fraction_ring_sub RatFunc.ofFractionRing_sub
protected irreducible_def neg : RatFunc K → RatFunc K
| ⟨p⟩ => ⟨-p⟩
#align ratfunc.neg RatFunc.neg
instance : Neg (RatFunc K) :=
⟨RatFunc.neg⟩
theorem ofFractionRing_neg (p : FractionRing K[X]) :
ofFractionRing (-p) = -ofFractionRing p := by simp only [Neg.neg, RatFunc.neg]
#align ratfunc.of_fraction_ring_neg RatFunc.ofFractionRing_neg
protected irreducible_def one : RatFunc K :=
⟨1⟩
#align ratfunc.one RatFunc.one
instance : One (RatFunc K) :=
⟨RatFunc.one⟩
-- Porting note: added `OfNat.ofNat`. using `simp?` produces `simp only [one_def]`
-- that does not close the goal
theorem ofFractionRing_one : (ofFractionRing 1 : RatFunc K) = 1 := by
simp only [One.one, OfNat.ofNat, RatFunc.one]
#align ratfunc.of_fraction_ring_one RatFunc.ofFractionRing_one
protected irreducible_def mul : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p * q⟩
#align ratfunc.mul RatFunc.mul
instance : Mul (RatFunc K) :=
⟨RatFunc.mul⟩
-- Porting note: added `HMul.hMul`. using `simp?` produces `simp only [mul_def]`
-- that does not close the goal
theorem ofFractionRing_mul (p q : FractionRing K[X]) :
ofFractionRing (p * q) = ofFractionRing p * ofFractionRing q := by
simp only [Mul.mul, HMul.hMul, RatFunc.mul]
#align ratfunc.of_fraction_ring_mul RatFunc.ofFractionRing_mul
section SMul
variable {R : Type*}
protected irreducible_def smul [SMul R (FractionRing K[X])] : R → RatFunc K → RatFunc K
| r, ⟨p⟩ => ⟨r • p⟩
#align ratfunc.smul RatFunc.smul
-- cannot reproduce
--@[nolint fails_quickly] -- Porting note: `linter 'fails_quickly' not found`
instance [SMul R (FractionRing K[X])] : SMul R (RatFunc K) :=
⟨RatFunc.smul⟩
-- Porting note: added `SMul.hSMul`. using `simp?` produces `simp only [smul_def]`
-- that does not close the goal
theorem ofFractionRing_smul [SMul R (FractionRing K[X])] (c : R) (p : FractionRing K[X]) :
ofFractionRing (c • p) = c • ofFractionRing p := by
simp only [SMul.smul, HSMul.hSMul, RatFunc.smul]
#align ratfunc.of_fraction_ring_smul RatFunc.ofFractionRing_smul
| Mathlib/FieldTheory/RatFunc/Basic.lean | 214 | 217 | theorem toFractionRing_smul [SMul R (FractionRing K[X])] (c : R) (p : RatFunc K) :
toFractionRing (c • p) = c • toFractionRing p := by |
cases p
rw [← ofFractionRing_smul]
| 2 | 7.389056 | 1 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
variable {K : Type u}
namespace RatFunc
section Field
variable [CommRing K]
protected irreducible_def zero : RatFunc K :=
⟨0⟩
#align ratfunc.zero RatFunc.zero
instance : Zero (RatFunc K) :=
⟨RatFunc.zero⟩
-- Porting note: added `OfNat.ofNat`. using `simp?` produces `simp only [zero_def]`
-- that does not close the goal
theorem ofFractionRing_zero : (ofFractionRing 0 : RatFunc K) = 0 := by
simp only [Zero.zero, OfNat.ofNat, RatFunc.zero]
#align ratfunc.of_fraction_ring_zero RatFunc.ofFractionRing_zero
protected irreducible_def add : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p + q⟩
#align ratfunc.add RatFunc.add
instance : Add (RatFunc K) :=
⟨RatFunc.add⟩
-- Porting note: added `HAdd.hAdd`. using `simp?` produces `simp only [add_def]`
-- that does not close the goal
theorem ofFractionRing_add (p q : FractionRing K[X]) :
ofFractionRing (p + q) = ofFractionRing p + ofFractionRing q := by
simp only [HAdd.hAdd, Add.add, RatFunc.add]
#align ratfunc.of_fraction_ring_add RatFunc.ofFractionRing_add
protected irreducible_def sub : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p - q⟩
#align ratfunc.sub RatFunc.sub
instance : Sub (RatFunc K) :=
⟨RatFunc.sub⟩
-- Porting note: added `HSub.hSub`. using `simp?` produces `simp only [sub_def]`
-- that does not close the goal
theorem ofFractionRing_sub (p q : FractionRing K[X]) :
ofFractionRing (p - q) = ofFractionRing p - ofFractionRing q := by
simp only [Sub.sub, HSub.hSub, RatFunc.sub]
#align ratfunc.of_fraction_ring_sub RatFunc.ofFractionRing_sub
protected irreducible_def neg : RatFunc K → RatFunc K
| ⟨p⟩ => ⟨-p⟩
#align ratfunc.neg RatFunc.neg
instance : Neg (RatFunc K) :=
⟨RatFunc.neg⟩
theorem ofFractionRing_neg (p : FractionRing K[X]) :
ofFractionRing (-p) = -ofFractionRing p := by simp only [Neg.neg, RatFunc.neg]
#align ratfunc.of_fraction_ring_neg RatFunc.ofFractionRing_neg
protected irreducible_def one : RatFunc K :=
⟨1⟩
#align ratfunc.one RatFunc.one
instance : One (RatFunc K) :=
⟨RatFunc.one⟩
-- Porting note: added `OfNat.ofNat`. using `simp?` produces `simp only [one_def]`
-- that does not close the goal
theorem ofFractionRing_one : (ofFractionRing 1 : RatFunc K) = 1 := by
simp only [One.one, OfNat.ofNat, RatFunc.one]
#align ratfunc.of_fraction_ring_one RatFunc.ofFractionRing_one
protected irreducible_def mul : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p * q⟩
#align ratfunc.mul RatFunc.mul
instance : Mul (RatFunc K) :=
⟨RatFunc.mul⟩
-- Porting note: added `HMul.hMul`. using `simp?` produces `simp only [mul_def]`
-- that does not close the goal
theorem ofFractionRing_mul (p q : FractionRing K[X]) :
ofFractionRing (p * q) = ofFractionRing p * ofFractionRing q := by
simp only [Mul.mul, HMul.hMul, RatFunc.mul]
#align ratfunc.of_fraction_ring_mul RatFunc.ofFractionRing_mul
section SMul
variable {R : Type*}
protected irreducible_def smul [SMul R (FractionRing K[X])] : R → RatFunc K → RatFunc K
| r, ⟨p⟩ => ⟨r • p⟩
#align ratfunc.smul RatFunc.smul
-- cannot reproduce
--@[nolint fails_quickly] -- Porting note: `linter 'fails_quickly' not found`
instance [SMul R (FractionRing K[X])] : SMul R (RatFunc K) :=
⟨RatFunc.smul⟩
-- Porting note: added `SMul.hSMul`. using `simp?` produces `simp only [smul_def]`
-- that does not close the goal
theorem ofFractionRing_smul [SMul R (FractionRing K[X])] (c : R) (p : FractionRing K[X]) :
ofFractionRing (c • p) = c • ofFractionRing p := by
simp only [SMul.smul, HSMul.hSMul, RatFunc.smul]
#align ratfunc.of_fraction_ring_smul RatFunc.ofFractionRing_smul
theorem toFractionRing_smul [SMul R (FractionRing K[X])] (c : R) (p : RatFunc K) :
toFractionRing (c • p) = c • toFractionRing p := by
cases p
rw [← ofFractionRing_smul]
#align ratfunc.to_fraction_ring_smul RatFunc.toFractionRing_smul
| Mathlib/FieldTheory/RatFunc/Basic.lean | 220 | 225 | theorem smul_eq_C_smul (x : RatFunc K) (r : K) : r • x = Polynomial.C r • x := by |
cases' x with x
-- Porting note: had to specify the induction principle manually
induction x using Localization.induction_on
rw [← ofFractionRing_smul, ← ofFractionRing_smul, Localization.smul_mk,
Localization.smul_mk, smul_eq_mul, Polynomial.smul_eq_C_mul]
| 5 | 148.413159 | 2 | 0.416667 | 12 | 404 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
variable {K : Type u}
namespace RatFunc
section Field
variable [CommRing K]
protected irreducible_def zero : RatFunc K :=
⟨0⟩
#align ratfunc.zero RatFunc.zero
instance : Zero (RatFunc K) :=
⟨RatFunc.zero⟩
-- Porting note: added `OfNat.ofNat`. using `simp?` produces `simp only [zero_def]`
-- that does not close the goal
theorem ofFractionRing_zero : (ofFractionRing 0 : RatFunc K) = 0 := by
simp only [Zero.zero, OfNat.ofNat, RatFunc.zero]
#align ratfunc.of_fraction_ring_zero RatFunc.ofFractionRing_zero
protected irreducible_def add : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p + q⟩
#align ratfunc.add RatFunc.add
instance : Add (RatFunc K) :=
⟨RatFunc.add⟩
-- Porting note: added `HAdd.hAdd`. using `simp?` produces `simp only [add_def]`
-- that does not close the goal
theorem ofFractionRing_add (p q : FractionRing K[X]) :
ofFractionRing (p + q) = ofFractionRing p + ofFractionRing q := by
simp only [HAdd.hAdd, Add.add, RatFunc.add]
#align ratfunc.of_fraction_ring_add RatFunc.ofFractionRing_add
protected irreducible_def sub : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p - q⟩
#align ratfunc.sub RatFunc.sub
instance : Sub (RatFunc K) :=
⟨RatFunc.sub⟩
-- Porting note: added `HSub.hSub`. using `simp?` produces `simp only [sub_def]`
-- that does not close the goal
theorem ofFractionRing_sub (p q : FractionRing K[X]) :
ofFractionRing (p - q) = ofFractionRing p - ofFractionRing q := by
simp only [Sub.sub, HSub.hSub, RatFunc.sub]
#align ratfunc.of_fraction_ring_sub RatFunc.ofFractionRing_sub
protected irreducible_def neg : RatFunc K → RatFunc K
| ⟨p⟩ => ⟨-p⟩
#align ratfunc.neg RatFunc.neg
instance : Neg (RatFunc K) :=
⟨RatFunc.neg⟩
theorem ofFractionRing_neg (p : FractionRing K[X]) :
ofFractionRing (-p) = -ofFractionRing p := by simp only [Neg.neg, RatFunc.neg]
#align ratfunc.of_fraction_ring_neg RatFunc.ofFractionRing_neg
protected irreducible_def one : RatFunc K :=
⟨1⟩
#align ratfunc.one RatFunc.one
instance : One (RatFunc K) :=
⟨RatFunc.one⟩
-- Porting note: added `OfNat.ofNat`. using `simp?` produces `simp only [one_def]`
-- that does not close the goal
theorem ofFractionRing_one : (ofFractionRing 1 : RatFunc K) = 1 := by
simp only [One.one, OfNat.ofNat, RatFunc.one]
#align ratfunc.of_fraction_ring_one RatFunc.ofFractionRing_one
protected irreducible_def mul : RatFunc K → RatFunc K → RatFunc K
| ⟨p⟩, ⟨q⟩ => ⟨p * q⟩
#align ratfunc.mul RatFunc.mul
instance : Mul (RatFunc K) :=
⟨RatFunc.mul⟩
-- Porting note: added `HMul.hMul`. using `simp?` produces `simp only [mul_def]`
-- that does not close the goal
theorem ofFractionRing_mul (p q : FractionRing K[X]) :
ofFractionRing (p * q) = ofFractionRing p * ofFractionRing q := by
simp only [Mul.mul, HMul.hMul, RatFunc.mul]
#align ratfunc.of_fraction_ring_mul RatFunc.ofFractionRing_mul
section SMul
variable {R : Type*}
protected irreducible_def smul [SMul R (FractionRing K[X])] : R → RatFunc K → RatFunc K
| r, ⟨p⟩ => ⟨r • p⟩
#align ratfunc.smul RatFunc.smul
-- cannot reproduce
--@[nolint fails_quickly] -- Porting note: `linter 'fails_quickly' not found`
instance [SMul R (FractionRing K[X])] : SMul R (RatFunc K) :=
⟨RatFunc.smul⟩
-- Porting note: added `SMul.hSMul`. using `simp?` produces `simp only [smul_def]`
-- that does not close the goal
theorem ofFractionRing_smul [SMul R (FractionRing K[X])] (c : R) (p : FractionRing K[X]) :
ofFractionRing (c • p) = c • ofFractionRing p := by
simp only [SMul.smul, HSMul.hSMul, RatFunc.smul]
#align ratfunc.of_fraction_ring_smul RatFunc.ofFractionRing_smul
theorem toFractionRing_smul [SMul R (FractionRing K[X])] (c : R) (p : RatFunc K) :
toFractionRing (c • p) = c • toFractionRing p := by
cases p
rw [← ofFractionRing_smul]
#align ratfunc.to_fraction_ring_smul RatFunc.toFractionRing_smul
theorem smul_eq_C_smul (x : RatFunc K) (r : K) : r • x = Polynomial.C r • x := by
cases' x with x
-- Porting note: had to specify the induction principle manually
induction x using Localization.induction_on
rw [← ofFractionRing_smul, ← ofFractionRing_smul, Localization.smul_mk,
Localization.smul_mk, smul_eq_mul, Polynomial.smul_eq_C_mul]
set_option linter.uppercaseLean3 false in
#align ratfunc.smul_eq_C_smul RatFunc.smul_eq_C_smul
section IsDomain
variable [IsDomain K]
variable [Monoid R] [DistribMulAction R K[X]]
variable [IsScalarTower R K[X] K[X]]
| Mathlib/FieldTheory/RatFunc/Basic.lean | 235 | 239 | theorem mk_smul (c : R) (p q : K[X]) : RatFunc.mk (c • p) q = c • RatFunc.mk p q := by |
by_cases hq : q = 0
· rw [hq, mk_zero, mk_zero, ← ofFractionRing_smul, smul_zero]
· rw [mk_eq_localization_mk _ hq, mk_eq_localization_mk _ hq, ← Localization.smul_mk, ←
ofFractionRing_smul]
| 4 | 54.59815 | 2 | 0.416667 | 12 | 404 |
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import data.list.chain from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
universe u v
open Nat
namespace List
variable {α : Type u} {β : Type v} {R r : α → α → Prop} {l l₁ l₂ : List α} {a b : α}
mk_iff_of_inductive_prop List.Chain List.chain_iff
#align list.chain_iff List.chain_iff
#align list.chain.nil List.Chain.nil
#align list.chain.cons List.Chain.cons
#align list.rel_of_chain_cons List.rel_of_chain_cons
#align list.chain_of_chain_cons List.chain_of_chain_cons
#align list.chain.imp' List.Chain.imp'
#align list.chain.imp List.Chain.imp
theorem Chain.iff {S : α → α → Prop} (H : ∀ a b, R a b ↔ S a b) {a : α} {l : List α} :
Chain R a l ↔ Chain S a l :=
⟨Chain.imp fun a b => (H a b).1, Chain.imp fun a b => (H a b).2⟩
#align list.chain.iff List.Chain.iff
theorem Chain.iff_mem {a : α} {l : List α} :
Chain R a l ↔ Chain (fun x y => x ∈ a :: l ∧ y ∈ l ∧ R x y) a l :=
⟨fun p => by
induction' p with _ a b l r _ IH <;> constructor <;>
[exact ⟨mem_cons_self _ _, mem_cons_self _ _, r⟩;
exact IH.imp fun a b ⟨am, bm, h⟩ => ⟨mem_cons_of_mem _ am, mem_cons_of_mem _ bm, h⟩],
Chain.imp fun a b h => h.2.2⟩
#align list.chain.iff_mem List.Chain.iff_mem
| Mathlib/Data/List/Chain.lean | 58 | 59 | theorem chain_singleton {a b : α} : Chain R a [b] ↔ R a b := by |
simp only [chain_cons, Chain.nil, and_true_iff]
| 1 | 2.718282 | 0 | 0.428571 | 7 | 405 |
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import data.list.chain from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
universe u v
open Nat
namespace List
variable {α : Type u} {β : Type v} {R r : α → α → Prop} {l l₁ l₂ : List α} {a b : α}
mk_iff_of_inductive_prop List.Chain List.chain_iff
#align list.chain_iff List.chain_iff
#align list.chain.nil List.Chain.nil
#align list.chain.cons List.Chain.cons
#align list.rel_of_chain_cons List.rel_of_chain_cons
#align list.chain_of_chain_cons List.chain_of_chain_cons
#align list.chain.imp' List.Chain.imp'
#align list.chain.imp List.Chain.imp
theorem Chain.iff {S : α → α → Prop} (H : ∀ a b, R a b ↔ S a b) {a : α} {l : List α} :
Chain R a l ↔ Chain S a l :=
⟨Chain.imp fun a b => (H a b).1, Chain.imp fun a b => (H a b).2⟩
#align list.chain.iff List.Chain.iff
theorem Chain.iff_mem {a : α} {l : List α} :
Chain R a l ↔ Chain (fun x y => x ∈ a :: l ∧ y ∈ l ∧ R x y) a l :=
⟨fun p => by
induction' p with _ a b l r _ IH <;> constructor <;>
[exact ⟨mem_cons_self _ _, mem_cons_self _ _, r⟩;
exact IH.imp fun a b ⟨am, bm, h⟩ => ⟨mem_cons_of_mem _ am, mem_cons_of_mem _ bm, h⟩],
Chain.imp fun a b h => h.2.2⟩
#align list.chain.iff_mem List.Chain.iff_mem
theorem chain_singleton {a b : α} : Chain R a [b] ↔ R a b := by
simp only [chain_cons, Chain.nil, and_true_iff]
#align list.chain_singleton List.chain_singleton
| Mathlib/Data/List/Chain.lean | 62 | 65 | theorem chain_split {a b : α} {l₁ l₂ : List α} :
Chain R a (l₁ ++ b :: l₂) ↔ Chain R a (l₁ ++ [b]) ∧ Chain R b l₂ := by |
induction' l₁ with x l₁ IH generalizing a <;>
simp only [*, nil_append, cons_append, Chain.nil, chain_cons, and_true_iff, and_assoc]
| 2 | 7.389056 | 1 | 0.428571 | 7 | 405 |
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import data.list.chain from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
universe u v
open Nat
namespace List
variable {α : Type u} {β : Type v} {R r : α → α → Prop} {l l₁ l₂ : List α} {a b : α}
mk_iff_of_inductive_prop List.Chain List.chain_iff
#align list.chain_iff List.chain_iff
#align list.chain.nil List.Chain.nil
#align list.chain.cons List.Chain.cons
#align list.rel_of_chain_cons List.rel_of_chain_cons
#align list.chain_of_chain_cons List.chain_of_chain_cons
#align list.chain.imp' List.Chain.imp'
#align list.chain.imp List.Chain.imp
theorem Chain.iff {S : α → α → Prop} (H : ∀ a b, R a b ↔ S a b) {a : α} {l : List α} :
Chain R a l ↔ Chain S a l :=
⟨Chain.imp fun a b => (H a b).1, Chain.imp fun a b => (H a b).2⟩
#align list.chain.iff List.Chain.iff
theorem Chain.iff_mem {a : α} {l : List α} :
Chain R a l ↔ Chain (fun x y => x ∈ a :: l ∧ y ∈ l ∧ R x y) a l :=
⟨fun p => by
induction' p with _ a b l r _ IH <;> constructor <;>
[exact ⟨mem_cons_self _ _, mem_cons_self _ _, r⟩;
exact IH.imp fun a b ⟨am, bm, h⟩ => ⟨mem_cons_of_mem _ am, mem_cons_of_mem _ bm, h⟩],
Chain.imp fun a b h => h.2.2⟩
#align list.chain.iff_mem List.Chain.iff_mem
theorem chain_singleton {a b : α} : Chain R a [b] ↔ R a b := by
simp only [chain_cons, Chain.nil, and_true_iff]
#align list.chain_singleton List.chain_singleton
theorem chain_split {a b : α} {l₁ l₂ : List α} :
Chain R a (l₁ ++ b :: l₂) ↔ Chain R a (l₁ ++ [b]) ∧ Chain R b l₂ := by
induction' l₁ with x l₁ IH generalizing a <;>
simp only [*, nil_append, cons_append, Chain.nil, chain_cons, and_true_iff, and_assoc]
#align list.chain_split List.chain_split
@[simp]
| Mathlib/Data/List/Chain.lean | 69 | 71 | theorem chain_append_cons_cons {a b c : α} {l₁ l₂ : List α} :
Chain R a (l₁ ++ b :: c :: l₂) ↔ Chain R a (l₁ ++ [b]) ∧ R b c ∧ Chain R c l₂ := by |
rw [chain_split, chain_cons]
| 1 | 2.718282 | 0 | 0.428571 | 7 | 405 |
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import data.list.chain from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
universe u v
open Nat
namespace List
variable {α : Type u} {β : Type v} {R r : α → α → Prop} {l l₁ l₂ : List α} {a b : α}
mk_iff_of_inductive_prop List.Chain List.chain_iff
#align list.chain_iff List.chain_iff
#align list.chain.nil List.Chain.nil
#align list.chain.cons List.Chain.cons
#align list.rel_of_chain_cons List.rel_of_chain_cons
#align list.chain_of_chain_cons List.chain_of_chain_cons
#align list.chain.imp' List.Chain.imp'
#align list.chain.imp List.Chain.imp
theorem Chain.iff {S : α → α → Prop} (H : ∀ a b, R a b ↔ S a b) {a : α} {l : List α} :
Chain R a l ↔ Chain S a l :=
⟨Chain.imp fun a b => (H a b).1, Chain.imp fun a b => (H a b).2⟩
#align list.chain.iff List.Chain.iff
theorem Chain.iff_mem {a : α} {l : List α} :
Chain R a l ↔ Chain (fun x y => x ∈ a :: l ∧ y ∈ l ∧ R x y) a l :=
⟨fun p => by
induction' p with _ a b l r _ IH <;> constructor <;>
[exact ⟨mem_cons_self _ _, mem_cons_self _ _, r⟩;
exact IH.imp fun a b ⟨am, bm, h⟩ => ⟨mem_cons_of_mem _ am, mem_cons_of_mem _ bm, h⟩],
Chain.imp fun a b h => h.2.2⟩
#align list.chain.iff_mem List.Chain.iff_mem
theorem chain_singleton {a b : α} : Chain R a [b] ↔ R a b := by
simp only [chain_cons, Chain.nil, and_true_iff]
#align list.chain_singleton List.chain_singleton
theorem chain_split {a b : α} {l₁ l₂ : List α} :
Chain R a (l₁ ++ b :: l₂) ↔ Chain R a (l₁ ++ [b]) ∧ Chain R b l₂ := by
induction' l₁ with x l₁ IH generalizing a <;>
simp only [*, nil_append, cons_append, Chain.nil, chain_cons, and_true_iff, and_assoc]
#align list.chain_split List.chain_split
@[simp]
theorem chain_append_cons_cons {a b c : α} {l₁ l₂ : List α} :
Chain R a (l₁ ++ b :: c :: l₂) ↔ Chain R a (l₁ ++ [b]) ∧ R b c ∧ Chain R c l₂ := by
rw [chain_split, chain_cons]
#align list.chain_append_cons_cons List.chain_append_cons_cons
theorem chain_iff_forall₂ :
∀ {a : α} {l : List α}, Chain R a l ↔ l = [] ∨ Forall₂ R (a :: dropLast l) l
| a, [] => by simp
| a, b :: l => by
by_cases h : l = [] <;>
simp [@chain_iff_forall₂ b l, dropLast, *]
#align list.chain_iff_forall₂ List.chain_iff_forall₂
| Mathlib/Data/List/Chain.lean | 82 | 83 | theorem chain_append_singleton_iff_forall₂ :
Chain R a (l ++ [b]) ↔ Forall₂ R (a :: l) (l ++ [b]) := by | simp [chain_iff_forall₂]
| 1 | 2.718282 | 0 | 0.428571 | 7 | 405 |
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import data.list.chain from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
universe u v
open Nat
namespace List
variable {α : Type u} {β : Type v} {R r : α → α → Prop} {l l₁ l₂ : List α} {a b : α}
mk_iff_of_inductive_prop List.Chain List.chain_iff
#align list.chain_iff List.chain_iff
#align list.chain.nil List.Chain.nil
#align list.chain.cons List.Chain.cons
#align list.rel_of_chain_cons List.rel_of_chain_cons
#align list.chain_of_chain_cons List.chain_of_chain_cons
#align list.chain.imp' List.Chain.imp'
#align list.chain.imp List.Chain.imp
theorem Chain.iff {S : α → α → Prop} (H : ∀ a b, R a b ↔ S a b) {a : α} {l : List α} :
Chain R a l ↔ Chain S a l :=
⟨Chain.imp fun a b => (H a b).1, Chain.imp fun a b => (H a b).2⟩
#align list.chain.iff List.Chain.iff
theorem Chain.iff_mem {a : α} {l : List α} :
Chain R a l ↔ Chain (fun x y => x ∈ a :: l ∧ y ∈ l ∧ R x y) a l :=
⟨fun p => by
induction' p with _ a b l r _ IH <;> constructor <;>
[exact ⟨mem_cons_self _ _, mem_cons_self _ _, r⟩;
exact IH.imp fun a b ⟨am, bm, h⟩ => ⟨mem_cons_of_mem _ am, mem_cons_of_mem _ bm, h⟩],
Chain.imp fun a b h => h.2.2⟩
#align list.chain.iff_mem List.Chain.iff_mem
theorem chain_singleton {a b : α} : Chain R a [b] ↔ R a b := by
simp only [chain_cons, Chain.nil, and_true_iff]
#align list.chain_singleton List.chain_singleton
theorem chain_split {a b : α} {l₁ l₂ : List α} :
Chain R a (l₁ ++ b :: l₂) ↔ Chain R a (l₁ ++ [b]) ∧ Chain R b l₂ := by
induction' l₁ with x l₁ IH generalizing a <;>
simp only [*, nil_append, cons_append, Chain.nil, chain_cons, and_true_iff, and_assoc]
#align list.chain_split List.chain_split
@[simp]
theorem chain_append_cons_cons {a b c : α} {l₁ l₂ : List α} :
Chain R a (l₁ ++ b :: c :: l₂) ↔ Chain R a (l₁ ++ [b]) ∧ R b c ∧ Chain R c l₂ := by
rw [chain_split, chain_cons]
#align list.chain_append_cons_cons List.chain_append_cons_cons
theorem chain_iff_forall₂ :
∀ {a : α} {l : List α}, Chain R a l ↔ l = [] ∨ Forall₂ R (a :: dropLast l) l
| a, [] => by simp
| a, b :: l => by
by_cases h : l = [] <;>
simp [@chain_iff_forall₂ b l, dropLast, *]
#align list.chain_iff_forall₂ List.chain_iff_forall₂
theorem chain_append_singleton_iff_forall₂ :
Chain R a (l ++ [b]) ↔ Forall₂ R (a :: l) (l ++ [b]) := by simp [chain_iff_forall₂]
#align list.chain_append_singleton_iff_forall₂ List.chain_append_singleton_iff_forall₂
| Mathlib/Data/List/Chain.lean | 86 | 88 | theorem chain_map (f : β → α) {b : β} {l : List β} :
Chain R (f b) (map f l) ↔ Chain (fun a b : β => R (f a) (f b)) b l := by |
induction l generalizing b <;> simp only [map, Chain.nil, chain_cons, *]
| 1 | 2.718282 | 0 | 0.428571 | 7 | 405 |
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import data.list.chain from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
universe u v
open Nat
namespace List
variable {α : Type u} {β : Type v} {R r : α → α → Prop} {l l₁ l₂ : List α} {a b : α}
mk_iff_of_inductive_prop List.Chain List.chain_iff
#align list.chain_iff List.chain_iff
#align list.chain.nil List.Chain.nil
#align list.chain.cons List.Chain.cons
#align list.rel_of_chain_cons List.rel_of_chain_cons
#align list.chain_of_chain_cons List.chain_of_chain_cons
#align list.chain.imp' List.Chain.imp'
#align list.chain.imp List.Chain.imp
theorem Chain.iff {S : α → α → Prop} (H : ∀ a b, R a b ↔ S a b) {a : α} {l : List α} :
Chain R a l ↔ Chain S a l :=
⟨Chain.imp fun a b => (H a b).1, Chain.imp fun a b => (H a b).2⟩
#align list.chain.iff List.Chain.iff
theorem Chain.iff_mem {a : α} {l : List α} :
Chain R a l ↔ Chain (fun x y => x ∈ a :: l ∧ y ∈ l ∧ R x y) a l :=
⟨fun p => by
induction' p with _ a b l r _ IH <;> constructor <;>
[exact ⟨mem_cons_self _ _, mem_cons_self _ _, r⟩;
exact IH.imp fun a b ⟨am, bm, h⟩ => ⟨mem_cons_of_mem _ am, mem_cons_of_mem _ bm, h⟩],
Chain.imp fun a b h => h.2.2⟩
#align list.chain.iff_mem List.Chain.iff_mem
theorem chain_singleton {a b : α} : Chain R a [b] ↔ R a b := by
simp only [chain_cons, Chain.nil, and_true_iff]
#align list.chain_singleton List.chain_singleton
theorem chain_split {a b : α} {l₁ l₂ : List α} :
Chain R a (l₁ ++ b :: l₂) ↔ Chain R a (l₁ ++ [b]) ∧ Chain R b l₂ := by
induction' l₁ with x l₁ IH generalizing a <;>
simp only [*, nil_append, cons_append, Chain.nil, chain_cons, and_true_iff, and_assoc]
#align list.chain_split List.chain_split
@[simp]
theorem chain_append_cons_cons {a b c : α} {l₁ l₂ : List α} :
Chain R a (l₁ ++ b :: c :: l₂) ↔ Chain R a (l₁ ++ [b]) ∧ R b c ∧ Chain R c l₂ := by
rw [chain_split, chain_cons]
#align list.chain_append_cons_cons List.chain_append_cons_cons
theorem chain_iff_forall₂ :
∀ {a : α} {l : List α}, Chain R a l ↔ l = [] ∨ Forall₂ R (a :: dropLast l) l
| a, [] => by simp
| a, b :: l => by
by_cases h : l = [] <;>
simp [@chain_iff_forall₂ b l, dropLast, *]
#align list.chain_iff_forall₂ List.chain_iff_forall₂
theorem chain_append_singleton_iff_forall₂ :
Chain R a (l ++ [b]) ↔ Forall₂ R (a :: l) (l ++ [b]) := by simp [chain_iff_forall₂]
#align list.chain_append_singleton_iff_forall₂ List.chain_append_singleton_iff_forall₂
theorem chain_map (f : β → α) {b : β} {l : List β} :
Chain R (f b) (map f l) ↔ Chain (fun a b : β => R (f a) (f b)) b l := by
induction l generalizing b <;> simp only [map, Chain.nil, chain_cons, *]
#align list.chain_map List.chain_map
theorem chain_of_chain_map {S : β → β → Prop} (f : α → β) (H : ∀ a b : α, S (f a) (f b) → R a b)
{a : α} {l : List α} (p : Chain S (f a) (map f l)) : Chain R a l :=
((chain_map f).1 p).imp H
#align list.chain_of_chain_map List.chain_of_chain_map
theorem chain_map_of_chain {S : β → β → Prop} (f : α → β) (H : ∀ a b : α, R a b → S (f a) (f b))
{a : α} {l : List α} (p : Chain R a l) : Chain S (f a) (map f l) :=
(chain_map f).2 <| p.imp H
#align list.chain_map_of_chain List.chain_map_of_chain
| Mathlib/Data/List/Chain.lean | 101 | 106 | theorem chain_pmap_of_chain {S : β → β → Prop} {p : α → Prop} {f : ∀ a, p a → β}
(H : ∀ a b ha hb, R a b → S (f a ha) (f b hb)) {a : α} {l : List α} (hl₁ : Chain R a l)
(ha : p a) (hl₂ : ∀ a ∈ l, p a) : Chain S (f a ha) (List.pmap f l hl₂) := by |
induction' l with lh lt l_ih generalizing a
· simp
· simp [H _ _ _ _ (rel_of_chain_cons hl₁), l_ih (chain_of_chain_cons hl₁)]
| 3 | 20.085537 | 1 | 0.428571 | 7 | 405 |
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import data.list.chain from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
universe u v
open Nat
namespace List
variable {α : Type u} {β : Type v} {R r : α → α → Prop} {l l₁ l₂ : List α} {a b : α}
mk_iff_of_inductive_prop List.Chain List.chain_iff
#align list.chain_iff List.chain_iff
#align list.chain.nil List.Chain.nil
#align list.chain.cons List.Chain.cons
#align list.rel_of_chain_cons List.rel_of_chain_cons
#align list.chain_of_chain_cons List.chain_of_chain_cons
#align list.chain.imp' List.Chain.imp'
#align list.chain.imp List.Chain.imp
theorem Chain.iff {S : α → α → Prop} (H : ∀ a b, R a b ↔ S a b) {a : α} {l : List α} :
Chain R a l ↔ Chain S a l :=
⟨Chain.imp fun a b => (H a b).1, Chain.imp fun a b => (H a b).2⟩
#align list.chain.iff List.Chain.iff
theorem Chain.iff_mem {a : α} {l : List α} :
Chain R a l ↔ Chain (fun x y => x ∈ a :: l ∧ y ∈ l ∧ R x y) a l :=
⟨fun p => by
induction' p with _ a b l r _ IH <;> constructor <;>
[exact ⟨mem_cons_self _ _, mem_cons_self _ _, r⟩;
exact IH.imp fun a b ⟨am, bm, h⟩ => ⟨mem_cons_of_mem _ am, mem_cons_of_mem _ bm, h⟩],
Chain.imp fun a b h => h.2.2⟩
#align list.chain.iff_mem List.Chain.iff_mem
theorem chain_singleton {a b : α} : Chain R a [b] ↔ R a b := by
simp only [chain_cons, Chain.nil, and_true_iff]
#align list.chain_singleton List.chain_singleton
theorem chain_split {a b : α} {l₁ l₂ : List α} :
Chain R a (l₁ ++ b :: l₂) ↔ Chain R a (l₁ ++ [b]) ∧ Chain R b l₂ := by
induction' l₁ with x l₁ IH generalizing a <;>
simp only [*, nil_append, cons_append, Chain.nil, chain_cons, and_true_iff, and_assoc]
#align list.chain_split List.chain_split
@[simp]
theorem chain_append_cons_cons {a b c : α} {l₁ l₂ : List α} :
Chain R a (l₁ ++ b :: c :: l₂) ↔ Chain R a (l₁ ++ [b]) ∧ R b c ∧ Chain R c l₂ := by
rw [chain_split, chain_cons]
#align list.chain_append_cons_cons List.chain_append_cons_cons
theorem chain_iff_forall₂ :
∀ {a : α} {l : List α}, Chain R a l ↔ l = [] ∨ Forall₂ R (a :: dropLast l) l
| a, [] => by simp
| a, b :: l => by
by_cases h : l = [] <;>
simp [@chain_iff_forall₂ b l, dropLast, *]
#align list.chain_iff_forall₂ List.chain_iff_forall₂
theorem chain_append_singleton_iff_forall₂ :
Chain R a (l ++ [b]) ↔ Forall₂ R (a :: l) (l ++ [b]) := by simp [chain_iff_forall₂]
#align list.chain_append_singleton_iff_forall₂ List.chain_append_singleton_iff_forall₂
theorem chain_map (f : β → α) {b : β} {l : List β} :
Chain R (f b) (map f l) ↔ Chain (fun a b : β => R (f a) (f b)) b l := by
induction l generalizing b <;> simp only [map, Chain.nil, chain_cons, *]
#align list.chain_map List.chain_map
theorem chain_of_chain_map {S : β → β → Prop} (f : α → β) (H : ∀ a b : α, S (f a) (f b) → R a b)
{a : α} {l : List α} (p : Chain S (f a) (map f l)) : Chain R a l :=
((chain_map f).1 p).imp H
#align list.chain_of_chain_map List.chain_of_chain_map
theorem chain_map_of_chain {S : β → β → Prop} (f : α → β) (H : ∀ a b : α, R a b → S (f a) (f b))
{a : α} {l : List α} (p : Chain R a l) : Chain S (f a) (map f l) :=
(chain_map f).2 <| p.imp H
#align list.chain_map_of_chain List.chain_map_of_chain
theorem chain_pmap_of_chain {S : β → β → Prop} {p : α → Prop} {f : ∀ a, p a → β}
(H : ∀ a b ha hb, R a b → S (f a ha) (f b hb)) {a : α} {l : List α} (hl₁ : Chain R a l)
(ha : p a) (hl₂ : ∀ a ∈ l, p a) : Chain S (f a ha) (List.pmap f l hl₂) := by
induction' l with lh lt l_ih generalizing a
· simp
· simp [H _ _ _ _ (rel_of_chain_cons hl₁), l_ih (chain_of_chain_cons hl₁)]
#align list.chain_pmap_of_chain List.chain_pmap_of_chain
| Mathlib/Data/List/Chain.lean | 109 | 114 | theorem chain_of_chain_pmap {S : β → β → Prop} {p : α → Prop} (f : ∀ a, p a → β) {l : List α}
(hl₁ : ∀ a ∈ l, p a) {a : α} (ha : p a) (hl₂ : Chain S (f a ha) (List.pmap f l hl₁))
(H : ∀ a b ha hb, S (f a ha) (f b hb) → R a b) : Chain R a l := by |
induction' l with lh lt l_ih generalizing a
· simp
· simp [H _ _ _ _ (rel_of_chain_cons hl₂), l_ih _ _ (chain_of_chain_cons hl₂)]
| 3 | 20.085537 | 1 | 0.428571 | 7 | 405 |
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