state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case intro
P : Type u_1
inst✝ : Preorder P
IF : PrimePair P
w : P
h : w ∈ ↑IF.F
⊢ IsProper IF.I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | apply isProper_of_not_mem (_ : w ∉ IF.I) | theorem I_isProper : IsProper IF.I := by
cases' IF.F.nonempty with w h
| Mathlib.Order.PrimeIdeal.67_0.4MyuCIeckR2MXpq | theorem I_isProper : IsProper IF.I | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝ : Preorder P
IF : PrimePair P
w : P
h : w ∈ ↑IF.F
⊢ w ∉ IF.I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | rwa [← IF.compl_I_eq_F] at h | theorem I_isProper : IsProper IF.I := by
cases' IF.F.nonempty with w h
apply isProper_of_not_mem (_ : w ∉ IF.I)
| Mathlib.Order.PrimeIdeal.67_0.4MyuCIeckR2MXpq | theorem I_isProper : IsProper IF.I | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝ : Preorder P
IF : PrimePair P
src✝ : IsProper IF.I := I_isProper IF
⊢ IsPFilter (↑IF.I)ᶜ | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | rw [IF.compl_I_eq_F] | theorem PrimePair.I_isPrime (IF : PrimePair P) : IsPrime IF.I :=
{ IF.I_isProper with
compl_filter := by
| Mathlib.Order.PrimeIdeal.109_0.4MyuCIeckR2MXpq | theorem PrimePair.I_isPrime (IF : PrimePair P) : IsPrime IF.I | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝ : Preorder P
IF : PrimePair P
src✝ : IsProper IF.I := I_isProper IF
⊢ IsPFilter ↑IF.F | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | exact IF.F.isPFilter | theorem PrimePair.I_isPrime (IF : PrimePair P) : IsPrime IF.I :=
{ IF.I_isProper with
compl_filter := by
rw [IF.compl_I_eq_F]
| Mathlib.Order.PrimeIdeal.109_0.4MyuCIeckR2MXpq | theorem PrimePair.I_isPrime (IF : PrimePair P) : IsPrime IF.I | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝ : SemilatticeInf P
x✝ y✝ : P
I : Ideal P
hI : IsPrime I
x y : P
⊢ x ⊓ y ∈ I → x ∈ I ∨ y ∈ I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | contrapose! | theorem IsPrime.mem_or_mem (hI : IsPrime I) {x y : P} : x ⊓ y ∈ I → x ∈ I ∨ y ∈ I := by
| Mathlib.Order.PrimeIdeal.123_0.4MyuCIeckR2MXpq | theorem IsPrime.mem_or_mem (hI : IsPrime I) {x y : P} : x ⊓ y ∈ I → x ∈ I ∨ y ∈ I | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝ : SemilatticeInf P
x✝ y✝ : P
I : Ideal P
hI : IsPrime I
x y : P
⊢ x ∉ I ∧ y ∉ I → x ⊓ y ∉ I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | let F := hI.compl_filter.toPFilter | theorem IsPrime.mem_or_mem (hI : IsPrime I) {x y : P} : x ⊓ y ∈ I → x ∈ I ∨ y ∈ I := by
contrapose!
| Mathlib.Order.PrimeIdeal.123_0.4MyuCIeckR2MXpq | theorem IsPrime.mem_or_mem (hI : IsPrime I) {x y : P} : x ⊓ y ∈ I → x ∈ I ∨ y ∈ I | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝ : SemilatticeInf P
x✝ y✝ : P
I : Ideal P
hI : IsPrime I
x y : P
F : PFilter P := IsPFilter.toPFilter (_ : IsPFilter (↑I)ᶜ)
⊢ x ∉ I ∧ y ∉ I → x ⊓ y ∉ I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | show x ∈ F ∧ y ∈ F → x ⊓ y ∈ F | theorem IsPrime.mem_or_mem (hI : IsPrime I) {x y : P} : x ⊓ y ∈ I → x ∈ I ∨ y ∈ I := by
contrapose!
let F := hI.compl_filter.toPFilter
| Mathlib.Order.PrimeIdeal.123_0.4MyuCIeckR2MXpq | theorem IsPrime.mem_or_mem (hI : IsPrime I) {x y : P} : x ⊓ y ∈ I → x ∈ I ∨ y ∈ I | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝ : SemilatticeInf P
x✝ y✝ : P
I : Ideal P
hI : IsPrime I
x y : P
F : PFilter P := IsPFilter.toPFilter (_ : IsPFilter (↑I)ᶜ)
⊢ x ∈ F ∧ y ∈ F → x ⊓ y ∈ F | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | exact fun h => inf_mem h.1 h.2 | theorem IsPrime.mem_or_mem (hI : IsPrime I) {x y : P} : x ⊓ y ∈ I → x ∈ I ∨ y ∈ I := by
contrapose!
let F := hI.compl_filter.toPFilter
show x ∈ F ∧ y ∈ F → x ⊓ y ∈ F
| Mathlib.Order.PrimeIdeal.123_0.4MyuCIeckR2MXpq | theorem IsPrime.mem_or_mem (hI : IsPrime I) {x y : P} : x ⊓ y ∈ I → x ∈ I ∨ y ∈ I | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝¹ : SemilatticeInf P
x y : P
I : Ideal P
inst✝ : IsProper I
hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I
⊢ IsPrime I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | rw [IsPrime_iff] | theorem IsPrime.of_mem_or_mem [IsProper I] (hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I) :
IsPrime I := by
| Mathlib.Order.PrimeIdeal.130_0.4MyuCIeckR2MXpq | theorem IsPrime.of_mem_or_mem [IsProper I] (hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I) :
IsPrime I | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝¹ : SemilatticeInf P
x y : P
I : Ideal P
inst✝ : IsProper I
hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I
⊢ IsProper I ∧ IsPFilter (↑I)ᶜ | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | use ‹_› | theorem IsPrime.of_mem_or_mem [IsProper I] (hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I) :
IsPrime I := by
rw [IsPrime_iff]
| Mathlib.Order.PrimeIdeal.130_0.4MyuCIeckR2MXpq | theorem IsPrime.of_mem_or_mem [IsProper I] (hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I) :
IsPrime I | Mathlib_Order_PrimeIdeal |
case right
P : Type u_1
inst✝¹ : SemilatticeInf P
x y : P
I : Ideal P
inst✝ : IsProper I
hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I
⊢ IsPFilter (↑I)ᶜ | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | refine .of_def ?_ ?_ ?_ | theorem IsPrime.of_mem_or_mem [IsProper I] (hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I) :
IsPrime I := by
rw [IsPrime_iff]
use ‹_›
| Mathlib.Order.PrimeIdeal.130_0.4MyuCIeckR2MXpq | theorem IsPrime.of_mem_or_mem [IsProper I] (hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I) :
IsPrime I | Mathlib_Order_PrimeIdeal |
case right.refine_1
P : Type u_1
inst✝¹ : SemilatticeInf P
x y : P
I : Ideal P
inst✝ : IsProper I
hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I
⊢ Set.Nonempty (↑I)ᶜ | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | exact Set.nonempty_compl.2 (I.IsProper_iff.1 ‹_›) | theorem IsPrime.of_mem_or_mem [IsProper I] (hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I) :
IsPrime I := by
rw [IsPrime_iff]
use ‹_›
refine .of_def ?_ ?_ ?_
· | Mathlib.Order.PrimeIdeal.130_0.4MyuCIeckR2MXpq | theorem IsPrime.of_mem_or_mem [IsProper I] (hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I) :
IsPrime I | Mathlib_Order_PrimeIdeal |
case right.refine_2
P : Type u_1
inst✝¹ : SemilatticeInf P
x y : P
I : Ideal P
inst✝ : IsProper I
hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I
⊢ DirectedOn (fun x x_1 => x ≥ x_1) (↑I)ᶜ | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | intro x hx y hy | theorem IsPrime.of_mem_or_mem [IsProper I] (hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I) :
IsPrime I := by
rw [IsPrime_iff]
use ‹_›
refine .of_def ?_ ?_ ?_
· exact Set.nonempty_compl.2 (I.IsProper_iff.1 ‹_›)
· | Mathlib.Order.PrimeIdeal.130_0.4MyuCIeckR2MXpq | theorem IsPrime.of_mem_or_mem [IsProper I] (hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I) :
IsPrime I | Mathlib_Order_PrimeIdeal |
case right.refine_2
P : Type u_1
inst✝¹ : SemilatticeInf P
x✝ y✝ : P
I : Ideal P
inst✝ : IsProper I
hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I
x : P
hx : x ∈ (↑I)ᶜ
y : P
hy : y ∈ (↑I)ᶜ
⊢ ∃ z ∈ (↑I)ᶜ, (fun x x_1 => x ≥ x_1) x z ∧ (fun x x_1 => x ≥ x_1) y z | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | exact ⟨x ⊓ y, fun h => (hI h).elim hx hy, inf_le_left, inf_le_right⟩ | theorem IsPrime.of_mem_or_mem [IsProper I] (hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I) :
IsPrime I := by
rw [IsPrime_iff]
use ‹_›
refine .of_def ?_ ?_ ?_
· exact Set.nonempty_compl.2 (I.IsProper_iff.1 ‹_›)
· intro x hx y hy
| Mathlib.Order.PrimeIdeal.130_0.4MyuCIeckR2MXpq | theorem IsPrime.of_mem_or_mem [IsProper I] (hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I) :
IsPrime I | Mathlib_Order_PrimeIdeal |
case right.refine_3
P : Type u_1
inst✝¹ : SemilatticeInf P
x y : P
I : Ideal P
inst✝ : IsProper I
hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I
⊢ ∀ {x y : P}, x ≤ y → x ∈ (↑I)ᶜ → y ∈ (↑I)ᶜ | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | exact @mem_compl_of_ge _ _ _ | theorem IsPrime.of_mem_or_mem [IsProper I] (hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I) :
IsPrime I := by
rw [IsPrime_iff]
use ‹_›
refine .of_def ?_ ?_ ?_
· exact Set.nonempty_compl.2 (I.IsProper_iff.1 ‹_›)
· intro x hx y hy
exact ⟨x ⊓ y, fun h => (hI h).elim hx hy, inf_le_left, inf_le_right⟩
· | Mathlib.Order.PrimeIdeal.130_0.4MyuCIeckR2MXpq | theorem IsPrime.of_mem_or_mem [IsProper I] (hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I) :
IsPrime I | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝¹ : DistribLattice P
I : Ideal P
inst✝ : IsMaximal I
⊢ IsPrime I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | rw [isPrime_iff_mem_or_mem] | instance (priority := 100) IsMaximal.isPrime [IsMaximal I] : IsPrime I := by
| Mathlib.Order.PrimeIdeal.151_0.4MyuCIeckR2MXpq | instance (priority | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝¹ : DistribLattice P
I : Ideal P
inst✝ : IsMaximal I
⊢ ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | intro x y | instance (priority := 100) IsMaximal.isPrime [IsMaximal I] : IsPrime I := by
rw [isPrime_iff_mem_or_mem]
| Mathlib.Order.PrimeIdeal.151_0.4MyuCIeckR2MXpq | instance (priority | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝¹ : DistribLattice P
I : Ideal P
inst✝ : IsMaximal I
x y : P
⊢ x ⊓ y ∈ I → x ∈ I ∨ y ∈ I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | contrapose! | instance (priority := 100) IsMaximal.isPrime [IsMaximal I] : IsPrime I := by
rw [isPrime_iff_mem_or_mem]
intro x y
| Mathlib.Order.PrimeIdeal.151_0.4MyuCIeckR2MXpq | instance (priority | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝¹ : DistribLattice P
I : Ideal P
inst✝ : IsMaximal I
x y : P
⊢ x ∉ I ∧ y ∉ I → x ⊓ y ∉ I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | rintro ⟨hx, hynI⟩ hxy | instance (priority := 100) IsMaximal.isPrime [IsMaximal I] : IsPrime I := by
rw [isPrime_iff_mem_or_mem]
intro x y
contrapose!
| Mathlib.Order.PrimeIdeal.151_0.4MyuCIeckR2MXpq | instance (priority | Mathlib_Order_PrimeIdeal |
case intro
P : Type u_1
inst✝¹ : DistribLattice P
I : Ideal P
inst✝ : IsMaximal I
x y : P
hx : x ∉ I
hynI : y ∉ I
hxy : x ⊓ y ∈ I
⊢ False | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | apply hynI | instance (priority := 100) IsMaximal.isPrime [IsMaximal I] : IsPrime I := by
rw [isPrime_iff_mem_or_mem]
intro x y
contrapose!
rintro ⟨hx, hynI⟩ hxy
| Mathlib.Order.PrimeIdeal.151_0.4MyuCIeckR2MXpq | instance (priority | Mathlib_Order_PrimeIdeal |
case intro
P : Type u_1
inst✝¹ : DistribLattice P
I : Ideal P
inst✝ : IsMaximal I
x y : P
hx : x ∉ I
hynI : y ∉ I
hxy : x ⊓ y ∈ I
⊢ y ∈ I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | let J := I ⊔ principal x | instance (priority := 100) IsMaximal.isPrime [IsMaximal I] : IsPrime I := by
rw [isPrime_iff_mem_or_mem]
intro x y
contrapose!
rintro ⟨hx, hynI⟩ hxy
apply hynI
| Mathlib.Order.PrimeIdeal.151_0.4MyuCIeckR2MXpq | instance (priority | Mathlib_Order_PrimeIdeal |
case intro
P : Type u_1
inst✝¹ : DistribLattice P
I : Ideal P
inst✝ : IsMaximal I
x y : P
hx : x ∉ I
hynI : y ∉ I
hxy : x ⊓ y ∈ I
J : Ideal P := I ⊔ principal x
⊢ y ∈ I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | have hJuniv : (J : Set P) = Set.univ :=
IsMaximal.maximal_proper (lt_sup_principal_of_not_mem ‹_›) | instance (priority := 100) IsMaximal.isPrime [IsMaximal I] : IsPrime I := by
rw [isPrime_iff_mem_or_mem]
intro x y
contrapose!
rintro ⟨hx, hynI⟩ hxy
apply hynI
let J := I ⊔ principal x
| Mathlib.Order.PrimeIdeal.151_0.4MyuCIeckR2MXpq | instance (priority | Mathlib_Order_PrimeIdeal |
case intro
P : Type u_1
inst✝¹ : DistribLattice P
I : Ideal P
inst✝ : IsMaximal I
x y : P
hx : x ∉ I
hynI : y ∉ I
hxy : x ⊓ y ∈ I
J : Ideal P := I ⊔ principal x
hJuniv : ↑J = Set.univ
⊢ y ∈ I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | have hyJ : y ∈ ↑J := Set.eq_univ_iff_forall.mp hJuniv y | instance (priority := 100) IsMaximal.isPrime [IsMaximal I] : IsPrime I := by
rw [isPrime_iff_mem_or_mem]
intro x y
contrapose!
rintro ⟨hx, hynI⟩ hxy
apply hynI
let J := I ⊔ principal x
have hJuniv : (J : Set P) = Set.univ :=
IsMaximal.maximal_proper (lt_sup_principal_of_not_mem ‹_›)
| Mathlib.Order.PrimeIdeal.151_0.4MyuCIeckR2MXpq | instance (priority | Mathlib_Order_PrimeIdeal |
case intro
P : Type u_1
inst✝¹ : DistribLattice P
I : Ideal P
inst✝ : IsMaximal I
x y : P
hx : x ∉ I
hynI : y ∉ I
hxy : x ⊓ y ∈ I
J : Ideal P := I ⊔ principal x
hJuniv : ↑J = Set.univ
hyJ : y ∈ ↑J
⊢ y ∈ I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | rw [coe_sup_eq] at hyJ | instance (priority := 100) IsMaximal.isPrime [IsMaximal I] : IsPrime I := by
rw [isPrime_iff_mem_or_mem]
intro x y
contrapose!
rintro ⟨hx, hynI⟩ hxy
apply hynI
let J := I ⊔ principal x
have hJuniv : (J : Set P) = Set.univ :=
IsMaximal.maximal_proper (lt_sup_principal_of_not_mem ‹_›)
have hyJ : y ∈ ↑... | Mathlib.Order.PrimeIdeal.151_0.4MyuCIeckR2MXpq | instance (priority | Mathlib_Order_PrimeIdeal |
case intro
P : Type u_1
inst✝¹ : DistribLattice P
I : Ideal P
inst✝ : IsMaximal I
x y : P
hx : x ∉ I
hynI : y ∉ I
hxy : x ⊓ y ∈ I
J : Ideal P := I ⊔ principal x
hJuniv : ↑J = Set.univ
hyJ : y ∈ {x_1 | ∃ i ∈ I, ∃ j ∈ principal x, x_1 = i ⊔ j}
⊢ y ∈ I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | rcases hyJ with ⟨a, ha, b, hb, hy⟩ | instance (priority := 100) IsMaximal.isPrime [IsMaximal I] : IsPrime I := by
rw [isPrime_iff_mem_or_mem]
intro x y
contrapose!
rintro ⟨hx, hynI⟩ hxy
apply hynI
let J := I ⊔ principal x
have hJuniv : (J : Set P) = Set.univ :=
IsMaximal.maximal_proper (lt_sup_principal_of_not_mem ‹_›)
have hyJ : y ∈ ↑... | Mathlib.Order.PrimeIdeal.151_0.4MyuCIeckR2MXpq | instance (priority | Mathlib_Order_PrimeIdeal |
case intro.intro.intro.intro.intro
P : Type u_1
inst✝¹ : DistribLattice P
I : Ideal P
inst✝ : IsMaximal I
x y : P
hx : x ∉ I
hynI : y ∉ I
hxy : x ⊓ y ∈ I
J : Ideal P := I ⊔ principal x
hJuniv : ↑J = Set.univ
a : P
ha : a ∈ I
b : P
hb : b ∈ principal x
hy : y = a ⊔ b
⊢ y ∈ I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | rw [hy] | instance (priority := 100) IsMaximal.isPrime [IsMaximal I] : IsPrime I := by
rw [isPrime_iff_mem_or_mem]
intro x y
contrapose!
rintro ⟨hx, hynI⟩ hxy
apply hynI
let J := I ⊔ principal x
have hJuniv : (J : Set P) = Set.univ :=
IsMaximal.maximal_proper (lt_sup_principal_of_not_mem ‹_›)
have hyJ : y ∈ ↑... | Mathlib.Order.PrimeIdeal.151_0.4MyuCIeckR2MXpq | instance (priority | Mathlib_Order_PrimeIdeal |
case intro.intro.intro.intro.intro
P : Type u_1
inst✝¹ : DistribLattice P
I : Ideal P
inst✝ : IsMaximal I
x y : P
hx : x ∉ I
hynI : y ∉ I
hxy : x ⊓ y ∈ I
J : Ideal P := I ⊔ principal x
hJuniv : ↑J = Set.univ
a : P
ha : a ∈ I
b : P
hb : b ∈ principal x
hy : y = a ⊔ b
⊢ a ⊔ b ∈ I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | refine' sup_mem ha (I.lower (le_inf hb _) hxy) | instance (priority := 100) IsMaximal.isPrime [IsMaximal I] : IsPrime I := by
rw [isPrime_iff_mem_or_mem]
intro x y
contrapose!
rintro ⟨hx, hynI⟩ hxy
apply hynI
let J := I ⊔ principal x
have hJuniv : (J : Set P) = Set.univ :=
IsMaximal.maximal_proper (lt_sup_principal_of_not_mem ‹_›)
have hyJ : y ∈ ↑... | Mathlib.Order.PrimeIdeal.151_0.4MyuCIeckR2MXpq | instance (priority | Mathlib_Order_PrimeIdeal |
case intro.intro.intro.intro.intro
P : Type u_1
inst✝¹ : DistribLattice P
I : Ideal P
inst✝ : IsMaximal I
x y : P
hx : x ∉ I
hynI : y ∉ I
hxy : x ⊓ y ∈ I
J : Ideal P := I ⊔ principal x
hJuniv : ↑J = Set.univ
a : P
ha : a ∈ I
b : P
hb : b ∈ principal x
hy : y = a ⊔ b
⊢ b ≤ y | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | rw [hy] | instance (priority := 100) IsMaximal.isPrime [IsMaximal I] : IsPrime I := by
rw [isPrime_iff_mem_or_mem]
intro x y
contrapose!
rintro ⟨hx, hynI⟩ hxy
apply hynI
let J := I ⊔ principal x
have hJuniv : (J : Set P) = Set.univ :=
IsMaximal.maximal_proper (lt_sup_principal_of_not_mem ‹_›)
have hyJ : y ∈ ↑... | Mathlib.Order.PrimeIdeal.151_0.4MyuCIeckR2MXpq | instance (priority | Mathlib_Order_PrimeIdeal |
case intro.intro.intro.intro.intro
P : Type u_1
inst✝¹ : DistribLattice P
I : Ideal P
inst✝ : IsMaximal I
x y : P
hx : x ∉ I
hynI : y ∉ I
hxy : x ⊓ y ∈ I
J : Ideal P := I ⊔ principal x
hJuniv : ↑J = Set.univ
a : P
ha : a ∈ I
b : P
hb : b ∈ principal x
hy : y = a ⊔ b
⊢ b ≤ a ⊔ b | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | exact le_sup_right | instance (priority := 100) IsMaximal.isPrime [IsMaximal I] : IsPrime I := by
rw [isPrime_iff_mem_or_mem]
intro x y
contrapose!
rintro ⟨hx, hynI⟩ hxy
apply hynI
let J := I ⊔ principal x
have hJuniv : (J : Set P) = Set.univ :=
IsMaximal.maximal_proper (lt_sup_principal_of_not_mem ‹_›)
have hyJ : y ∈ ↑... | Mathlib.Order.PrimeIdeal.151_0.4MyuCIeckR2MXpq | instance (priority | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝ : BooleanAlgebra P
x : P
I : Ideal P
hI : IsPrime I
⊢ x ∈ I ∨ xᶜ ∈ I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | apply hI.mem_or_mem | theorem IsPrime.mem_or_compl_mem (hI : IsPrime I) : x ∈ I ∨ xᶜ ∈ I := by
| Mathlib.Order.PrimeIdeal.175_0.4MyuCIeckR2MXpq | theorem IsPrime.mem_or_compl_mem (hI : IsPrime I) : x ∈ I ∨ xᶜ ∈ I | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝ : BooleanAlgebra P
x : P
I : Ideal P
hI : IsPrime I
⊢ x ⊓ xᶜ ∈ I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | rw [inf_compl_eq_bot] | theorem IsPrime.mem_or_compl_mem (hI : IsPrime I) : x ∈ I ∨ xᶜ ∈ I := by
apply hI.mem_or_mem
| Mathlib.Order.PrimeIdeal.175_0.4MyuCIeckR2MXpq | theorem IsPrime.mem_or_compl_mem (hI : IsPrime I) : x ∈ I ∨ xᶜ ∈ I | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝ : BooleanAlgebra P
x : P
I : Ideal P
hI : IsPrime I
⊢ ⊥ ∈ I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | exact I.bot_mem | theorem IsPrime.mem_or_compl_mem (hI : IsPrime I) : x ∈ I ∨ xᶜ ∈ I := by
apply hI.mem_or_mem
rw [inf_compl_eq_bot]
| Mathlib.Order.PrimeIdeal.175_0.4MyuCIeckR2MXpq | theorem IsPrime.mem_or_compl_mem (hI : IsPrime I) : x ∈ I ∨ xᶜ ∈ I | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝¹ : BooleanAlgebra P
x : P
I : Ideal P
inst✝ : IsProper I
h : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I
⊢ IsPrime I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | simp only [isPrime_iff_mem_or_mem, or_iff_not_imp_left] | theorem isPrime_of_mem_or_compl_mem [IsProper I] (h : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I) : IsPrime I := by
| Mathlib.Order.PrimeIdeal.185_0.4MyuCIeckR2MXpq | theorem isPrime_of_mem_or_compl_mem [IsProper I] (h : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I) : IsPrime I | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝¹ : BooleanAlgebra P
x : P
I : Ideal P
inst✝ : IsProper I
h : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I
⊢ ∀ {x y : P}, x ⊓ y ∈ I → x ∉ I → y ∈ I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | intro x y hxy hxI | theorem isPrime_of_mem_or_compl_mem [IsProper I] (h : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I) : IsPrime I := by
simp only [isPrime_iff_mem_or_mem, or_iff_not_imp_left]
| Mathlib.Order.PrimeIdeal.185_0.4MyuCIeckR2MXpq | theorem isPrime_of_mem_or_compl_mem [IsProper I] (h : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I) : IsPrime I | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝¹ : BooleanAlgebra P
x✝ : P
I : Ideal P
inst✝ : IsProper I
h : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I
x y : P
hxy : x ⊓ y ∈ I
hxI : x ∉ I
⊢ y ∈ I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | have hxcI : xᶜ ∈ I := h.resolve_left hxI | theorem isPrime_of_mem_or_compl_mem [IsProper I] (h : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I) : IsPrime I := by
simp only [isPrime_iff_mem_or_mem, or_iff_not_imp_left]
intro x y hxy hxI
| Mathlib.Order.PrimeIdeal.185_0.4MyuCIeckR2MXpq | theorem isPrime_of_mem_or_compl_mem [IsProper I] (h : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I) : IsPrime I | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝¹ : BooleanAlgebra P
x✝ : P
I : Ideal P
inst✝ : IsProper I
h : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I
x y : P
hxy : x ⊓ y ∈ I
hxI : x ∉ I
hxcI : xᶜ ∈ I
⊢ y ∈ I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | have ass : x ⊓ y ⊔ y ⊓ xᶜ ∈ I := sup_mem hxy (I.lower inf_le_right hxcI) | theorem isPrime_of_mem_or_compl_mem [IsProper I] (h : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I) : IsPrime I := by
simp only [isPrime_iff_mem_or_mem, or_iff_not_imp_left]
intro x y hxy hxI
have hxcI : xᶜ ∈ I := h.resolve_left hxI
| Mathlib.Order.PrimeIdeal.185_0.4MyuCIeckR2MXpq | theorem isPrime_of_mem_or_compl_mem [IsProper I] (h : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I) : IsPrime I | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝¹ : BooleanAlgebra P
x✝ : P
I : Ideal P
inst✝ : IsProper I
h : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I
x y : P
hxy : x ⊓ y ∈ I
hxI : x ∉ I
hxcI : xᶜ ∈ I
ass : x ⊓ y ⊔ y ⊓ xᶜ ∈ I
⊢ y ∈ I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | rwa [inf_comm, sup_inf_inf_compl] at ass | theorem isPrime_of_mem_or_compl_mem [IsProper I] (h : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I) : IsPrime I := by
simp only [isPrime_iff_mem_or_mem, or_iff_not_imp_left]
intro x y hxy hxI
have hxcI : xᶜ ∈ I := h.resolve_left hxI
have ass : x ⊓ y ⊔ y ⊓ xᶜ ∈ I := sup_mem hxy (I.lower inf_le_right hxcI)
| Mathlib.Order.PrimeIdeal.185_0.4MyuCIeckR2MXpq | theorem isPrime_of_mem_or_compl_mem [IsProper I] (h : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I) : IsPrime I | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝¹ : BooleanAlgebra P
x : P
I : Ideal P
inst✝ : IsPrime I
⊢ IsMaximal I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | simp only [IsMaximal_iff, Set.eq_univ_iff_forall, IsPrime.toIsProper, true_and] | instance (priority := 100) IsPrime.isMaximal [IsPrime I] : IsMaximal I := by
| Mathlib.Order.PrimeIdeal.197_0.4MyuCIeckR2MXpq | instance (priority | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝¹ : BooleanAlgebra P
x : P
I : Ideal P
inst✝ : IsPrime I
⊢ ∀ ⦃J : Ideal P⦄, I < J → ∀ (x : P), x ∈ ↑J | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | intro J hIJ x | instance (priority := 100) IsPrime.isMaximal [IsPrime I] : IsMaximal I := by
simp only [IsMaximal_iff, Set.eq_univ_iff_forall, IsPrime.toIsProper, true_and]
| Mathlib.Order.PrimeIdeal.197_0.4MyuCIeckR2MXpq | instance (priority | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝¹ : BooleanAlgebra P
x✝ : P
I : Ideal P
inst✝ : IsPrime I
J : Ideal P
hIJ : I < J
x : P
⊢ x ∈ ↑J | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | rcases Set.exists_of_ssubset hIJ with ⟨y, hyJ, hyI⟩ | instance (priority := 100) IsPrime.isMaximal [IsPrime I] : IsMaximal I := by
simp only [IsMaximal_iff, Set.eq_univ_iff_forall, IsPrime.toIsProper, true_and]
intro J hIJ x
| Mathlib.Order.PrimeIdeal.197_0.4MyuCIeckR2MXpq | instance (priority | Mathlib_Order_PrimeIdeal |
case intro.intro
P : Type u_1
inst✝¹ : BooleanAlgebra P
x✝ : P
I : Ideal P
inst✝ : IsPrime I
J : Ideal P
hIJ : I < J
x y : P
hyJ : y ∈ ↑J
hyI : y ∉ ↑I
⊢ x ∈ ↑J | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | suffices ass : x ⊓ y ⊔ x ⊓ yᶜ ∈ J by rwa [sup_inf_inf_compl] at ass | instance (priority := 100) IsPrime.isMaximal [IsPrime I] : IsMaximal I := by
simp only [IsMaximal_iff, Set.eq_univ_iff_forall, IsPrime.toIsProper, true_and]
intro J hIJ x
rcases Set.exists_of_ssubset hIJ with ⟨y, hyJ, hyI⟩
| Mathlib.Order.PrimeIdeal.197_0.4MyuCIeckR2MXpq | instance (priority | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝¹ : BooleanAlgebra P
x✝ : P
I : Ideal P
inst✝ : IsPrime I
J : Ideal P
hIJ : I < J
x y : P
hyJ : y ∈ ↑J
hyI : y ∉ ↑I
ass : x ⊓ y ⊔ x ⊓ yᶜ ∈ J
⊢ x ∈ ↑J | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | rwa [sup_inf_inf_compl] at ass | instance (priority := 100) IsPrime.isMaximal [IsPrime I] : IsMaximal I := by
simp only [IsMaximal_iff, Set.eq_univ_iff_forall, IsPrime.toIsProper, true_and]
intro J hIJ x
rcases Set.exists_of_ssubset hIJ with ⟨y, hyJ, hyI⟩
suffices ass : x ⊓ y ⊔ x ⊓ yᶜ ∈ J by | Mathlib.Order.PrimeIdeal.197_0.4MyuCIeckR2MXpq | instance (priority | Mathlib_Order_PrimeIdeal |
case intro.intro
P : Type u_1
inst✝¹ : BooleanAlgebra P
x✝ : P
I : Ideal P
inst✝ : IsPrime I
J : Ideal P
hIJ : I < J
x y : P
hyJ : y ∈ ↑J
hyI : y ∉ ↑I
⊢ x ⊓ y ⊔ x ⊓ yᶜ ∈ J | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | exact
sup_mem (J.lower inf_le_right hyJ)
(hIJ.le <| I.lower inf_le_right <| IsPrime.mem_compl_of_not_mem ‹_› hyI) | instance (priority := 100) IsPrime.isMaximal [IsPrime I] : IsMaximal I := by
simp only [IsMaximal_iff, Set.eq_univ_iff_forall, IsPrime.toIsProper, true_and]
intro J hIJ x
rcases Set.exists_of_ssubset hIJ with ⟨y, hyJ, hyI⟩
suffices ass : x ⊓ y ⊔ x ⊓ yᶜ ∈ J by rwa [sup_inf_inf_compl] at ass
| Mathlib.Order.PrimeIdeal.197_0.4MyuCIeckR2MXpq | instance (priority | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝ : Preorder P
IF : Ideal.PrimePair P
⊢ IsIdeal (↑IF.F)ᶜ | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | rw [IF.compl_F_eq_I] | theorem _root_.Order.Ideal.PrimePair.F_isPrime (IF : Ideal.PrimePair P) : IsPrime IF.F :=
{
compl_ideal := by
| Mathlib.Order.PrimeIdeal.230_0.4MyuCIeckR2MXpq | theorem _root_.Order.Ideal.PrimePair.F_isPrime (IF : Ideal.PrimePair P) : IsPrime IF.F | Mathlib_Order_PrimeIdeal |
P : Type u_1
inst✝ : Preorder P
IF : Ideal.PrimePair P
⊢ IsIdeal ↑IF.I | /-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | exact IF.I.isIdeal | theorem _root_.Order.Ideal.PrimePair.F_isPrime (IF : Ideal.PrimePair P) : IsPrime IF.F :=
{
compl_ideal := by
rw [IF.compl_F_eq_I]
| Mathlib.Order.PrimeIdeal.230_0.4MyuCIeckR2MXpq | theorem _root_.Order.Ideal.PrimePair.F_isPrime (IF : Ideal.PrimePair P) : IsPrime IF.F | Mathlib_Order_PrimeIdeal |
α : Type u_1
inst✝ : LinearOrderedRing α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h : decide (Int.mul num✝¹ (Int.ofNat db) ≤ Int.mul num✝ (Int.ofNat da)) = true
⊢ ↑num✝¹ * ⅟↑da ≤ ↑num✝ * ⅟↑db | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | have h := Int.cast_mono (α := α) <| of_decide_eq_true h | theorem isRat_le_true [LinearOrderedRing α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db →
decide (Int.mul na (.ofNat db) ≤ Int.mul nb (.ofNat da)) → a ≤ b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
| Mathlib.Tactic.NormNum.Ineq.42_0.k3BeWIU6eBBZx1h | theorem isRat_le_true [LinearOrderedRing α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db →
decide (Int.mul na (.ofNat db) ≤ Int.mul nb (.ofNat da)) → a ≤ b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h | Mathlib_Tactic_NormNum_Ineq |
α : Type u_1
inst✝ : LinearOrderedRing α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h✝ : decide (Int.mul num✝¹ (Int.ofNat db) ≤ Int.mul num✝ (Int.ofNat da)) = true
h : (fun x => ↑x) (Int.mul num✝¹ (Int.ofNat db)) ≤ (fun x => ↑x) (Int.mul num✝ (Int.ofNat da))
⊢ ↑num✝¹ * ⅟↑da ≤ ↑num✝ * ⅟↑db | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | have ha : 0 ≤ ⅟(da : α) := invOf_nonneg.mpr <| Nat.cast_nonneg da | theorem isRat_le_true [LinearOrderedRing α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db →
decide (Int.mul na (.ofNat db) ≤ Int.mul nb (.ofNat da)) → a ≤ b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h := Int.cast_mono (α := α) <| of_decide_eq_true h
| Mathlib.Tactic.NormNum.Ineq.42_0.k3BeWIU6eBBZx1h | theorem isRat_le_true [LinearOrderedRing α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db →
decide (Int.mul na (.ofNat db) ≤ Int.mul nb (.ofNat da)) → a ≤ b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h | Mathlib_Tactic_NormNum_Ineq |
α : Type u_1
inst✝ : LinearOrderedRing α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h✝ : decide (Int.mul num✝¹ (Int.ofNat db) ≤ Int.mul num✝ (Int.ofNat da)) = true
h : (fun x => ↑x) (Int.mul num✝¹ (Int.ofNat db)) ≤ (fun x => ↑x) (Int.mul num✝ (Int.ofNat da))
ha : 0 ≤ ⅟↑da
⊢ ↑num✝¹ * ⅟↑da ≤ ↑n... | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | have hb : 0 ≤ ⅟(db : α) := invOf_nonneg.mpr <| Nat.cast_nonneg db | theorem isRat_le_true [LinearOrderedRing α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db →
decide (Int.mul na (.ofNat db) ≤ Int.mul nb (.ofNat da)) → a ≤ b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h := Int.cast_mono (α := α) <| of_decide_eq_true h
have ha : 0 ≤... | Mathlib.Tactic.NormNum.Ineq.42_0.k3BeWIU6eBBZx1h | theorem isRat_le_true [LinearOrderedRing α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db →
decide (Int.mul na (.ofNat db) ≤ Int.mul nb (.ofNat da)) → a ≤ b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h | Mathlib_Tactic_NormNum_Ineq |
α : Type u_1
inst✝ : LinearOrderedRing α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h✝ : decide (Int.mul num✝¹ (Int.ofNat db) ≤ Int.mul num✝ (Int.ofNat da)) = true
h : (fun x => ↑x) (Int.mul num✝¹ (Int.ofNat db)) ≤ (fun x => ↑x) (Int.mul num✝ (Int.ofNat da))
ha : 0 ≤ ⅟↑da
hb : 0 ≤ ⅟↑db
⊢ ↑num... | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | have h := (mul_le_mul_of_nonneg_left · hb) <| mul_le_mul_of_nonneg_right h ha | theorem isRat_le_true [LinearOrderedRing α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db →
decide (Int.mul na (.ofNat db) ≤ Int.mul nb (.ofNat da)) → a ≤ b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h := Int.cast_mono (α := α) <| of_decide_eq_true h
have ha : 0 ≤... | Mathlib.Tactic.NormNum.Ineq.42_0.k3BeWIU6eBBZx1h | theorem isRat_le_true [LinearOrderedRing α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db →
decide (Int.mul na (.ofNat db) ≤ Int.mul nb (.ofNat da)) → a ≤ b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h | Mathlib_Tactic_NormNum_Ineq |
α : Type u_1
inst✝ : LinearOrderedRing α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h✝¹ : decide (Int.mul num✝¹ (Int.ofNat db) ≤ Int.mul num✝ (Int.ofNat da)) = true
h✝ : (fun x => ↑x) (Int.mul num✝¹ (Int.ofNat db)) ≤ (fun x => ↑x) (Int.mul num✝ (Int.ofNat da))
ha : 0 ≤ ⅟↑da
hb : 0 ≤ ⅟↑db
h :
... | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | rw [← mul_assoc, Int.commute_cast] at h | theorem isRat_le_true [LinearOrderedRing α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db →
decide (Int.mul na (.ofNat db) ≤ Int.mul nb (.ofNat da)) → a ≤ b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h := Int.cast_mono (α := α) <| of_decide_eq_true h
have ha : 0 ≤... | Mathlib.Tactic.NormNum.Ineq.42_0.k3BeWIU6eBBZx1h | theorem isRat_le_true [LinearOrderedRing α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db →
decide (Int.mul na (.ofNat db) ≤ Int.mul nb (.ofNat da)) → a ≤ b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h | Mathlib_Tactic_NormNum_Ineq |
α : Type u_1
inst✝ : LinearOrderedRing α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h✝¹ : decide (Int.mul num✝¹ (Int.ofNat db) ≤ Int.mul num✝ (Int.ofNat da)) = true
h✝ : (fun x => ↑x) (Int.mul num✝¹ (Int.ofNat db)) ≤ (fun x => ↑x) (Int.mul num✝ (Int.ofNat da))
ha : 0 ≤ ⅟↑da
hb : 0 ≤ ⅟↑db
h : ... | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | simp at h | theorem isRat_le_true [LinearOrderedRing α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db →
decide (Int.mul na (.ofNat db) ≤ Int.mul nb (.ofNat da)) → a ≤ b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h := Int.cast_mono (α := α) <| of_decide_eq_true h
have ha : 0 ≤... | Mathlib.Tactic.NormNum.Ineq.42_0.k3BeWIU6eBBZx1h | theorem isRat_le_true [LinearOrderedRing α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db →
decide (Int.mul na (.ofNat db) ≤ Int.mul nb (.ofNat da)) → a ≤ b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h | Mathlib_Tactic_NormNum_Ineq |
α : Type u_1
inst✝ : LinearOrderedRing α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h✝¹ : decide (Int.mul num✝¹ (Int.ofNat db) ≤ Int.mul num✝ (Int.ofNat da)) = true
h✝ : (fun x => ↑x) (Int.mul num✝¹ (Int.ofNat db)) ≤ (fun x => ↑x) (Int.mul num✝ (Int.ofNat da))
ha : 0 ≤ ⅟↑da
hb : 0 ≤ ⅟↑db
h : ... | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | rwa [Int.commute_cast] at h | theorem isRat_le_true [LinearOrderedRing α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db →
decide (Int.mul na (.ofNat db) ≤ Int.mul nb (.ofNat da)) → a ≤ b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h := Int.cast_mono (α := α) <| of_decide_eq_true h
have ha : 0 ≤... | Mathlib.Tactic.NormNum.Ineq.42_0.k3BeWIU6eBBZx1h | theorem isRat_le_true [LinearOrderedRing α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db →
decide (Int.mul na (.ofNat db) ≤ Int.mul nb (.ofNat da)) → a ≤ b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h | Mathlib_Tactic_NormNum_Ineq |
α : Type u_1
inst✝¹ : LinearOrderedRing α
inst✝ : Nontrivial α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h : decide (num✝¹ * ↑db < num✝ * ↑da) = true
⊢ ↑num✝¹ * ⅟↑da < ↑num✝ * ⅟↑db | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | have h := Int.cast_strictMono (α := α) <| of_decide_eq_true h | theorem isRat_lt_true [LinearOrderedRing α] [Nontrivial α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db → decide (na * db < nb * da) → a < b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
| Mathlib.Tactic.NormNum.Ineq.53_0.k3BeWIU6eBBZx1h | theorem isRat_lt_true [LinearOrderedRing α] [Nontrivial α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db → decide (na * db < nb * da) → a < b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h | Mathlib_Tactic_NormNum_Ineq |
α : Type u_1
inst✝¹ : LinearOrderedRing α
inst✝ : Nontrivial α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h✝ : decide (num✝¹ * ↑db < num✝ * ↑da) = true
h : (fun x => ↑x) (num✝¹ * ↑db) < (fun x => ↑x) (num✝ * ↑da)
⊢ ↑num✝¹ * ⅟↑da < ↑num✝ * ⅟↑db | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | have ha : 0 < ⅟(da : α) := pos_invOf_of_invertible_cast da | theorem isRat_lt_true [LinearOrderedRing α] [Nontrivial α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db → decide (na * db < nb * da) → a < b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h := Int.cast_strictMono (α := α) <| of_decide_eq_true h
| Mathlib.Tactic.NormNum.Ineq.53_0.k3BeWIU6eBBZx1h | theorem isRat_lt_true [LinearOrderedRing α] [Nontrivial α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db → decide (na * db < nb * da) → a < b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h | Mathlib_Tactic_NormNum_Ineq |
α : Type u_1
inst✝¹ : LinearOrderedRing α
inst✝ : Nontrivial α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h✝ : decide (num✝¹ * ↑db < num✝ * ↑da) = true
h : (fun x => ↑x) (num✝¹ * ↑db) < (fun x => ↑x) (num✝ * ↑da)
ha : 0 < ⅟↑da
⊢ ↑num✝¹ * ⅟↑da < ↑num✝ * ⅟↑db | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | have hb : 0 < ⅟(db : α) := pos_invOf_of_invertible_cast db | theorem isRat_lt_true [LinearOrderedRing α] [Nontrivial α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db → decide (na * db < nb * da) → a < b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h := Int.cast_strictMono (α := α) <| of_decide_eq_true h
have ha : 0 < ⅟(da : α) :=... | Mathlib.Tactic.NormNum.Ineq.53_0.k3BeWIU6eBBZx1h | theorem isRat_lt_true [LinearOrderedRing α] [Nontrivial α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db → decide (na * db < nb * da) → a < b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h | Mathlib_Tactic_NormNum_Ineq |
α : Type u_1
inst✝¹ : LinearOrderedRing α
inst✝ : Nontrivial α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h✝ : decide (num✝¹ * ↑db < num✝ * ↑da) = true
h : (fun x => ↑x) (num✝¹ * ↑db) < (fun x => ↑x) (num✝ * ↑da)
ha : 0 < ⅟↑da
hb : 0 < ⅟↑db
⊢ ↑num✝¹ * ⅟↑da < ↑num✝ * ⅟↑db | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | have h := (mul_lt_mul_of_pos_left · hb) <| mul_lt_mul_of_pos_right h ha | theorem isRat_lt_true [LinearOrderedRing α] [Nontrivial α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db → decide (na * db < nb * da) → a < b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h := Int.cast_strictMono (α := α) <| of_decide_eq_true h
have ha : 0 < ⅟(da : α) :=... | Mathlib.Tactic.NormNum.Ineq.53_0.k3BeWIU6eBBZx1h | theorem isRat_lt_true [LinearOrderedRing α] [Nontrivial α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db → decide (na * db < nb * da) → a < b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h | Mathlib_Tactic_NormNum_Ineq |
α : Type u_1
inst✝¹ : LinearOrderedRing α
inst✝ : Nontrivial α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h✝¹ : decide (num✝¹ * ↑db < num✝ * ↑da) = true
h✝ : (fun x => ↑x) (num✝¹ * ↑db) < (fun x => ↑x) (num✝ * ↑da)
ha : 0 < ⅟↑da
hb : 0 < ⅟↑db
h : ⅟↑db * ((fun x => ↑x) (num✝¹ * ↑db) * ⅟↑da) < ... | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | rw [← mul_assoc, Int.commute_cast] at h | theorem isRat_lt_true [LinearOrderedRing α] [Nontrivial α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db → decide (na * db < nb * da) → a < b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h := Int.cast_strictMono (α := α) <| of_decide_eq_true h
have ha : 0 < ⅟(da : α) :=... | Mathlib.Tactic.NormNum.Ineq.53_0.k3BeWIU6eBBZx1h | theorem isRat_lt_true [LinearOrderedRing α] [Nontrivial α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db → decide (na * db < nb * da) → a < b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h | Mathlib_Tactic_NormNum_Ineq |
α : Type u_1
inst✝¹ : LinearOrderedRing α
inst✝ : Nontrivial α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h✝¹ : decide (num✝¹ * ↑db < num✝ * ↑da) = true
h✝ : (fun x => ↑x) (num✝¹ * ↑db) < (fun x => ↑x) (num✝ * ↑da)
ha : 0 < ⅟↑da
hb : 0 < ⅟↑db
h : ↑(num✝¹ * ↑db) * ⅟↑db * ⅟↑da < ⅟↑db * ((fun x ... | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | simp? at h says simp only [Int.cast_mul, Int.cast_ofNat, mul_mul_invOf_self_cancel'] at h | theorem isRat_lt_true [LinearOrderedRing α] [Nontrivial α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db → decide (na * db < nb * da) → a < b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h := Int.cast_strictMono (α := α) <| of_decide_eq_true h
have ha : 0 < ⅟(da : α) :=... | Mathlib.Tactic.NormNum.Ineq.53_0.k3BeWIU6eBBZx1h | theorem isRat_lt_true [LinearOrderedRing α] [Nontrivial α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db → decide (na * db < nb * da) → a < b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h | Mathlib_Tactic_NormNum_Ineq |
α : Type u_1
inst✝¹ : LinearOrderedRing α
inst✝ : Nontrivial α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h✝¹ : decide (num✝¹ * ↑db < num✝ * ↑da) = true
h✝ : (fun x => ↑x) (num✝¹ * ↑db) < (fun x => ↑x) (num✝ * ↑da)
ha : 0 < ⅟↑da
hb : 0 < ⅟↑db
h : ↑(num✝¹ * ↑db) * ⅟↑db * ⅟↑da < ⅟↑db * ((fun x ... | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | simp only [Int.cast_mul, Int.cast_ofNat, mul_mul_invOf_self_cancel'] at h | theorem isRat_lt_true [LinearOrderedRing α] [Nontrivial α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db → decide (na * db < nb * da) → a < b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h := Int.cast_strictMono (α := α) <| of_decide_eq_true h
have ha : 0 < ⅟(da : α) :=... | Mathlib.Tactic.NormNum.Ineq.53_0.k3BeWIU6eBBZx1h | theorem isRat_lt_true [LinearOrderedRing α] [Nontrivial α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db → decide (na * db < nb * da) → a < b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h | Mathlib_Tactic_NormNum_Ineq |
α : Type u_1
inst✝¹ : LinearOrderedRing α
inst✝ : Nontrivial α
num✝¹ num✝ : ℤ
da db : ℕ
inv✝¹ : Invertible ↑da
inv✝ : Invertible ↑db
h✝¹ : decide (num✝¹ * ↑db < num✝ * ↑da) = true
h✝ : (fun x => ↑x) (num✝¹ * ↑db) < (fun x => ↑x) (num✝ * ↑da)
ha : 0 < ⅟↑da
hb : 0 < ⅟↑db
h : ↑num✝¹ * ⅟↑da < ⅟↑db * ↑num✝
⊢ ↑num✝¹ * ⅟↑da <... | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | rwa [Int.commute_cast] at h | theorem isRat_lt_true [LinearOrderedRing α] [Nontrivial α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db → decide (na * db < nb * da) → a < b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h := Int.cast_strictMono (α := α) <| of_decide_eq_true h
have ha : 0 < ⅟(da : α) :=... | Mathlib.Tactic.NormNum.Ineq.53_0.k3BeWIU6eBBZx1h | theorem isRat_lt_true [LinearOrderedRing α] [Nontrivial α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
IsRat a na da → IsRat b nb db → decide (na * db < nb * da) → a < b
| _, _, _, _, da, db, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
have h | Mathlib_Tactic_NormNum_Ineq |
«$β» : Type := Prop
«$e» : «$β»
«$f» $a✝ $b✝ : Expr
«$u» : Level
«$α» : Type u
«$a» «$b» : «$α»
«$ra» : AddMonoidWithOne «$α»
«$na» : ℕ
«$pa» : IsNat «$a» «$na»
«$rb» : AddMonoidWithOne «$α»
«$nb» : ℕ
«$pb» : IsNat «$b» «$nb»
«$_i» : OrderedSemiring «$α»
⊢ AddMonoidWithOne «$α» | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | clear! $ra $rb | /-- The `norm_num` extension which identifies expressions of the form `a ≤ b`,
such that `norm_num` successfully recognises both `a` and `b`. -/
@[norm_num _ ≤ _] def evalLE : NormNumExt where eval {v β} e := do
haveI' : v =QL 0 := ⟨⟩; haveI' : $β =Q Prop := ⟨⟩
let .app (.app f a) b ← whnfR e | failure
let ⟨u, α,... | Mathlib.Tactic.NormNum.Ineq.99_0.k3BeWIU6eBBZx1h | /-- The `norm_num` extension which identifies expressions of the form `a ≤ b`,
such that `norm_num` successfully recognises both `a` and `b`. -/
@[norm_num _ ≤ _] def evalLE : NormNumExt where eval {v β} e | Mathlib_Tactic_NormNum_Ineq |
«$β» : Type := Prop
«$e» : «$β»
«$f» $a✝ $b✝ : Expr
«$u» : Level
«$α» : Type u
«$a» «$b» : «$α»
«$na» «$nb» : ℕ
«$_i» : OrderedSemiring «$α»
⊢ AddMonoidWithOne «$α» | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | infer_instance | /-- The `norm_num` extension which identifies expressions of the form `a ≤ b`,
such that `norm_num` successfully recognises both `a` and `b`. -/
@[norm_num _ ≤ _] def evalLE : NormNumExt where eval {v β} e := do
haveI' : v =QL 0 := ⟨⟩; haveI' : $β =Q Prop := ⟨⟩
let .app (.app f a) b ← whnfR e | failure
let ⟨u, α,... | Mathlib.Tactic.NormNum.Ineq.99_0.k3BeWIU6eBBZx1h | /-- The `norm_num` extension which identifies expressions of the form `a ≤ b`,
such that `norm_num` successfully recognises both `a` and `b`. -/
@[norm_num _ ≤ _] def evalLE : NormNumExt where eval {v β} e | Mathlib_Tactic_NormNum_Ineq |
«$β» : Type := Prop
«$e» : «$β»
«$f» $a✝ $b✝ : Expr
«$u» : Level
«$α» : Type u
«$a» «$b» : «$α»
«$na» : ℕ
«$rb» : AddMonoidWithOne «$α»
«$nb» : ℕ
«$pb» : IsNat «$b» «$nb»
«$_i» : OrderedSemiring «$α»
«$ra» : AddMonoidWithOne «$α» := inferInstance
«$pa» : IsNat «$a» «$na»
⊢ AddMonoidWithOne «$α» | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | clear! $ra $rb | /-- The `norm_num` extension which identifies expressions of the form `a ≤ b`,
such that `norm_num` successfully recognises both `a` and `b`. -/
@[norm_num _ ≤ _] def evalLE : NormNumExt where eval {v β} e := do
haveI' : v =QL 0 := ⟨⟩; haveI' : $β =Q Prop := ⟨⟩
let .app (.app f a) b ← whnfR e | failure
let ⟨u, α,... | Mathlib.Tactic.NormNum.Ineq.99_0.k3BeWIU6eBBZx1h | /-- The `norm_num` extension which identifies expressions of the form `a ≤ b`,
such that `norm_num` successfully recognises both `a` and `b`. -/
@[norm_num _ ≤ _] def evalLE : NormNumExt where eval {v β} e | Mathlib_Tactic_NormNum_Ineq |
«$β» : Type := Prop
«$e» : «$β»
«$f» $a✝ $b✝ : Expr
«$u» : Level
«$α» : Type u
«$a» «$b» : «$α»
«$na» «$nb» : ℕ
«$_i» : OrderedSemiring «$α»
⊢ AddMonoidWithOne «$α» | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | infer_instance | /-- The `norm_num` extension which identifies expressions of the form `a ≤ b`,
such that `norm_num` successfully recognises both `a` and `b`. -/
@[norm_num _ ≤ _] def evalLE : NormNumExt where eval {v β} e := do
haveI' : v =QL 0 := ⟨⟩; haveI' : $β =Q Prop := ⟨⟩
let .app (.app f a) b ← whnfR e | failure
let ⟨u, α,... | Mathlib.Tactic.NormNum.Ineq.99_0.k3BeWIU6eBBZx1h | /-- The `norm_num` extension which identifies expressions of the form `a ≤ b`,
such that `norm_num` successfully recognises both `a` and `b`. -/
@[norm_num _ ≤ _] def evalLE : NormNumExt where eval {v β} e | Mathlib_Tactic_NormNum_Ineq |
«$β» : Type := Prop
«$e» : «$β»
«$f» $a✝ $b✝ : Expr
«$u» : Level
«$α» : Type u
«$a» «$b» : «$α»
«$ra» : AddMonoidWithOne «$α»
«$na» : ℕ
«$pa» : IsNat «$a» «$na»
«$rb» : AddMonoidWithOne «$α»
«$nb» : ℕ
«$pb» : IsNat «$b» «$nb»
«$_i» : OrderedSemiring «$α»
⊢ AddMonoidWithOne «$α» | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | clear! $ra $rb | /-- The `norm_num` extension which identifies expressions of the form `a < b`,
such that `norm_num` successfully recognises both `a` and `b`. -/
@[norm_num _ < _] def evalLT : NormNumExt where eval {v β} e := do
haveI' : v =QL 0 := ⟨⟩; haveI' : $β =Q Prop := ⟨⟩
let .app (.app f a) b ← whnfR e | failure
let ⟨u, α,... | Mathlib.Tactic.NormNum.Ineq.154_0.k3BeWIU6eBBZx1h | /-- The `norm_num` extension which identifies expressions of the form `a < b`,
such that `norm_num` successfully recognises both `a` and `b`. -/
@[norm_num _ < _] def evalLT : NormNumExt where eval {v β} e | Mathlib_Tactic_NormNum_Ineq |
«$β» : Type := Prop
«$e» : «$β»
«$f» $a✝ $b✝ : Expr
«$u» : Level
«$α» : Type u
«$a» «$b» : «$α»
«$na» «$nb» : ℕ
«$_i» : OrderedSemiring «$α»
⊢ AddMonoidWithOne «$α» | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | infer_instance | /-- The `norm_num` extension which identifies expressions of the form `a < b`,
such that `norm_num` successfully recognises both `a` and `b`. -/
@[norm_num _ < _] def evalLT : NormNumExt where eval {v β} e := do
haveI' : v =QL 0 := ⟨⟩; haveI' : $β =Q Prop := ⟨⟩
let .app (.app f a) b ← whnfR e | failure
let ⟨u, α,... | Mathlib.Tactic.NormNum.Ineq.154_0.k3BeWIU6eBBZx1h | /-- The `norm_num` extension which identifies expressions of the form `a < b`,
such that `norm_num` successfully recognises both `a` and `b`. -/
@[norm_num _ < _] def evalLT : NormNumExt where eval {v β} e | Mathlib_Tactic_NormNum_Ineq |
«$β» : Type := Prop
«$e» : «$β»
«$f» $a✝ $b✝ : Expr
«$u» : Level
«$α» : Type u
«$a» «$b» : «$α»
«$na» : ℕ
«$rb» : AddMonoidWithOne «$α»
«$nb» : ℕ
«$pb» : IsNat «$b» «$nb»
«$_i» : OrderedSemiring «$α»
«$ra» : AddMonoidWithOne «$α» := inferInstance
«$pa» : IsNat «$a» «$na»
⊢ AddMonoidWithOne «$α» | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | clear! $ra $rb | /-- The `norm_num` extension which identifies expressions of the form `a < b`,
such that `norm_num` successfully recognises both `a` and `b`. -/
@[norm_num _ < _] def evalLT : NormNumExt where eval {v β} e := do
haveI' : v =QL 0 := ⟨⟩; haveI' : $β =Q Prop := ⟨⟩
let .app (.app f a) b ← whnfR e | failure
let ⟨u, α,... | Mathlib.Tactic.NormNum.Ineq.154_0.k3BeWIU6eBBZx1h | /-- The `norm_num` extension which identifies expressions of the form `a < b`,
such that `norm_num` successfully recognises both `a` and `b`. -/
@[norm_num _ < _] def evalLT : NormNumExt where eval {v β} e | Mathlib_Tactic_NormNum_Ineq |
«$β» : Type := Prop
«$e» : «$β»
«$f» $a✝ $b✝ : Expr
«$u» : Level
«$α» : Type u
«$a» «$b» : «$α»
«$na» «$nb» : ℕ
«$_i» : OrderedSemiring «$α»
⊢ AddMonoidWithOne «$α» | /-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Eq
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
/-!
# `norm_num` extensions for inequalities.
-/... | infer_instance | /-- The `norm_num` extension which identifies expressions of the form `a < b`,
such that `norm_num` successfully recognises both `a` and `b`. -/
@[norm_num _ < _] def evalLT : NormNumExt where eval {v β} e := do
haveI' : v =QL 0 := ⟨⟩; haveI' : $β =Q Prop := ⟨⟩
let .app (.app f a) b ← whnfR e | failure
let ⟨u, α,... | Mathlib.Tactic.NormNum.Ineq.154_0.k3BeWIU6eBBZx1h | /-- The `norm_num` extension which identifies expressions of the form `a < b`,
such that `norm_num` successfully recognises both `a` and `b`. -/
@[norm_num _ < _] def evalLT : NormNumExt where eval {v β} e | Mathlib_Tactic_NormNum_Ineq |
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
f : α → ℝ
hf : Integrable f
⊢ SignedMeasure.rnDeriv (VectorMeasure.trim (Measure.withDensityᵥ μ f) hm) (Measure.trim μ hm) =ᵐ[μ] μ[f|m] | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | refine' ae_eq_condexp_of_forall_set_integral_eq hm hf _ _ _ | theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
(hf : Integrable f μ) :
SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] := by
| Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.40_0.pyZGtJVYgCCwDLj | theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
(hf : Integrable f μ) :
SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
case refine'_1
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
f : α → ℝ
hf : Integrable f
⊢ ∀ (s : Set α),
MeasurableSet s →
↑↑μ s < ⊤ →
IntegrableOn (SignedMeasure.rnDeriv (VectorMeasure.trim (Measure.withDensityᵥ μ f) hm) (Measure.trim μ hm)) s | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | exact fun _ _ _ => (integrable_of_integrable_trim hm
(SignedMeasure.integrable_rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm))).integrableOn | theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
(hf : Integrable f μ) :
SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] := by
refine' ae_eq_condexp_of_forall_set_integral_eq hm hf _ _ _
· | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.40_0.pyZGtJVYgCCwDLj | theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
(hf : Integrable f μ) :
SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
case refine'_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
f : α → ℝ
hf : Integrable f
⊢ ∀ (s : Set α),
MeasurableSet s →
↑↑μ s < ⊤ →
∫ (x : α) in s,
SignedMeasure.rnDeriv (VectorMeasure.trim (Measure.withDensityᵥ μ f) hm) (Measure.... | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | intro s hs _ | theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
(hf : Integrable f μ) :
SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] := by
refine' ae_eq_condexp_of_forall_set_integral_eq hm hf _ _ _
· exact fun _ _ _ => (integrable_of_integrable_trim hm... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.40_0.pyZGtJVYgCCwDLj | theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
(hf : Integrable f μ) :
SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
case refine'_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
f : α → ℝ
hf : Integrable f
s : Set α
hs : MeasurableSet s
a✝ : ↑↑μ s < ⊤
⊢ ∫ (x : α) in s, SignedMeasure.rnDeriv (VectorMeasure.trim (Measure.withDensityᵥ μ f) hm) (Measure.trim μ hm) x ∂μ =
∫ (x : ... | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | conv_rhs => rw [← hf.withDensityᵥ_trim_eq_integral hm hs,
← SignedMeasure.withDensityᵥ_rnDeriv_eq ((μ.withDensityᵥ f).trim hm) (μ.trim hm)
(hf.withDensityᵥ_trim_absolutelyContinuous hm)] | theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
(hf : Integrable f μ) :
SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] := by
refine' ae_eq_condexp_of_forall_set_integral_eq hm hf _ _ _
· exact fun _ _ _ => (integrable_of_integrable_trim hm... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.40_0.pyZGtJVYgCCwDLj | theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
(hf : Integrable f μ) :
SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
f : α → ℝ
hf : Integrable f
s : Set α
hs : MeasurableSet s
a✝ : ↑↑μ s < ⊤
| ∫ (x : α) in s, f x ∂μ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | rw [← hf.withDensityᵥ_trim_eq_integral hm hs,
← SignedMeasure.withDensityᵥ_rnDeriv_eq ((μ.withDensityᵥ f).trim hm) (μ.trim hm)
(hf.withDensityᵥ_trim_absolutelyContinuous hm)] | theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
(hf : Integrable f μ) :
SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] := by
refine' ae_eq_condexp_of_forall_set_integral_eq hm hf _ _ _
· exact fun _ _ _ => (integrable_of_integrable_trim hm... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.40_0.pyZGtJVYgCCwDLj | theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
(hf : Integrable f μ) :
SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
f : α → ℝ
hf : Integrable f
s : Set α
hs : MeasurableSet s
a✝ : ↑↑μ s < ⊤
| ∫ (x : α) in s, f x ∂μ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | rw [← hf.withDensityᵥ_trim_eq_integral hm hs,
← SignedMeasure.withDensityᵥ_rnDeriv_eq ((μ.withDensityᵥ f).trim hm) (μ.trim hm)
(hf.withDensityᵥ_trim_absolutelyContinuous hm)] | theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
(hf : Integrable f μ) :
SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] := by
refine' ae_eq_condexp_of_forall_set_integral_eq hm hf _ _ _
· exact fun _ _ _ => (integrable_of_integrable_trim hm... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.40_0.pyZGtJVYgCCwDLj | theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
(hf : Integrable f μ) :
SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
f : α → ℝ
hf : Integrable f
s : Set α
hs : MeasurableSet s
a✝ : ↑↑μ s < ⊤
| ∫ (x : α) in s, f x ∂μ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | rw [← hf.withDensityᵥ_trim_eq_integral hm hs,
← SignedMeasure.withDensityᵥ_rnDeriv_eq ((μ.withDensityᵥ f).trim hm) (μ.trim hm)
(hf.withDensityᵥ_trim_absolutelyContinuous hm)] | theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
(hf : Integrable f μ) :
SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] := by
refine' ae_eq_condexp_of_forall_set_integral_eq hm hf _ _ _
· exact fun _ _ _ => (integrable_of_integrable_trim hm... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.40_0.pyZGtJVYgCCwDLj | theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
(hf : Integrable f μ) :
SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
case refine'_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
f : α → ℝ
hf : Integrable f
s : Set α
hs : MeasurableSet s
a✝ : ↑↑μ s < ⊤
⊢ ∫ (x : α) in s, SignedMeasure.rnDeriv (VectorMeasure.trim (Measure.withDensityᵥ μ f) hm) (Measure.trim μ hm) x ∂μ =
↑(Measu... | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | rw [withDensityᵥ_apply
(SignedMeasure.integrable_rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm)) hs,
← set_integral_trim hm _ hs] | theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
(hf : Integrable f μ) :
SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] := by
refine' ae_eq_condexp_of_forall_set_integral_eq hm hf _ _ _
· exact fun _ _ _ => (integrable_of_integrable_trim hm... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.40_0.pyZGtJVYgCCwDLj | theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
(hf : Integrable f μ) :
SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
f : α → ℝ
hf : Integrable f
s : Set α
hs : MeasurableSet s
a✝ : ↑↑μ s < ⊤
⊢ StronglyMeasurable fun x =>
SignedMeasure.rnDeriv (VectorMeasure.trim (Measure.withDensityᵥ μ f) hm) (Measure.trim μ hm) x | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | exact (SignedMeasure.measurable_rnDeriv _ _).stronglyMeasurable | theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
(hf : Integrable f μ) :
SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] := by
refine' ae_eq_condexp_of_forall_set_integral_eq hm hf _ _ _
· exact fun _ _ _ => (integrable_of_integrable_trim hm... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.40_0.pyZGtJVYgCCwDLj | theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
(hf : Integrable f μ) :
SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
case refine'_3
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
f : α → ℝ
hf : Integrable f
⊢ AEStronglyMeasurable' m (SignedMeasure.rnDeriv (VectorMeasure.trim (Measure.withDensityᵥ μ f) hm) (Measure.trim μ hm))
μ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | exact StronglyMeasurable.aeStronglyMeasurable'
(SignedMeasure.measurable_rnDeriv _ _).stronglyMeasurable | theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
(hf : Integrable f μ) :
SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] := by
refine' ae_eq_condexp_of_forall_set_integral_eq hm hf _ _ _
· exact fun _ _ _ => (integrable_of_integrable_trim hm... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.40_0.pyZGtJVYgCCwDLj | theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
(hf : Integrable f μ) :
SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
⊢ snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | by_cases hf : Integrable f μ | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
| Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f
⊢ snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ
case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : ¬Integrable f
⊢ snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | swap | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
by_cases hf : Integrable f μ
| Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : ¬Integrable f
⊢ snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | rw [condexp_undef hf, snorm_zero] | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
by_cases hf : Integrable f μ
swap; · | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : ¬Integrable f
⊢ 0 ≤ snorm f 1 μ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | exact zero_le _ | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
by_cases hf : Integrable f μ
swap; · rw [condexp_undef hf, snorm_zero]; | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f
⊢ snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | by_cases hm : m ≤ m0 | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
by_cases hf : Integrable f μ
swap; · rw [condexp_undef hf, snorm_zero]; exact zero_le _
| Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f
hm : m ≤ m0
⊢ snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ
case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f
hm : ¬m ≤ m0
⊢ snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | swap | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
by_cases hf : Integrable f μ
swap; · rw [condexp_undef hf, snorm_zero]; exact zero_le _
by_cases hm : m ≤ m0
| Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f
hm : ¬m ≤ m0
⊢ snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | rw [condexp_of_not_le hm, snorm_zero] | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
by_cases hf : Integrable f μ
swap; · rw [condexp_undef hf, snorm_zero]; exact zero_le _
by_cases hm : m ≤ m0
swap; · | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f
hm : ¬m ≤ m0
⊢ 0 ≤ snorm f 1 μ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | exact zero_le _ | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
by_cases hf : Integrable f μ
swap; · rw [condexp_undef hf, snorm_zero]; exact zero_le _
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm, snorm_zero]; | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f
hm : m ≤ m0
⊢ snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | by_cases hsig : SigmaFinite (μ.trim hm) | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
by_cases hf : Integrable f μ
swap; · rw [condexp_undef hf, snorm_zero]; exact zero_le _
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm, snorm_zero]; exact zero_le _
| Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f
hm : m ≤ m0
hsig : SigmaFinite (Measure.trim μ hm)
⊢ snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ
case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f
hm : m ≤ m0
hsig : ¬SigmaFinite (Measure.trim μ hm)
⊢ sn... | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | swap | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
by_cases hf : Integrable f μ
swap; · rw [condexp_undef hf, snorm_zero]; exact zero_le _
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm, snorm_zero]; exact zero_le _
by_cases hsig : SigmaFinite (μ.trim hm)
| Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f
hm : m ≤ m0
hsig : ¬SigmaFinite (Measure.trim μ hm)
⊢ snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | rw [condexp_of_not_sigmaFinite hm hsig, snorm_zero] | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
by_cases hf : Integrable f μ
swap; · rw [condexp_undef hf, snorm_zero]; exact zero_le _
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm, snorm_zero]; exact zero_le _
by_cases hsig : SigmaFinite (μ.trim hm)
swap; · | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f
hm : m ≤ m0
hsig : ¬SigmaFinite (Measure.trim μ hm)
⊢ 0 ≤ snorm f 1 μ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | exact zero_le _ | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
by_cases hf : Integrable f μ
swap; · rw [condexp_undef hf, snorm_zero]; exact zero_le _
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm, snorm_zero]; exact zero_le _
by_cases hsig : SigmaFinite (μ.trim hm)
swap; · ... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f
hm : m ≤ m0
hsig : SigmaFinite (Measure.trim μ hm)
⊢ snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | calc
snorm (μ[f|m]) 1 μ ≤ snorm (μ[(|f|)|m]) 1 μ := by
refine' snorm_mono_ae _
filter_upwards [condexp_mono hf hf.abs
(ae_of_all μ (fun x => le_abs_self (f x) : ∀ x, f x ≤ |f x|)),
EventuallyLE.trans (condexp_neg f).symm.le
(condexp_mono hf.neg hf.abs
(ae_of_all μ (fu... | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
by_cases hf : Integrable f μ
swap; · rw [condexp_undef hf, snorm_zero]; exact zero_le _
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm, snorm_zero]; exact zero_le _
by_cases hsig : SigmaFinite (μ.trim hm)
swap; · ... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f
hm : m ≤ m0
hsig : SigmaFinite (Measure.trim μ hm)
⊢ snorm (μ[f|m]) 1 μ ≤ snorm (μ[|f||m]) 1 μ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | refine' snorm_mono_ae _ | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
by_cases hf : Integrable f μ
swap; · rw [condexp_undef hf, snorm_zero]; exact zero_le _
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm, snorm_zero]; exact zero_le _
by_cases hsig : SigmaFinite (μ.trim hm)
swap; · ... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f
hm : m ≤ m0
hsig : SigmaFinite (Measure.trim μ hm)
⊢ ∀ᵐ (x : α) ∂μ, ‖(μ[f|m]) x‖ ≤ ‖(μ[|f||m]) x‖ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | filter_upwards [condexp_mono hf hf.abs
(ae_of_all μ (fun x => le_abs_self (f x) : ∀ x, f x ≤ |f x|)),
EventuallyLE.trans (condexp_neg f).symm.le
(condexp_mono hf.neg hf.abs
(ae_of_all μ (fun x => neg_le_abs_self (f x): ∀ x, -f x ≤ |f x|)))] with x hx₁ hx₂ | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
by_cases hf : Integrable f μ
swap; · rw [condexp_undef hf, snorm_zero]; exact zero_le _
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm, snorm_zero]; exact zero_le _
by_cases hsig : SigmaFinite (μ.trim hm)
swap; · ... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
case h
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f
hm : m ≤ m0
hsig : SigmaFinite (Measure.trim μ hm)
x : α
hx₁ : (μ[f|?m.6495]) x ≤ (μ[fun a => |f a||?m.6495]) x
hx₂ : (-μ[f|?m.5997]) x ≤ (μ[fun a => |f a||?m.5997]) x
⊢ ‖(μ[f|m]) x‖ ≤ ‖(μ[|f||m]) x‖ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | exact abs_le_abs hx₁ hx₂ | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
by_cases hf : Integrable f μ
swap; · rw [condexp_undef hf, snorm_zero]; exact zero_le _
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm, snorm_zero]; exact zero_le _
by_cases hsig : SigmaFinite (μ.trim hm)
swap; · ... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f
hm : m ≤ m0
hsig : SigmaFinite (Measure.trim μ hm)
⊢ snorm (μ[|f||m]) 1 μ = snorm f 1 μ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | rw [snorm_one_eq_lintegral_nnnorm, snorm_one_eq_lintegral_nnnorm, ←
ENNReal.toReal_eq_toReal (ne_of_lt integrable_condexp.2) (ne_of_lt hf.2), ←
integral_norm_eq_lintegral_nnnorm
(stronglyMeasurable_condexp.mono hm).aestronglyMeasurable,
← integral_norm_eq_lintegral_nnnorm hf.1] | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
by_cases hf : Integrable f μ
swap; · rw [condexp_undef hf, snorm_zero]; exact zero_le _
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm, snorm_zero]; exact zero_le _
by_cases hsig : SigmaFinite (μ.trim hm)
swap; · ... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f
hm : m ≤ m0
hsig : SigmaFinite (Measure.trim μ hm)
⊢ ∫ (x : α), ‖(μ[|f||m]) x‖ ∂μ = ∫ (x : α), ‖f x‖ ∂μ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | simp_rw [Real.norm_eq_abs] | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
by_cases hf : Integrable f μ
swap; · rw [condexp_undef hf, snorm_zero]; exact zero_le _
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm, snorm_zero]; exact zero_le _
by_cases hsig : SigmaFinite (μ.trim hm)
swap; · ... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f
hm : m ≤ m0
hsig : SigmaFinite (Measure.trim μ hm)
⊢ ∫ (x : α), |(μ[|f||m]) x| ∂μ = ∫ (x : α), |f x| ∂μ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | rw [← integral_condexp hm hf.abs] | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
by_cases hf : Integrable f μ
swap; · rw [condexp_undef hf, snorm_zero]; exact zero_le _
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm, snorm_zero]; exact zero_le _
by_cases hsig : SigmaFinite (μ.trim hm)
swap; · ... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f
hm : m ≤ m0
hsig : SigmaFinite (Measure.trim μ hm)
⊢ ∫ (x : α), |(μ[|f||m]) x| ∂μ = ∫ (x : α), (μ[fun a => |f a||m]) x ∂μ | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | refine' integral_congr_ae _ | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
by_cases hf : Integrable f μ
swap; · rw [condexp_undef hf, snorm_zero]; exact zero_le _
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm, snorm_zero]; exact zero_le _
by_cases hsig : SigmaFinite (μ.trim hm)
swap; · ... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f
hm : m ≤ m0
hsig : SigmaFinite (Measure.trim μ hm)
⊢ (fun x => |(μ[|f||m]) x|) =ᵐ[μ] fun x => (μ[fun a => |f a||m]) x | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | have : 0 ≤ᵐ[μ] μ[(|f|)|m] := by
rw [← condexp_zero]
exact condexp_mono (integrable_zero _ _ _) hf.abs
(ae_of_all μ (fun x => abs_nonneg (f x) : ∀ x, 0 ≤ |f x|)) | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
by_cases hf : Integrable f μ
swap; · rw [condexp_undef hf, snorm_zero]; exact zero_le _
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm, snorm_zero]; exact zero_le _
by_cases hsig : SigmaFinite (μ.trim hm)
swap; · ... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f
hm : m ≤ m0
hsig : SigmaFinite (Measure.trim μ hm)
⊢ 0 ≤ᵐ[μ] μ[|f||m] | /-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | rw [← condexp_zero] | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ := by
by_cases hf : Integrable f μ
swap; · rw [condexp_undef hf, snorm_zero]; exact zero_le _
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm, snorm_zero]; exact zero_le _
by_cases hsig : SigmaFinite (μ.trim hm)
swap; · ... | Mathlib.MeasureTheory.Function.ConditionalExpectation.Real.60_0.pyZGtJVYgCCwDLj | theorem snorm_one_condexp_le_snorm (f : α → ℝ) : snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ | Mathlib_MeasureTheory_Function_ConditionalExpectation_Real |
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