state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case mp
V : Type u
G : SimpleGraph V
n : ℕ
⊢ Partitionable G n → Colorable G n | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | rintro ⟨P, hf, hc⟩ | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n := by
constructor
· | Mathlib.Combinatorics.SimpleGraph.Partition.141_0.83yZsHNZsmSPOSw | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n | Mathlib_Combinatorics_SimpleGraph_Partition |
case mp.intro.intro
V : Type u
G : SimpleGraph V
n : ℕ
P : Partition G
hf : Set.Finite P.parts
hc : Finset.card (Set.Finite.toFinset hf) ≤ n
⊢ Colorable G n | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | have : Fintype P.parts := hf.fintype | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n := by
constructor
· rintro ⟨P, hf, hc⟩
| Mathlib.Combinatorics.SimpleGraph.Partition.141_0.83yZsHNZsmSPOSw | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n | Mathlib_Combinatorics_SimpleGraph_Partition |
case mp.intro.intro
V : Type u
G : SimpleGraph V
n : ℕ
P : Partition G
hf : Set.Finite P.parts
hc : Finset.card (Set.Finite.toFinset hf) ≤ n
this : Fintype ↑P.parts
⊢ Colorable G n | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | rw [Set.Finite.card_toFinset hf] at hc | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n := by
constructor
· rintro ⟨P, hf, hc⟩
have : Fintype P.parts := hf.fintype
| Mathlib.Combinatorics.SimpleGraph.Partition.141_0.83yZsHNZsmSPOSw | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n | Mathlib_Combinatorics_SimpleGraph_Partition |
case mp.intro.intro
V : Type u
G : SimpleGraph V
n : ℕ
P : Partition G
hf : Set.Finite P.parts
this : Fintype ↑P.parts
hc : Fintype.card ↑P.parts ≤ n
⊢ Colorable G n | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | apply P.to_colorable.mono hc | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n := by
constructor
· rintro ⟨P, hf, hc⟩
have : Fintype P.parts := hf.fintype
rw [Set.Finite.card_toFinset hf] at hc
| Mathlib.Combinatorics.SimpleGraph.Partition.141_0.83yZsHNZsmSPOSw | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n | Mathlib_Combinatorics_SimpleGraph_Partition |
case mpr
V : Type u
G : SimpleGraph V
n : ℕ
⊢ Colorable G n → Partitionable G n | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | rintro ⟨C⟩ | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n := by
constructor
· rintro ⟨P, hf, hc⟩
have : Fintype P.parts := hf.fintype
rw [Set.Finite.card_toFinset hf] at hc
apply P.to_colorable.mono hc
· | Mathlib.Combinatorics.SimpleGraph.Partition.141_0.83yZsHNZsmSPOSw | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n | Mathlib_Combinatorics_SimpleGraph_Partition |
case mpr.intro
V : Type u
G : SimpleGraph V
n : ℕ
C : Coloring G (Fin n)
⊢ Partitionable G n | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | refine' ⟨C.toPartition, C.colorClasses_finite, le_trans _ (Fintype.card_fin n).le⟩ | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n := by
constructor
· rintro ⟨P, hf, hc⟩
have : Fintype P.parts := hf.fintype
rw [Set.Finite.card_toFinset hf] at hc
apply P.to_colorable.mono hc
· rintro ⟨C⟩
| Mathlib.Combinatorics.SimpleGraph.Partition.141_0.83yZsHNZsmSPOSw | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n | Mathlib_Combinatorics_SimpleGraph_Partition |
case mpr.intro
V : Type u
G : SimpleGraph V
n : ℕ
C : Coloring G (Fin n)
⊢ Finset.card (Set.Finite.toFinset (_ : Set.Finite (Coloring.colorClasses C))) ≤ Fintype.card (Fin n) | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | generalize_proofs h | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n := by
constructor
· rintro ⟨P, hf, hc⟩
have : Fintype P.parts := hf.fintype
rw [Set.Finite.card_toFinset hf] at hc
apply P.to_colorable.mono hc
· rintro ⟨C⟩
refine' ⟨C.toPartition, C.colorClasses_finite, le_trans _ (Fi... | Mathlib.Combinatorics.SimpleGraph.Partition.141_0.83yZsHNZsmSPOSw | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n | Mathlib_Combinatorics_SimpleGraph_Partition |
case mpr.intro
V : Type u
G : SimpleGraph V
n : ℕ
C : Coloring G (Fin n)
h : Set.Finite (Coloring.colorClasses C)
⊢ Finset.card (Set.Finite.toFinset h) ≤ Fintype.card (Fin n) | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | haveI : Fintype C.colorClasses := C.colorClasses_finite.fintype | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n := by
constructor
· rintro ⟨P, hf, hc⟩
have : Fintype P.parts := hf.fintype
rw [Set.Finite.card_toFinset hf] at hc
apply P.to_colorable.mono hc
· rintro ⟨C⟩
refine' ⟨C.toPartition, C.colorClasses_finite, le_trans _ (Fi... | Mathlib.Combinatorics.SimpleGraph.Partition.141_0.83yZsHNZsmSPOSw | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n | Mathlib_Combinatorics_SimpleGraph_Partition |
case mpr.intro
V : Type u
G : SimpleGraph V
n : ℕ
C : Coloring G (Fin n)
h : Set.Finite (Coloring.colorClasses C)
this : Fintype ↑(Coloring.colorClasses C)
⊢ Finset.card (Set.Finite.toFinset h) ≤ Fintype.card (Fin n) | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | rw [h.card_toFinset] | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n := by
constructor
· rintro ⟨P, hf, hc⟩
have : Fintype P.parts := hf.fintype
rw [Set.Finite.card_toFinset hf] at hc
apply P.to_colorable.mono hc
· rintro ⟨C⟩
refine' ⟨C.toPartition, C.colorClasses_finite, le_trans _ (Fi... | Mathlib.Combinatorics.SimpleGraph.Partition.141_0.83yZsHNZsmSPOSw | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n | Mathlib_Combinatorics_SimpleGraph_Partition |
case mpr.intro
V : Type u
G : SimpleGraph V
n : ℕ
C : Coloring G (Fin n)
h : Set.Finite (Coloring.colorClasses C)
this : Fintype ↑(Coloring.colorClasses C)
⊢ Fintype.card ↑(Coloring.colorClasses C) ≤ Fintype.card (Fin n) | /-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c7... | exact C.card_colorClasses_le | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n := by
constructor
· rintro ⟨P, hf, hc⟩
have : Fintype P.parts := hf.fintype
rw [Set.Finite.card_toFinset hf] at hc
apply P.to_colorable.mono hc
· rintro ⟨C⟩
refine' ⟨C.toPartition, C.colorClasses_finite, le_trans _ (Fi... | Mathlib.Combinatorics.SimpleGraph.Partition.141_0.83yZsHNZsmSPOSw | theorem partitionable_iff_colorable {n : ℕ} : G.Partitionable n ↔ G.Colorable n | Mathlib_Combinatorics_SimpleGraph_Partition |
α : Type u_1
inst✝⁴ : Zero α
inst✝³ : TopologicalSpace α
inst✝² : PartialOrder α
inst✝¹ : DecidableRel fun x x_1 => x < x_1
inst✝ : OrderTopology α
a : α
h : 0 < a
⊢ ContinuousAt (⇑SignType.sign) a | /-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Sign
import Mathlib.Topology.Order.Basic
#align_import topology.instances.sign from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99... | refine' (continuousAt_const : ContinuousAt (fun _ => (1 : SignType)) a).congr _ | theorem continuousAt_sign_of_pos {a : α} (h : 0 < a) : ContinuousAt SignType.sign a := by
| Mathlib.Topology.Instances.Sign.32_0.LAngzpnhtVqlFPY | theorem continuousAt_sign_of_pos {a : α} (h : 0 < a) : ContinuousAt SignType.sign a | Mathlib_Topology_Instances_Sign |
α : Type u_1
inst✝⁴ : Zero α
inst✝³ : TopologicalSpace α
inst✝² : PartialOrder α
inst✝¹ : DecidableRel fun x x_1 => x < x_1
inst✝ : OrderTopology α
a : α
h : 0 < a
⊢ (fun x => 1) =ᶠ[nhds a] ⇑SignType.sign | /-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Sign
import Mathlib.Topology.Order.Basic
#align_import topology.instances.sign from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99... | rw [Filter.EventuallyEq, eventually_nhds_iff] | theorem continuousAt_sign_of_pos {a : α} (h : 0 < a) : ContinuousAt SignType.sign a := by
refine' (continuousAt_const : ContinuousAt (fun _ => (1 : SignType)) a).congr _
| Mathlib.Topology.Instances.Sign.32_0.LAngzpnhtVqlFPY | theorem continuousAt_sign_of_pos {a : α} (h : 0 < a) : ContinuousAt SignType.sign a | Mathlib_Topology_Instances_Sign |
α : Type u_1
inst✝⁴ : Zero α
inst✝³ : TopologicalSpace α
inst✝² : PartialOrder α
inst✝¹ : DecidableRel fun x x_1 => x < x_1
inst✝ : OrderTopology α
a : α
h : 0 < a
⊢ ∃ t, (∀ x ∈ t, 1 = SignType.sign x) ∧ IsOpen t ∧ a ∈ t | /-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Sign
import Mathlib.Topology.Order.Basic
#align_import topology.instances.sign from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99... | exact ⟨{ x | 0 < x }, fun x hx => (sign_pos hx).symm, isOpen_lt' 0, h⟩ | theorem continuousAt_sign_of_pos {a : α} (h : 0 < a) : ContinuousAt SignType.sign a := by
refine' (continuousAt_const : ContinuousAt (fun _ => (1 : SignType)) a).congr _
rw [Filter.EventuallyEq, eventually_nhds_iff]
| Mathlib.Topology.Instances.Sign.32_0.LAngzpnhtVqlFPY | theorem continuousAt_sign_of_pos {a : α} (h : 0 < a) : ContinuousAt SignType.sign a | Mathlib_Topology_Instances_Sign |
α : Type u_1
inst✝⁴ : Zero α
inst✝³ : TopologicalSpace α
inst✝² : PartialOrder α
inst✝¹ : DecidableRel fun x x_1 => x < x_1
inst✝ : OrderTopology α
a : α
h : a < 0
⊢ ContinuousAt (⇑SignType.sign) a | /-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Sign
import Mathlib.Topology.Order.Basic
#align_import topology.instances.sign from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99... | refine' (continuousAt_const : ContinuousAt (fun x => (-1 : SignType)) a).congr _ | theorem continuousAt_sign_of_neg {a : α} (h : a < 0) : ContinuousAt SignType.sign a := by
| Mathlib.Topology.Instances.Sign.38_0.LAngzpnhtVqlFPY | theorem continuousAt_sign_of_neg {a : α} (h : a < 0) : ContinuousAt SignType.sign a | Mathlib_Topology_Instances_Sign |
α : Type u_1
inst✝⁴ : Zero α
inst✝³ : TopologicalSpace α
inst✝² : PartialOrder α
inst✝¹ : DecidableRel fun x x_1 => x < x_1
inst✝ : OrderTopology α
a : α
h : a < 0
⊢ (fun x => -1) =ᶠ[nhds a] ⇑SignType.sign | /-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Sign
import Mathlib.Topology.Order.Basic
#align_import topology.instances.sign from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99... | rw [Filter.EventuallyEq, eventually_nhds_iff] | theorem continuousAt_sign_of_neg {a : α} (h : a < 0) : ContinuousAt SignType.sign a := by
refine' (continuousAt_const : ContinuousAt (fun x => (-1 : SignType)) a).congr _
| Mathlib.Topology.Instances.Sign.38_0.LAngzpnhtVqlFPY | theorem continuousAt_sign_of_neg {a : α} (h : a < 0) : ContinuousAt SignType.sign a | Mathlib_Topology_Instances_Sign |
α : Type u_1
inst✝⁴ : Zero α
inst✝³ : TopologicalSpace α
inst✝² : PartialOrder α
inst✝¹ : DecidableRel fun x x_1 => x < x_1
inst✝ : OrderTopology α
a : α
h : a < 0
⊢ ∃ t, (∀ x ∈ t, -1 = SignType.sign x) ∧ IsOpen t ∧ a ∈ t | /-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Sign
import Mathlib.Topology.Order.Basic
#align_import topology.instances.sign from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99... | exact ⟨{ x | x < 0 }, fun x hx => (sign_neg hx).symm, isOpen_gt' 0, h⟩ | theorem continuousAt_sign_of_neg {a : α} (h : a < 0) : ContinuousAt SignType.sign a := by
refine' (continuousAt_const : ContinuousAt (fun x => (-1 : SignType)) a).congr _
rw [Filter.EventuallyEq, eventually_nhds_iff]
| Mathlib.Topology.Instances.Sign.38_0.LAngzpnhtVqlFPY | theorem continuousAt_sign_of_neg {a : α} (h : a < 0) : ContinuousAt SignType.sign a | Mathlib_Topology_Instances_Sign |
α : Type u_1
inst✝³ : Zero α
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a : α
h : a ≠ 0
⊢ ContinuousAt (⇑SignType.sign) a | /-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Sign
import Mathlib.Topology.Order.Basic
#align_import topology.instances.sign from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99... | rcases h.lt_or_lt with (h_neg | h_pos) | theorem continuousAt_sign_of_ne_zero {a : α} (h : a ≠ 0) : ContinuousAt SignType.sign a := by
| Mathlib.Topology.Instances.Sign.50_0.LAngzpnhtVqlFPY | theorem continuousAt_sign_of_ne_zero {a : α} (h : a ≠ 0) : ContinuousAt SignType.sign a | Mathlib_Topology_Instances_Sign |
case inl
α : Type u_1
inst✝³ : Zero α
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a : α
h : a ≠ 0
h_neg : a < 0
⊢ ContinuousAt (⇑SignType.sign) a | /-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Sign
import Mathlib.Topology.Order.Basic
#align_import topology.instances.sign from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99... | exact continuousAt_sign_of_neg h_neg | theorem continuousAt_sign_of_ne_zero {a : α} (h : a ≠ 0) : ContinuousAt SignType.sign a := by
rcases h.lt_or_lt with (h_neg | h_pos)
· | Mathlib.Topology.Instances.Sign.50_0.LAngzpnhtVqlFPY | theorem continuousAt_sign_of_ne_zero {a : α} (h : a ≠ 0) : ContinuousAt SignType.sign a | Mathlib_Topology_Instances_Sign |
case inr
α : Type u_1
inst✝³ : Zero α
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
a : α
h : a ≠ 0
h_pos : 0 < a
⊢ ContinuousAt (⇑SignType.sign) a | /-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Sign
import Mathlib.Topology.Order.Basic
#align_import topology.instances.sign from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99... | exact continuousAt_sign_of_pos h_pos | theorem continuousAt_sign_of_ne_zero {a : α} (h : a ≠ 0) : ContinuousAt SignType.sign a := by
rcases h.lt_or_lt with (h_neg | h_pos)
· exact continuousAt_sign_of_neg h_neg
· | Mathlib.Topology.Instances.Sign.50_0.LAngzpnhtVqlFPY | theorem continuousAt_sign_of_ne_zero {a : α} (h : a ≠ 0) : ContinuousAt SignType.sign a | Mathlib_Topology_Instances_Sign |
α : Type u_1
a b c d x y z : α
inst✝⁶ : LinearOrderedCommMonoidWithZero α
β : Type u_2
inst✝⁵ : Zero β
inst✝⁴ : One β
inst✝³ : Mul β
inst✝² : Pow β ℕ
inst✝¹ : Sup β
inst✝ : Inf β
f : β → α
hf : Injective f
zero : f 0 = 0
one : f 1 = 1
mul : ∀ (x y : β), f (x * y) = f x * f y
npow : ∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ ... | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | simp only [zero, one, LinearOrderedCommMonoidWithZero.zero_le_one] | /-- Pullback a `LinearOrderedCommMonoidWithZero` under an injective map.
See note [reducible non-instances]. -/
@[reducible]
def Function.Injective.linearOrderedCommMonoidWithZero {β : Type*} [Zero β] [One β] [Mul β]
[Pow β ℕ] [Sup β] [Inf β] (f : β → α) (hf : Function.Injective f) (zero : f 0 = 0)
(one : f 1 =... | Mathlib.Algebra.Order.WithZero.78_0.E09o7sC17CChiNO | /-- Pullback a `LinearOrderedCommMonoidWithZero` under an injective map.
See note [reducible non-instances]. -/
@[reducible]
def Function.Injective.linearOrderedCommMonoidWithZero {β : Type*} [Zero β] [One β] [Mul β]
[Pow β ℕ] [Sup β] [Inf β] (f : β → α) (hf : Function.Injective f) (zero : f 0 = 0)
(one : f 1 =... | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommMonoidWithZero α
⊢ 0 ≤ a | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | simpa only [mul_zero, mul_one] using mul_le_mul_left' zero_le_one a | @[simp]
theorem zero_le' : 0 ≤ a := by | Mathlib.Algebra.Order.WithZero.92_0.E09o7sC17CChiNO | @[simp]
theorem zero_le' : 0 ≤ a | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
h : c ≠ 0
hab : a * c ≤ b * c
⊢ a ≤ b | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | simpa only [mul_inv_cancel_right₀ h] using mul_le_mul_right' hab c⁻¹ | theorem le_of_le_mul_right (h : c ≠ 0) (hab : a * c ≤ b * c) : a ≤ b := by
| Mathlib.Algebra.Order.WithZero.136_0.E09o7sC17CChiNO | theorem le_of_le_mul_right (h : c ≠ 0) (hab : a * c ≤ b * c) : a ≤ b | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
h : c ≠ 0
hab : a * c ≤ b
⊢ a * c ≤ b * c⁻¹ * c | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | simpa [h] using hab | theorem le_mul_inv_of_mul_le (h : c ≠ 0) (hab : a * c ≤ b) : a ≤ b * c⁻¹ :=
le_of_le_mul_right h (by | Mathlib.Algebra.Order.WithZero.140_0.E09o7sC17CChiNO | theorem le_mul_inv_of_mul_le (h : c ≠ 0) (hab : a * c ≤ b) : a ≤ b * c⁻¹ | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
hab : a ≤ b * c
⊢ a * c⁻¹ ≤ b | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | by_cases h : c = 0 | theorem mul_inv_le_of_le_mul (hab : a ≤ b * c) : a * c⁻¹ ≤ b := by
| Mathlib.Algebra.Order.WithZero.144_0.E09o7sC17CChiNO | theorem mul_inv_le_of_le_mul (hab : a ≤ b * c) : a * c⁻¹ ≤ b | Mathlib_Algebra_Order_WithZero |
case pos
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
hab : a ≤ b * c
h : c = 0
⊢ a * c⁻¹ ≤ b | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | simp [h] | theorem mul_inv_le_of_le_mul (hab : a ≤ b * c) : a * c⁻¹ ≤ b := by
by_cases h : c = 0
· | Mathlib.Algebra.Order.WithZero.144_0.E09o7sC17CChiNO | theorem mul_inv_le_of_le_mul (hab : a ≤ b * c) : a * c⁻¹ ≤ b | Mathlib_Algebra_Order_WithZero |
case neg
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
hab : a ≤ b * c
h : ¬c = 0
⊢ a * c⁻¹ ≤ b | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | exact le_of_le_mul_right h (by simpa [h] using hab) | theorem mul_inv_le_of_le_mul (hab : a ≤ b * c) : a * c⁻¹ ≤ b := by
by_cases h : c = 0
· simp [h]
· | Mathlib.Algebra.Order.WithZero.144_0.E09o7sC17CChiNO | theorem mul_inv_le_of_le_mul (hab : a ≤ b * c) : a * c⁻¹ ≤ b | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
hab : a ≤ b * c
h : ¬c = 0
⊢ a * c⁻¹ * c ≤ b * c | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | simpa [h] using hab | theorem mul_inv_le_of_le_mul (hab : a ≤ b * c) : a * c⁻¹ ≤ b := by
by_cases h : c = 0
· simp [h]
· exact le_of_le_mul_right h (by | Mathlib.Algebra.Order.WithZero.144_0.E09o7sC17CChiNO | theorem mul_inv_le_of_le_mul (hab : a ≤ b * c) : a * c⁻¹ ≤ b | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a✝ b✝ c✝ d✝ x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
a b c d : α
hb : b ≠ 0
hd : d ≠ 0
⊢ a * b⁻¹ ≤ c * d⁻¹ ↔ a * d ≤ c * b | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | rw [mul_inv_le_iff₀ hb, mul_right_comm, le_mul_inv_iff₀ hd] | theorem div_le_div₀ (a b c d : α) (hb : b ≠ 0) (hd : d ≠ 0) : a * b⁻¹ ≤ c * d⁻¹ ↔ a * d ≤ c * b :=
by | Mathlib.Algebra.Order.WithZero.166_0.E09o7sC17CChiNO | theorem div_le_div₀ (a b c d : α) (hb : b ≠ 0) (hd : d ≠ 0) : a * b⁻¹ ≤ c * d⁻¹ ↔ a * d ≤ c * b | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
hab : a ≤ b
hb : b ≠ 0
hcd : c < d
hd : d ≠ 0
ha : a = 0
⊢ a * c < b * d | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | rw [ha, zero_mul, zero_lt_iff] | theorem mul_lt_mul_of_lt_of_le₀ (hab : a ≤ b) (hb : b ≠ 0) (hcd : c < d) : a * c < b * d :=
have hd : d ≠ 0 := ne_zero_of_lt hcd
if ha : a = 0 then by
| Mathlib.Algebra.Order.WithZero.175_0.E09o7sC17CChiNO | theorem mul_lt_mul_of_lt_of_le₀ (hab : a ≤ b) (hb : b ≠ 0) (hcd : c < d) : a * c < b * d | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
hab : a ≤ b
hb : b ≠ 0
hcd : c < d
hd : d ≠ 0
ha : a = 0
⊢ b * d ≠ 0 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | exact mul_ne_zero hb hd | theorem mul_lt_mul_of_lt_of_le₀ (hab : a ≤ b) (hb : b ≠ 0) (hcd : c < d) : a * c < b * d :=
have hd : d ≠ 0 := ne_zero_of_lt hcd
if ha : a = 0 then by
rw [ha, zero_mul, zero_lt_iff]
| Mathlib.Algebra.Order.WithZero.175_0.E09o7sC17CChiNO | theorem mul_lt_mul_of_lt_of_le₀ (hab : a ≤ b) (hb : b ≠ 0) (hcd : c < d) : a * c < b * d | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
hab : a ≤ b
hb : b ≠ 0
hcd : c < d
hd : d ≠ 0
ha : ¬a = 0
hc : c = 0
⊢ a * c < b * d | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | rw [hc, mul_zero, zero_lt_iff] | theorem mul_lt_mul_of_lt_of_le₀ (hab : a ≤ b) (hb : b ≠ 0) (hcd : c < d) : a * c < b * d :=
have hd : d ≠ 0 := ne_zero_of_lt hcd
if ha : a = 0 then by
rw [ha, zero_mul, zero_lt_iff]
exact mul_ne_zero hb hd
else
if hc : c = 0 then by
| Mathlib.Algebra.Order.WithZero.175_0.E09o7sC17CChiNO | theorem mul_lt_mul_of_lt_of_le₀ (hab : a ≤ b) (hb : b ≠ 0) (hcd : c < d) : a * c < b * d | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
hab : a ≤ b
hb : b ≠ 0
hcd : c < d
hd : d ≠ 0
ha : ¬a = 0
hc : c = 0
⊢ b * d ≠ 0 | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | exact mul_ne_zero hb hd | theorem mul_lt_mul_of_lt_of_le₀ (hab : a ≤ b) (hb : b ≠ 0) (hcd : c < d) : a * c < b * d :=
have hd : d ≠ 0 := ne_zero_of_lt hcd
if ha : a = 0 then by
rw [ha, zero_mul, zero_lt_iff]
exact mul_ne_zero hb hd
else
if hc : c = 0 then by
rw [hc, mul_zero, zero_lt_iff]
| Mathlib.Algebra.Order.WithZero.175_0.E09o7sC17CChiNO | theorem mul_lt_mul_of_lt_of_le₀ (hab : a ≤ b) (hb : b ≠ 0) (hcd : c < d) : a * c < b * d | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
h : x < y * z
⊢ x * z⁻¹ < y | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | contrapose! h | theorem mul_inv_lt_of_lt_mul₀ (h : x < y * z) : x * z⁻¹ < y := by
| Mathlib.Algebra.Order.WithZero.193_0.E09o7sC17CChiNO | theorem mul_inv_lt_of_lt_mul₀ (h : x < y * z) : x * z⁻¹ < y | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
h : y ≤ x * z⁻¹
⊢ y * z ≤ x | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | simpa only [inv_inv] using mul_inv_le_of_le_mul h | theorem mul_inv_lt_of_lt_mul₀ (h : x < y * z) : x * z⁻¹ < y := by
contrapose! h
| Mathlib.Algebra.Order.WithZero.193_0.E09o7sC17CChiNO | theorem mul_inv_lt_of_lt_mul₀ (h : x < y * z) : x * z⁻¹ < y | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
h : x < y * z
⊢ y⁻¹ * x < z | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | rw [mul_comm] at * | theorem inv_mul_lt_of_lt_mul₀ (h : x < y * z) : y⁻¹ * x < z := by
| Mathlib.Algebra.Order.WithZero.198_0.E09o7sC17CChiNO | theorem inv_mul_lt_of_lt_mul₀ (h : x < y * z) : y⁻¹ * x < z | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
h : x < z * y
⊢ x * y⁻¹ < z | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | exact mul_inv_lt_of_lt_mul₀ h | theorem inv_mul_lt_of_lt_mul₀ (h : x < y * z) : y⁻¹ * x < z := by
rw [mul_comm] at *
| Mathlib.Algebra.Order.WithZero.198_0.E09o7sC17CChiNO | theorem inv_mul_lt_of_lt_mul₀ (h : x < y * z) : y⁻¹ * x < z | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c✝ d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
c : α
h : a < b
hc : c ≠ 0
⊢ a * c < b * c | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | contrapose! h | theorem mul_lt_right₀ (c : α) (h : a < b) (hc : c ≠ 0) : a * c < b * c := by
| Mathlib.Algebra.Order.WithZero.203_0.E09o7sC17CChiNO | theorem mul_lt_right₀ (c : α) (h : a < b) (hc : c ≠ 0) : a * c < b * c | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c✝ d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
c : α
hc : c ≠ 0
h : b * c ≤ a * c
⊢ b ≤ a | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | exact le_of_le_mul_right hc h | theorem mul_lt_right₀ (c : α) (h : a < b) (hc : c ≠ 0) : a * c < b * c := by
contrapose! h
| Mathlib.Algebra.Order.WithZero.203_0.E09o7sC17CChiNO | theorem mul_lt_right₀ (c : α) (h : a < b) (hc : c ≠ 0) : a * c < b * c | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
h : a * b < c * d
hc : 0 < c
hh : c ≤ a
⊢ b < d | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | have ha : a ≠ 0 := ne_of_gt (lt_of_lt_of_le hc hh) | theorem lt_of_mul_lt_mul_of_le₀ (h : a * b < c * d) (hc : 0 < c) (hh : c ≤ a) : b < d := by
| Mathlib.Algebra.Order.WithZero.220_0.E09o7sC17CChiNO | theorem lt_of_mul_lt_mul_of_le₀ (h : a * b < c * d) (hc : 0 < c) (hh : c ≤ a) : b < d | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
h : a * b < c * d
hc : 0 < c
hh : c ≤ a
ha : a ≠ 0
⊢ b < d | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | simp_rw [← inv_le_inv₀ ha (ne_of_gt hc)] at hh | theorem lt_of_mul_lt_mul_of_le₀ (h : a * b < c * d) (hc : 0 < c) (hh : c ≤ a) : b < d := by
have ha : a ≠ 0 := ne_of_gt (lt_of_lt_of_le hc hh)
| Mathlib.Algebra.Order.WithZero.220_0.E09o7sC17CChiNO | theorem lt_of_mul_lt_mul_of_le₀ (h : a * b < c * d) (hc : 0 < c) (hh : c ≤ a) : b < d | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
h : a * b < c * d
hc : 0 < c
ha : a ≠ 0
hh : a⁻¹ ≤ c⁻¹
⊢ b < d | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | have := mul_lt_mul_of_lt_of_le₀ hh (inv_ne_zero (ne_of_gt hc)) h | theorem lt_of_mul_lt_mul_of_le₀ (h : a * b < c * d) (hc : 0 < c) (hh : c ≤ a) : b < d := by
have ha : a ≠ 0 := ne_of_gt (lt_of_lt_of_le hc hh)
simp_rw [← inv_le_inv₀ ha (ne_of_gt hc)] at hh
| Mathlib.Algebra.Order.WithZero.220_0.E09o7sC17CChiNO | theorem lt_of_mul_lt_mul_of_le₀ (h : a * b < c * d) (hc : 0 < c) (hh : c ≤ a) : b < d | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
h : a * b < c * d
hc : 0 < c
ha : a ≠ 0
hh : a⁻¹ ≤ c⁻¹
this : a⁻¹ * (a * b) < c⁻¹ * (c * d)
⊢ b < d | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | simpa [inv_mul_cancel_left₀ ha, inv_mul_cancel_left₀ (ne_of_gt hc)] using this | theorem lt_of_mul_lt_mul_of_le₀ (h : a * b < c * d) (hc : 0 < c) (hh : c ≤ a) : b < d := by
have ha : a ≠ 0 := ne_of_gt (lt_of_lt_of_le hc hh)
simp_rw [← inv_le_inv₀ ha (ne_of_gt hc)] at hh
have := mul_lt_mul_of_lt_of_le₀ hh (inv_ne_zero (ne_of_gt hc)) h
| Mathlib.Algebra.Order.WithZero.220_0.E09o7sC17CChiNO | theorem lt_of_mul_lt_mul_of_le₀ (h : a * b < c * d) (hc : 0 < c) (hh : c ≤ a) : b < d | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
ha : a ≠ 0
⊢ a * b ≤ a * c ↔ b ≤ c | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | simp only [mul_comm a] | theorem mul_le_mul_left₀ (ha : a ≠ 0) : a * b ≤ a * c ↔ b ≤ c := by
| Mathlib.Algebra.Order.WithZero.231_0.E09o7sC17CChiNO | theorem mul_le_mul_left₀ (ha : a ≠ 0) : a * b ≤ a * c ↔ b ≤ c | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
ha : a ≠ 0
⊢ b * a ≤ c * a ↔ b ≤ c | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | exact mul_le_mul_right₀ ha | theorem mul_le_mul_left₀ (ha : a ≠ 0) : a * b ≤ a * c ↔ b ≤ c := by
simp only [mul_comm a]
| Mathlib.Algebra.Order.WithZero.231_0.E09o7sC17CChiNO | theorem mul_le_mul_left₀ (ha : a ≠ 0) : a * b ≤ a * c ↔ b ≤ c | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
hc : c ≠ 0
⊢ a / c ≤ b / c ↔ a ≤ b | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | rw [div_eq_mul_inv, div_eq_mul_inv, mul_le_mul_right₀ (inv_ne_zero hc)] | theorem div_le_div_right₀ (hc : c ≠ 0) : a / c ≤ b / c ↔ a ≤ b := by
| Mathlib.Algebra.Order.WithZero.236_0.E09o7sC17CChiNO | theorem div_le_div_right₀ (hc : c ≠ 0) : a / c ≤ b / c ↔ a ≤ b | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
⊢ a / b ≤ a / c ↔ c ≤ b | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | simp only [div_eq_mul_inv, mul_le_mul_left₀ ha, inv_le_inv₀ hb hc] | theorem div_le_div_left₀ (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0) : a / b ≤ a / c ↔ c ≤ b := by
| Mathlib.Algebra.Order.WithZero.240_0.E09o7sC17CChiNO | theorem div_le_div_left₀ (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0) : a / b ≤ a / c ↔ c ≤ b | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
hc : c ≠ 0
⊢ a ≤ b / c ↔ a * c ≤ b | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | rw [div_eq_mul_inv, le_mul_inv_iff₀ hc] | theorem le_div_iff₀ (hc : c ≠ 0) : a ≤ b / c ↔ a * c ≤ b := by
| Mathlib.Algebra.Order.WithZero.244_0.E09o7sC17CChiNO | theorem le_div_iff₀ (hc : c ≠ 0) : a ≤ b / c ↔ a * c ≤ b | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
hc : c ≠ 0
⊢ a / c ≤ b ↔ a ≤ b * c | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | rw [div_eq_mul_inv, mul_inv_le_iff₀ hc] | theorem div_le_iff₀ (hc : c ≠ 0) : a / c ≤ b ↔ a ≤ b * c := by
| Mathlib.Algebra.Order.WithZero.248_0.E09o7sC17CChiNO | theorem div_le_iff₀ (hc : c ≠ 0) : a / c ≤ b ↔ a ≤ b * c | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a✝ b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
a : α
ha : a ≠ 0
⊢ symm (mulLeft₀' ha) = mulLeft₀' (_ : a⁻¹ ≠ 0) | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | ext | theorem OrderIso.mulLeft₀'_symm {a : α} (ha : a ≠ 0) :
(OrderIso.mulLeft₀' ha).symm = OrderIso.mulLeft₀' (inv_ne_zero ha) := by
| Mathlib.Algebra.Order.WithZero.262_0.E09o7sC17CChiNO | theorem OrderIso.mulLeft₀'_symm {a : α} (ha : a ≠ 0) :
(OrderIso.mulLeft₀' ha).symm = OrderIso.mulLeft₀' (inv_ne_zero ha) | Mathlib_Algebra_Order_WithZero |
case h.h
α : Type u_1
a✝ b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
a : α
ha : a ≠ 0
x✝ : α
⊢ (symm (mulLeft₀' ha)) x✝ = (mulLeft₀' (_ : a⁻¹ ≠ 0)) x✝ | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | rfl | theorem OrderIso.mulLeft₀'_symm {a : α} (ha : a ≠ 0) :
(OrderIso.mulLeft₀' ha).symm = OrderIso.mulLeft₀' (inv_ne_zero ha) := by
ext
| Mathlib.Algebra.Order.WithZero.262_0.E09o7sC17CChiNO | theorem OrderIso.mulLeft₀'_symm {a : α} (ha : a ≠ 0) :
(OrderIso.mulLeft₀' ha).symm = OrderIso.mulLeft₀' (inv_ne_zero ha) | Mathlib_Algebra_Order_WithZero |
α : Type u_1
a✝ b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
a : α
ha : a ≠ 0
⊢ symm (mulRight₀' ha) = mulRight₀' (_ : a⁻¹ ≠ 0) | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | ext | theorem OrderIso.mulRight₀'_symm {a : α} (ha : a ≠ 0) :
(OrderIso.mulRight₀' ha).symm = OrderIso.mulRight₀' (inv_ne_zero ha) := by
| Mathlib.Algebra.Order.WithZero.278_0.E09o7sC17CChiNO | theorem OrderIso.mulRight₀'_symm {a : α} (ha : a ≠ 0) :
(OrderIso.mulRight₀' ha).symm = OrderIso.mulRight₀' (inv_ne_zero ha) | Mathlib_Algebra_Order_WithZero |
case h.h
α : Type u_1
a✝ b c d x y z : α
inst✝ : LinearOrderedCommGroupWithZero α
a : α
ha : a ≠ 0
x✝ : α
⊢ (symm (mulRight₀' ha)) x✝ = (mulRight₀' (_ : a⁻¹ ≠ 0)) x✝ | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Group.Units
import Ma... | rfl | theorem OrderIso.mulRight₀'_symm {a : α} (ha : a ≠ 0) :
(OrderIso.mulRight₀' ha).symm = OrderIso.mulRight₀' (inv_ne_zero ha) := by
ext
| Mathlib.Algebra.Order.WithZero.278_0.E09o7sC17CChiNO | theorem OrderIso.mulRight₀'_symm {a : α} (ha : a ≠ 0) :
(OrderIso.mulRight₀' ha).symm = OrderIso.mulRight₀' (inv_ne_zero ha) | Mathlib_Algebra_Order_WithZero |
m : ℝ
⊢ gaussianPdfReal m 0 = 0 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | ext1 x | @[simp]
lemma gaussianPdfReal_zero_var (m : ℝ) : gaussianPdfReal m 0 = 0 := by
| Mathlib.Probability.Distributions.Gaussian.50_0.7VInOP4QlYS5vmc | @[simp]
lemma gaussianPdfReal_zero_var (m : ℝ) : gaussianPdfReal m 0 = 0 | Mathlib_Probability_Distributions_Gaussian |
case h
m x : ℝ
⊢ gaussianPdfReal m 0 x = OfNat.ofNat 0 x | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp [gaussianPdfReal] | @[simp]
lemma gaussianPdfReal_zero_var (m : ℝ) : gaussianPdfReal m 0 = 0 := by
ext1 x
| Mathlib.Probability.Distributions.Gaussian.50_0.7VInOP4QlYS5vmc | @[simp]
lemma gaussianPdfReal_zero_var (m : ℝ) : gaussianPdfReal m 0 = 0 | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
x : ℝ
hv : v ≠ 0
⊢ 0 < gaussianPdfReal μ v x | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [gaussianPdfReal] | /-- The gaussian pdf is positive when the variance is not zero. -/
lemma gaussianPdfReal_pos (μ : ℝ) (v : ℝ≥0) (x : ℝ) (hv : v ≠ 0) : 0 < gaussianPdfReal μ v x := by
| Mathlib.Probability.Distributions.Gaussian.55_0.7VInOP4QlYS5vmc | /-- The gaussian pdf is positive when the variance is not zero. -/
lemma gaussianPdfReal_pos (μ : ℝ) (v : ℝ≥0) (x : ℝ) (hv : v ≠ 0) : 0 < gaussianPdfReal μ v x | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
x : ℝ
hv : v ≠ 0
⊢ 0 < (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v)) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | positivity | /-- The gaussian pdf is positive when the variance is not zero. -/
lemma gaussianPdfReal_pos (μ : ℝ) (v : ℝ≥0) (x : ℝ) (hv : v ≠ 0) : 0 < gaussianPdfReal μ v x := by
rw [gaussianPdfReal]
| Mathlib.Probability.Distributions.Gaussian.55_0.7VInOP4QlYS5vmc | /-- The gaussian pdf is positive when the variance is not zero. -/
lemma gaussianPdfReal_pos (μ : ℝ) (v : ℝ≥0) (x : ℝ) (hv : v ≠ 0) : 0 < gaussianPdfReal μ v x | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
x : ℝ
⊢ 0 ≤ gaussianPdfReal μ v x | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [gaussianPdfReal] | /--The gaussian pdf is nonnegative. -/
lemma gaussianPdfReal_nonneg (μ : ℝ) (v : ℝ≥0) (x : ℝ) : 0 ≤ gaussianPdfReal μ v x := by
| Mathlib.Probability.Distributions.Gaussian.60_0.7VInOP4QlYS5vmc | /--The gaussian pdf is nonnegative. -/
lemma gaussianPdfReal_nonneg (μ : ℝ) (v : ℝ≥0) (x : ℝ) : 0 ≤ gaussianPdfReal μ v x | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
x : ℝ
⊢ 0 ≤ (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v)) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | positivity | /--The gaussian pdf is nonnegative. -/
lemma gaussianPdfReal_nonneg (μ : ℝ) (v : ℝ≥0) (x : ℝ) : 0 ≤ gaussianPdfReal μ v x := by
rw [gaussianPdfReal]
| Mathlib.Probability.Distributions.Gaussian.60_0.7VInOP4QlYS5vmc | /--The gaussian pdf is nonnegative. -/
lemma gaussianPdfReal_nonneg (μ : ℝ) (v : ℝ≥0) (x : ℝ) : 0 ≤ gaussianPdfReal μ v x | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
⊢ Integrable (gaussianPdfReal μ v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [gaussianPdfReal_def] | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) := by
| Mathlib.Probability.Distributions.Gaussian.74_0.7VInOP4QlYS5vmc | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
⊢ Integrable fun x => (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v)) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | by_cases hv : v = 0 | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) := by
rw [gaussianPdfReal_def]
| Mathlib.Probability.Distributions.Gaussian.74_0.7VInOP4QlYS5vmc | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) | Mathlib_Probability_Distributions_Gaussian |
case pos
μ : ℝ
v : ℝ≥0
hv : v = 0
⊢ Integrable fun x => (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v)) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp [hv] | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) := by
rw [gaussianPdfReal_def]
by_cases hv : v = 0
· | Mathlib.Probability.Distributions.Gaussian.74_0.7VInOP4QlYS5vmc | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) | Mathlib_Probability_Distributions_Gaussian |
case neg
μ : ℝ
v : ℝ≥0
hv : ¬v = 0
⊢ Integrable fun x => (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v)) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | let g : ℝ → ℝ := fun x ↦ (Real.sqrt (2 * π * v))⁻¹ * rexp (- x ^ 2 / (2 * v)) | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) := by
rw [gaussianPdfReal_def]
by_cases hv : v = 0
· simp [hv]
| Mathlib.Probability.Distributions.Gaussian.74_0.7VInOP4QlYS5vmc | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) | Mathlib_Probability_Distributions_Gaussian |
case neg
μ : ℝ
v : ℝ≥0
hv : ¬v = 0
g : ℝ → ℝ := fun x => (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-x ^ 2 / (2 * ↑v))
⊢ Integrable fun x => (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v)) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | have hg : Integrable g := by
suffices g = fun x ↦ (Real.sqrt (2 * π * v))⁻¹ * rexp (- (2 * v)⁻¹ * x ^ 2) by
rw [this]
refine (integrable_exp_neg_mul_sq ?_).const_mul (Real.sqrt (2 * π * v))⁻¹
simp [lt_of_le_of_ne (zero_le _) (Ne.symm hv)]
ext x
simp only [gt_iff_lt, zero_lt_two, zero_le_mu... | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) := by
rw [gaussianPdfReal_def]
by_cases hv : v = 0
· simp [hv]
let g : ℝ → ℝ := fun x ↦ (Real.sqrt (2 * π * v))⁻¹ * rexp (- x ^ 2 / (2 * v))
| Mathlib.Probability.Distributions.Gaussian.74_0.7VInOP4QlYS5vmc | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
hv : ¬v = 0
g : ℝ → ℝ := fun x => (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-x ^ 2 / (2 * ↑v))
⊢ Integrable g | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | suffices g = fun x ↦ (Real.sqrt (2 * π * v))⁻¹ * rexp (- (2 * v)⁻¹ * x ^ 2) by
rw [this]
refine (integrable_exp_neg_mul_sq ?_).const_mul (Real.sqrt (2 * π * v))⁻¹
simp [lt_of_le_of_ne (zero_le _) (Ne.symm hv)] | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) := by
rw [gaussianPdfReal_def]
by_cases hv : v = 0
· simp [hv]
let g : ℝ → ℝ := fun x ↦ (Real.sqrt (2 * π * v))⁻¹ * rexp (- x ^ 2 / (2 * v))
have hg : Integrable g := by
| Mathlib.Probability.Distributions.Gaussian.74_0.7VInOP4QlYS5vmc | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
hv : ¬v = 0
g : ℝ → ℝ := fun x => (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-x ^ 2 / (2 * ↑v))
this : g = fun x => (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-↑(2 * v)⁻¹ * x ^ 2)
⊢ Integrable g | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [this] | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) := by
rw [gaussianPdfReal_def]
by_cases hv : v = 0
· simp [hv]
let g : ℝ → ℝ := fun x ↦ (Real.sqrt (2 * π * v))⁻¹ * rexp (- x ^ 2 / (2 * v))
have hg : Integrable g := by
suffices g = fun x ↦ (Real.sqrt (2 * π * v))⁻¹... | Mathlib.Probability.Distributions.Gaussian.74_0.7VInOP4QlYS5vmc | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
hv : ¬v = 0
g : ℝ → ℝ := fun x => (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-x ^ 2 / (2 * ↑v))
this : g = fun x => (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-↑(2 * v)⁻¹ * x ^ 2)
⊢ Integrable fun x => (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-↑(2 * v)⁻¹ * x ^ 2) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | refine (integrable_exp_neg_mul_sq ?_).const_mul (Real.sqrt (2 * π * v))⁻¹ | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) := by
rw [gaussianPdfReal_def]
by_cases hv : v = 0
· simp [hv]
let g : ℝ → ℝ := fun x ↦ (Real.sqrt (2 * π * v))⁻¹ * rexp (- x ^ 2 / (2 * v))
have hg : Integrable g := by
suffices g = fun x ↦ (Real.sqrt (2 * π * v))⁻¹... | Mathlib.Probability.Distributions.Gaussian.74_0.7VInOP4QlYS5vmc | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
hv : ¬v = 0
g : ℝ → ℝ := fun x => (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-x ^ 2 / (2 * ↑v))
this : g = fun x => (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-↑(2 * v)⁻¹ * x ^ 2)
⊢ 0 < ↑(2 * v)⁻¹ | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp [lt_of_le_of_ne (zero_le _) (Ne.symm hv)] | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) := by
rw [gaussianPdfReal_def]
by_cases hv : v = 0
· simp [hv]
let g : ℝ → ℝ := fun x ↦ (Real.sqrt (2 * π * v))⁻¹ * rexp (- x ^ 2 / (2 * v))
have hg : Integrable g := by
suffices g = fun x ↦ (Real.sqrt (2 * π * v))⁻¹... | Mathlib.Probability.Distributions.Gaussian.74_0.7VInOP4QlYS5vmc | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
hv : ¬v = 0
g : ℝ → ℝ := fun x => (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-x ^ 2 / (2 * ↑v))
⊢ g = fun x => (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-↑(2 * v)⁻¹ * x ^ 2) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | ext x | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) := by
rw [gaussianPdfReal_def]
by_cases hv : v = 0
· simp [hv]
let g : ℝ → ℝ := fun x ↦ (Real.sqrt (2 * π * v))⁻¹ * rexp (- x ^ 2 / (2 * v))
have hg : Integrable g := by
suffices g = fun x ↦ (Real.sqrt (2 * π * v))⁻¹... | Mathlib.Probability.Distributions.Gaussian.74_0.7VInOP4QlYS5vmc | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) | Mathlib_Probability_Distributions_Gaussian |
case h
μ : ℝ
v : ℝ≥0
hv : ¬v = 0
g : ℝ → ℝ := fun x => (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-x ^ 2 / (2 * ↑v))
x : ℝ
⊢ g x = (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-↑(2 * v)⁻¹ * x ^ 2) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp only [gt_iff_lt, zero_lt_two, zero_le_mul_left, NNReal.zero_le_coe, Real.sqrt_mul',
mul_inv_rev, NNReal.coe_mul, NNReal.coe_inv, NNReal.coe_ofNat, neg_mul, mul_eq_mul_left_iff,
Real.exp_eq_exp, mul_eq_zero, inv_eq_zero, Real.sqrt_eq_zero, NNReal.coe_eq_zero, hv,
false_or] | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) := by
rw [gaussianPdfReal_def]
by_cases hv : v = 0
· simp [hv]
let g : ℝ → ℝ := fun x ↦ (Real.sqrt (2 * π * v))⁻¹ * rexp (- x ^ 2 / (2 * v))
have hg : Integrable g := by
suffices g = fun x ↦ (Real.sqrt (2 * π * v))⁻¹... | Mathlib.Probability.Distributions.Gaussian.74_0.7VInOP4QlYS5vmc | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) | Mathlib_Probability_Distributions_Gaussian |
case h
μ : ℝ
v : ℝ≥0
hv : ¬v = 0
g : ℝ → ℝ := fun x => (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-x ^ 2 / (2 * ↑v))
x : ℝ
⊢ -x ^ 2 / (2 * ↑v) = -((↑v)⁻¹ * 2⁻¹ * x ^ 2) ∨ Real.sqrt (2 * π) = 0 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [mul_comm] | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) := by
rw [gaussianPdfReal_def]
by_cases hv : v = 0
· simp [hv]
let g : ℝ → ℝ := fun x ↦ (Real.sqrt (2 * π * v))⁻¹ * rexp (- x ^ 2 / (2 * v))
have hg : Integrable g := by
suffices g = fun x ↦ (Real.sqrt (2 * π * v))⁻¹... | Mathlib.Probability.Distributions.Gaussian.74_0.7VInOP4QlYS5vmc | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) | Mathlib_Probability_Distributions_Gaussian |
case h
μ : ℝ
v : ℝ≥0
hv : ¬v = 0
g : ℝ → ℝ := fun x => (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-x ^ 2 / (2 * ↑v))
x : ℝ
⊢ -x ^ 2 / (↑v * 2) = -((↑v)⁻¹ * 2⁻¹ * x ^ 2) ∨ Real.sqrt (2 * π) = 0 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | left | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) := by
rw [gaussianPdfReal_def]
by_cases hv : v = 0
· simp [hv]
let g : ℝ → ℝ := fun x ↦ (Real.sqrt (2 * π * v))⁻¹ * rexp (- x ^ 2 / (2 * v))
have hg : Integrable g := by
suffices g = fun x ↦ (Real.sqrt (2 * π * v))⁻¹... | Mathlib.Probability.Distributions.Gaussian.74_0.7VInOP4QlYS5vmc | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) | Mathlib_Probability_Distributions_Gaussian |
case h.h
μ : ℝ
v : ℝ≥0
hv : ¬v = 0
g : ℝ → ℝ := fun x => (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-x ^ 2 / (2 * ↑v))
x : ℝ
⊢ -x ^ 2 / (↑v * 2) = -((↑v)⁻¹ * 2⁻¹ * x ^ 2) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | field_simp | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) := by
rw [gaussianPdfReal_def]
by_cases hv : v = 0
· simp [hv]
let g : ℝ → ℝ := fun x ↦ (Real.sqrt (2 * π * v))⁻¹ * rexp (- x ^ 2 / (2 * v))
have hg : Integrable g := by
suffices g = fun x ↦ (Real.sqrt (2 * π * v))⁻¹... | Mathlib.Probability.Distributions.Gaussian.74_0.7VInOP4QlYS5vmc | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) | Mathlib_Probability_Distributions_Gaussian |
case neg
μ : ℝ
v : ℝ≥0
hv : ¬v = 0
g : ℝ → ℝ := fun x => (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-x ^ 2 / (2 * ↑v))
hg : Integrable g
⊢ Integrable fun x => (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v)) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | exact Integrable.comp_sub_right hg μ | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) := by
rw [gaussianPdfReal_def]
by_cases hv : v = 0
· simp [hv]
let g : ℝ → ℝ := fun x ↦ (Real.sqrt (2 * π * v))⁻¹ * rexp (- x ^ 2 / (2 * v))
have hg : Integrable g := by
suffices g = fun x ↦ (Real.sqrt (2 * π * v))⁻¹... | Mathlib.Probability.Distributions.Gaussian.74_0.7VInOP4QlYS5vmc | lemma integrable_gaussianPdfReal (μ : ℝ) (v : ℝ≥0) :
Integrable (gaussianPdfReal μ v) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
h : v ≠ 0
⊢ ∫⁻ (x : ℝ), ENNReal.ofReal (gaussianPdfReal μ v x) = 1 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [← ENNReal.toReal_eq_one_iff] | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 := by
| Mathlib.Probability.Distributions.Gaussian.95_0.7VInOP4QlYS5vmc | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
h : v ≠ 0
⊢ ENNReal.toReal (∫⁻ (x : ℝ), ENNReal.ofReal (gaussianPdfReal μ v x)) = 1 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | have hfm : AEStronglyMeasurable (gaussianPdfReal μ v) volume :=
(stronglyMeasurable_gaussianPdfReal μ v).aestronglyMeasurable | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 := by
rw [← ENNReal.toReal_eq_one_iff]
| Mathlib.Probability.Distributions.Gaussian.95_0.7VInOP4QlYS5vmc | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
h : v ≠ 0
hfm : AEStronglyMeasurable (gaussianPdfReal μ v) ℙ
⊢ ENNReal.toReal (∫⁻ (x : ℝ), ENNReal.ofReal (gaussianPdfReal μ v x)) = 1 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | have hf : 0 ≤ₐₛ gaussianPdfReal μ v := ae_of_all _ (gaussianPdfReal_nonneg μ v) | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 := by
rw [← ENNReal.toReal_eq_one_iff]
have hfm : AEStronglyMeasurable (gaussianPdfReal μ v) volume :=
... | Mathlib.Probability.Distributions.Gaussian.95_0.7VInOP4QlYS5vmc | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
h : v ≠ 0
hfm : AEStronglyMeasurable (gaussianPdfReal μ v) ℙ
hf : 0 ≤ᵐ[ℙ] gaussianPdfReal μ v
⊢ ENNReal.toReal (∫⁻ (x : ℝ), ENNReal.ofReal (gaussianPdfReal μ v x)) = 1 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [← integral_eq_lintegral_of_nonneg_ae hf hfm] | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 := by
rw [← ENNReal.toReal_eq_one_iff]
have hfm : AEStronglyMeasurable (gaussianPdfReal μ v) volume :=
... | Mathlib.Probability.Distributions.Gaussian.95_0.7VInOP4QlYS5vmc | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
h : v ≠ 0
hfm : AEStronglyMeasurable (gaussianPdfReal μ v) ℙ
hf : 0 ≤ᵐ[ℙ] gaussianPdfReal μ v
⊢ ∫ (a : ℝ), gaussianPdfReal μ v a = 1 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp only [gaussianPdfReal, gt_iff_lt, zero_lt_two, zero_le_mul_right, ge_iff_le, one_div,
Nat.cast_ofNat, integral_mul_left] | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 := by
rw [← ENNReal.toReal_eq_one_iff]
have hfm : AEStronglyMeasurable (gaussianPdfReal μ v) volume :=
... | Mathlib.Probability.Distributions.Gaussian.95_0.7VInOP4QlYS5vmc | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
h : v ≠ 0
hfm : AEStronglyMeasurable (gaussianPdfReal μ v) ℙ
hf : 0 ≤ᵐ[ℙ] gaussianPdfReal μ v
⊢ (Real.sqrt (2 * π * ↑v))⁻¹ * ∫ (a : ℝ), rexp (-(a - μ) ^ 2 / (2 * ↑v)) = 1 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [integral_sub_right_eq_self (μ := volume) (fun a ↦ rexp (-a ^ 2 / ((2 : ℝ) * v))) μ] | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 := by
rw [← ENNReal.toReal_eq_one_iff]
have hfm : AEStronglyMeasurable (gaussianPdfReal μ v) volume :=
... | Mathlib.Probability.Distributions.Gaussian.95_0.7VInOP4QlYS5vmc | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
h : v ≠ 0
hfm : AEStronglyMeasurable (gaussianPdfReal μ v) ℙ
hf : 0 ≤ᵐ[ℙ] gaussianPdfReal μ v
⊢ (Real.sqrt (2 * π * ↑v))⁻¹ * ∫ (x : ℝ), rexp (-x ^ 2 / (2 * ↑v)) = 1 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp only [gt_iff_lt, zero_lt_two, zero_le_mul_right, ge_iff_le, div_eq_inv_mul, mul_inv_rev,
mul_neg] | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 := by
rw [← ENNReal.toReal_eq_one_iff]
have hfm : AEStronglyMeasurable (gaussianPdfReal μ v) volume :=
... | Mathlib.Probability.Distributions.Gaussian.95_0.7VInOP4QlYS5vmc | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
h : v ≠ 0
hfm : AEStronglyMeasurable (gaussianPdfReal μ v) ℙ
hf : 0 ≤ᵐ[ℙ] gaussianPdfReal μ v
⊢ (Real.sqrt (2 * π * ↑v))⁻¹ * ∫ (x : ℝ), rexp (-((↑v)⁻¹ * 2⁻¹ * x ^ 2)) = 1 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp_rw [← neg_mul] | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 := by
rw [← ENNReal.toReal_eq_one_iff]
have hfm : AEStronglyMeasurable (gaussianPdfReal μ v) volume :=
... | Mathlib.Probability.Distributions.Gaussian.95_0.7VInOP4QlYS5vmc | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
h : v ≠ 0
hfm : AEStronglyMeasurable (gaussianPdfReal μ v) ℙ
hf : 0 ≤ᵐ[ℙ] gaussianPdfReal μ v
⊢ (Real.sqrt (2 * π * ↑v))⁻¹ * ∫ (x : ℝ), rexp (-(↑v)⁻¹ * 2⁻¹ * x ^ 2) = 1 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [neg_mul, integral_gaussian, ← Real.sqrt_inv, ← Real.sqrt_mul] | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 := by
rw [← ENNReal.toReal_eq_one_iff]
have hfm : AEStronglyMeasurable (gaussianPdfReal μ v) volume :=
... | Mathlib.Probability.Distributions.Gaussian.95_0.7VInOP4QlYS5vmc | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
h : v ≠ 0
hfm : AEStronglyMeasurable (gaussianPdfReal μ v) ℙ
hf : 0 ≤ᵐ[ℙ] gaussianPdfReal μ v
⊢ Real.sqrt ((2 * π * ↑v)⁻¹ * (π / ((↑v)⁻¹ * 2⁻¹))) = 1 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | field_simp | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 := by
rw [← ENNReal.toReal_eq_one_iff]
have hfm : AEStronglyMeasurable (gaussianPdfReal μ v) volume :=
... | Mathlib.Probability.Distributions.Gaussian.95_0.7VInOP4QlYS5vmc | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
h : v ≠ 0
hfm : AEStronglyMeasurable (gaussianPdfReal μ v) ℙ
hf : 0 ≤ᵐ[ℙ] gaussianPdfReal μ v
⊢ Real.sqrt π * (Real.sqrt ↑v * Real.sqrt 2) = Real.sqrt 2 * Real.sqrt π * Real.sqrt ↑v | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | ring | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 := by
rw [← ENNReal.toReal_eq_one_iff]
have hfm : AEStronglyMeasurable (gaussianPdfReal μ v) volume :=
... | Mathlib.Probability.Distributions.Gaussian.95_0.7VInOP4QlYS5vmc | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 | Mathlib_Probability_Distributions_Gaussian |
case hx
μ : ℝ
v : ℝ≥0
h : v ≠ 0
hfm : AEStronglyMeasurable (gaussianPdfReal μ v) ℙ
hf : 0 ≤ᵐ[ℙ] gaussianPdfReal μ v
⊢ 0 ≤ (2 * π * ↑v)⁻¹ | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | positivity | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 := by
rw [← ENNReal.toReal_eq_one_iff]
have hfm : AEStronglyMeasurable (gaussianPdfReal μ v) volume :=
... | Mathlib.Probability.Distributions.Gaussian.95_0.7VInOP4QlYS5vmc | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPdfReal μ v x) = 1 | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
⊢ ∫ (x : ℝ), gaussianPdfReal μ v x = 1 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | have h := lintegral_gaussianPdfReal_eq_one μ hv | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma integral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
∫ x, gaussianPdfReal μ v x = 1 := by
| Mathlib.Probability.Distributions.Gaussian.114_0.7VInOP4QlYS5vmc | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma integral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
∫ x, gaussianPdfReal μ v x = 1 | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
h : ∫⁻ (x : ℝ), ENNReal.ofReal (gaussianPdfReal μ v x) = 1
⊢ ∫ (x : ℝ), gaussianPdfReal μ v x = 1 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [← ofReal_integral_eq_lintegral_ofReal (integrable_gaussianPdfReal _ _)
(ae_of_all _ (gaussianPdfReal_nonneg _ _)), ← ENNReal.ofReal_one] at h | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma integral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
∫ x, gaussianPdfReal μ v x = 1 := by
have h := lintegral_gaussianPdfReal_eq_one μ hv
| Mathlib.Probability.Distributions.Gaussian.114_0.7VInOP4QlYS5vmc | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma integral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
∫ x, gaussianPdfReal μ v x = 1 | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
h : ENNReal.ofReal (∫ (x : ℝ), gaussianPdfReal μ v x) = ENNReal.ofReal 1
⊢ ∫ (x : ℝ), gaussianPdfReal μ v x = 1 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rwa [← ENNReal.ofReal_eq_ofReal_iff (integral_nonneg (gaussianPdfReal_nonneg _ _)) zero_le_one] | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma integral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
∫ x, gaussianPdfReal μ v x = 1 := by
have h := lintegral_gaussianPdfReal_eq_one μ hv
rw [← ofReal_integral_eq_lintegral_ofReal (integrable_gaussianPdfReal _... | Mathlib.Probability.Distributions.Gaussian.114_0.7VInOP4QlYS5vmc | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma integral_gaussianPdfReal_eq_one (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
∫ x, gaussianPdfReal μ v x = 1 | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
x y : ℝ
⊢ gaussianPdfReal μ v (x - y) = gaussianPdfReal (μ + y) v x | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp only [gaussianPdfReal] | lemma gaussianPdfReal_sub {μ : ℝ} {v : ℝ≥0} (x y : ℝ) :
gaussianPdfReal μ v (x - y) = gaussianPdfReal (μ + y) v x := by
| Mathlib.Probability.Distributions.Gaussian.122_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_sub {μ : ℝ} {v : ℝ≥0} (x y : ℝ) :
gaussianPdfReal μ v (x - y) = gaussianPdfReal (μ + y) v x | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
x y : ℝ
⊢ (Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-(x - y - μ) ^ 2 / (2 * ↑v)) =
(Real.sqrt (2 * π * ↑v))⁻¹ * rexp (-(x - (μ + y)) ^ 2 / (2 * ↑v)) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [sub_add_eq_sub_sub_swap] | lemma gaussianPdfReal_sub {μ : ℝ} {v : ℝ≥0} (x y : ℝ) :
gaussianPdfReal μ v (x - y) = gaussianPdfReal (μ + y) v x := by
simp only [gaussianPdfReal]
| Mathlib.Probability.Distributions.Gaussian.122_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_sub {μ : ℝ} {v : ℝ≥0} (x y : ℝ) :
gaussianPdfReal μ v (x - y) = gaussianPdfReal (μ + y) v x | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
x y : ℝ
⊢ gaussianPdfReal μ v (x + y) = gaussianPdfReal (μ - y) v x | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [sub_eq_add_neg, ← gaussianPdfReal_sub, sub_eq_add_neg, neg_neg] | lemma gaussianPdfReal_add {μ : ℝ} {v : ℝ≥0} (x y : ℝ) :
gaussianPdfReal μ v (x + y) = gaussianPdfReal (μ - y) v x := by
| Mathlib.Probability.Distributions.Gaussian.127_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_add {μ : ℝ} {v : ℝ≥0} (x y : ℝ) :
gaussianPdfReal μ v (x + y) = gaussianPdfReal (μ - y) v x | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) ({ val := c ^ 2, property := (_ : 0 ≤ c ^ 2) } * v) x | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp only [gaussianPdfReal._eq_1, gt_iff_lt, zero_lt_two, zero_le_mul_left, NNReal.zero_le_coe,
Real.sqrt_mul', one_div, mul_inv_rev, NNReal.coe_mul, NNReal.coe_mk, NNReal.coe_pos] | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x := by
| Mathlib.Probability.Distributions.Gaussian.131_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ (Real.sqrt ↑v)⁻¹ * (Real.sqrt (2 * π))⁻¹ * rexp (-(c⁻¹ * x - μ) ^ 2 / (2 * ↑v)) =
|c| * ((Real.sqrt (2 * π * (c ^ 2 * ↑v)))⁻¹ * rexp (-(x - c * μ) ^ 2 / (2 * (c ^ 2 * ↑v)))) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [← mul_assoc] | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x := by
simp only [gaussianPdfReal._eq_1, gt_iff_lt, zero_lt_two, zero_le_mul_left, NNReal.zero_le_coe,
Real.sqrt_mul', one_div, mul_inv_rev, NN... | Mathlib.Probability.Distributions.Gaussian.131_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ (Real.sqrt ↑v)⁻¹ * (Real.sqrt (2 * π))⁻¹ * rexp (-(c⁻¹ * x - μ) ^ 2 / (2 * ↑v)) =
|c| * (Real.sqrt (2 * π * (c ^ 2 * ↑v)))⁻¹ * rexp (-(x - c * μ) ^ 2 / (2 * (c ^ 2 * ↑v))) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | refine congr_arg₂ _ ?_ ?_ | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x := by
simp only [gaussianPdfReal._eq_1, gt_iff_lt, zero_lt_two, zero_le_mul_left, NNReal.zero_le_coe,
Real.sqrt_mul', one_div, mul_inv_rev, NN... | Mathlib.Probability.Distributions.Gaussian.131_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x | Mathlib_Probability_Distributions_Gaussian |
case refine_1
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ (Real.sqrt ↑v)⁻¹ * (Real.sqrt (2 * π))⁻¹ = |c| * (Real.sqrt (2 * π * (c ^ 2 * ↑v)))⁻¹ | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | field_simp | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x := by
simp only [gaussianPdfReal._eq_1, gt_iff_lt, zero_lt_two, zero_le_mul_left, NNReal.zero_le_coe,
Real.sqrt_mul', one_div, mul_inv_rev, NN... | Mathlib.Probability.Distributions.Gaussian.131_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x | Mathlib_Probability_Distributions_Gaussian |
case refine_1
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ 1 / (Real.sqrt ↑v * (Real.sqrt 2 * Real.sqrt π)) =
|c| / (Real.sqrt 2 * Real.sqrt π * (Real.sqrt (c ^ 2) * Real.sqrt ↑v)) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [Real.sqrt_sq_eq_abs] | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x := by
simp only [gaussianPdfReal._eq_1, gt_iff_lt, zero_lt_two, zero_le_mul_left, NNReal.zero_le_coe,
Real.sqrt_mul', one_div, mul_inv_rev, NN... | Mathlib.Probability.Distributions.Gaussian.131_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x | Mathlib_Probability_Distributions_Gaussian |
case refine_1
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ 1 / (Real.sqrt ↑v * (Real.sqrt 2 * Real.sqrt π)) = |c| / (Real.sqrt 2 * Real.sqrt π * (|c| * Real.sqrt ↑v)) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | ring_nf | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x := by
simp only [gaussianPdfReal._eq_1, gt_iff_lt, zero_lt_two, zero_le_mul_left, NNReal.zero_le_coe,
Real.sqrt_mul', one_div, mul_inv_rev, NN... | Mathlib.Probability.Distributions.Gaussian.131_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x | Mathlib_Probability_Distributions_Gaussian |
case refine_1
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ (Real.sqrt ↑v)⁻¹ * (Real.sqrt 2)⁻¹ * (Real.sqrt π)⁻¹ =
(Real.sqrt ↑v)⁻¹ * (Real.sqrt 2)⁻¹ * (Real.sqrt π)⁻¹ * |c| * |c|⁻¹ | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | calc (Real.sqrt ↑v)⁻¹ * (Real.sqrt 2)⁻¹ * (Real.sqrt π)⁻¹
= (Real.sqrt ↑v)⁻¹ * (Real.sqrt 2)⁻¹ * (Real.sqrt π)⁻¹ * (|c| * |c|⁻¹) := by
rw [mul_inv_cancel, mul_one]
simp only [ne_eq, abs_eq_zero, hc, not_false_eq_true]
_ = (Real.sqrt ↑v)⁻¹ * (Real.sqrt 2)⁻¹ * (Real.sqrt π)⁻¹ * |c| * |c|⁻¹ :... | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x := by
simp only [gaussianPdfReal._eq_1, gt_iff_lt, zero_lt_two, zero_le_mul_left, NNReal.zero_le_coe,
Real.sqrt_mul', one_div, mul_inv_rev, NN... | Mathlib.Probability.Distributions.Gaussian.131_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ (Real.sqrt ↑v)⁻¹ * (Real.sqrt 2)⁻¹ * (Real.sqrt π)⁻¹ =
(Real.sqrt ↑v)⁻¹ * (Real.sqrt 2)⁻¹ * (Real.sqrt π)⁻¹ * (|c| * |c|⁻¹) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [mul_inv_cancel, mul_one] | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x := by
simp only [gaussianPdfReal._eq_1, gt_iff_lt, zero_lt_two, zero_le_mul_left, NNReal.zero_le_coe,
Real.sqrt_mul', one_div, mul_inv_rev, NN... | Mathlib.Probability.Distributions.Gaussian.131_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ |c| ≠ 0 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp only [ne_eq, abs_eq_zero, hc, not_false_eq_true] | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x := by
simp only [gaussianPdfReal._eq_1, gt_iff_lt, zero_lt_two, zero_le_mul_left, NNReal.zero_le_coe,
Real.sqrt_mul', one_div, mul_inv_rev, NN... | Mathlib.Probability.Distributions.Gaussian.131_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x | Mathlib_Probability_Distributions_Gaussian |
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