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 pos
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
this : ∀ i ∈ s, f i * g i = 0
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | have : ∑ i in s, f i * g i = ∑ i in s, 0 := sum_congr rfl this | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case pos
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
this✝ : ∀ i ∈ s, f i * g i = 0
this : ∑ i in s, f i * g i = ∑ i in s, 0
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | simp [this] | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : ¬((∑ i in s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i in s, g i ^ q) ^ (1 / q) = 0)
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | push_neg at H | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤ | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case pos
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | cases' H' with H' H' | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case pos.inl
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H] | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case pos.inr
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∑ i in s, g i ^ q) ^ (1 / q) = ⊤
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | simp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H] | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : ¬((∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤)
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤ | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case H'
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : ¬((∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ q) ^ (1 / q) = ⊤)
⊢ (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤ | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | simpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos,
ENNReal.sum_eq_top_iff, not_or] using H' | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
⊢ ∑ i in s, f i * g i ≤ (∑ i in s, f i ^ p) ^ (1 / p) * (∑ i in s, g i ^ q) ^ (1 / q) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | have :=
ENNReal.coe_le_coe.2
(@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
(fun i => ENNReal.toNNReal (g i)) _ _ hpq) | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
↑(∑ i in s, (fun i => ENNReal.toNNReal (f i)) i * (fun i => ENNReal.toNNReal (g i)) i) ≤
↑((∑ i in ... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f ... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f ... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | convert this using 1 <;> [skip; congr 2] | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f ... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | convert this using 1 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_3
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal ... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | skip | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_4
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal ... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | congr 2 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_3
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal ... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | skip | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_4.e_a.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.t... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | skip | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_4.e_a.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.t... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | simp | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_4.e_a.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.t... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | skip | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_4.e_a.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.t... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | simp | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_3
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal ... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | refine Finset.sum_congr rfl fun i hi => ?_ | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_3
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.toNNReal ... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm] | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_4.e_a.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.t... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | refine Finset.sum_congr rfl fun i hi => ?_ | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_4.e_a.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.t... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm] | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_4.e_a.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.t... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | refine Finset.sum_congr rfl fun i hi => ?_ | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_4.e_a.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : Real.IsConjugateExponent p q
H : (∑ i in s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i in s, g i ^ q) ^ (1 / q) ≠ 0
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
∑ x in s, ↑(ENNReal.toNNReal (f x)) * ↑(ENNReal.toNNReal (g x)) ≤
(∑ x in s, ↑(ENNReal.t... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm] | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib.Analysis.MeanInequalities.739_0.4hD1oATDjTWuML9 | /-- Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. -/
theorem inner_le_Lp_mul_Lq (hpq : p.IsConjugateExponent q) :
∑ i in s, f i * g i ≤ (∑ i in... | Mathlib_Analysis_MeanInequalities |
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
⊢ (∑ i in s, f i) ^ p ≤ ↑(card s) ^ (p - 1) * ∑ i in s, f i ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | cases' eq_or_lt_of_le hp with hp hp | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ... | Mathlib.Analysis.MeanInequalities.768_0.4hD1oATDjTWuML9 | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ... | Mathlib_Analysis_MeanInequalities |
case inl
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp✝ : 1 ≤ p
hp : 1 = p
⊢ (∑ i in s, f i) ^ p ≤ ↑(card s) ^ (p - 1) * ∑ i in s, f i ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | simp [← hp] | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ... | Mathlib.Analysis.MeanInequalities.768_0.4hD1oATDjTWuML9 | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ... | Mathlib_Analysis_MeanInequalities |
case inr
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
⊢ (∑ i in s, f i) ^ p ≤ ↑(card s) ^ (p - 1) * ∑ i in s, f i ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | let q : ℝ := p / (p - 1) | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ... | Mathlib.Analysis.MeanInequalities.768_0.4hD1oATDjTWuML9 | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ... | Mathlib_Analysis_MeanInequalities |
case inr
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q✝ : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
⊢ (∑ i in s, f i) ^ p ≤ ↑(card s) ^ (p - 1) * ∑ i in s, f i ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | have hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp] | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ... | Mathlib.Analysis.MeanInequalities.768_0.4hD1oATDjTWuML9 | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ... | Mathlib_Analysis_MeanInequalities |
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q✝ : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
⊢ Real.IsConjugateExponent p q | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | rw [Real.isConjugateExponent_iff hp] | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ... | Mathlib.Analysis.MeanInequalities.768_0.4hD1oATDjTWuML9 | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ... | Mathlib_Analysis_MeanInequalities |
case inr
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q✝ : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
hpq : Real.IsConjugateExponent p q
⊢ (∑ i in s, f i) ^ p ≤ ↑(card s) ^ (p - 1) * ∑ i in s, f i ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ... | Mathlib.Analysis.MeanInequalities.768_0.4hD1oATDjTWuML9 | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ... | Mathlib_Analysis_MeanInequalities |
case inr
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q✝ : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
hpq : Real.IsConjugateExponent p q
hp₁ : 1 / p * p = 1
⊢ (∑ i in s, f i) ^ p ≤ ↑(card s) ^ (p - 1) * ∑ i in s, f i ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | have hq : 1 / q * p = p - 1 := by
rw [← hpq.div_conj_eq_sub_one]
ring | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ... | Mathlib.Analysis.MeanInequalities.768_0.4hD1oATDjTWuML9 | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ... | Mathlib_Analysis_MeanInequalities |
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q✝ : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
hpq : Real.IsConjugateExponent p q
hp₁ : 1 / p * p = 1
⊢ 1 / q * p = p - 1 | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | rw [← hpq.div_conj_eq_sub_one] | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ... | Mathlib.Analysis.MeanInequalities.768_0.4hD1oATDjTWuML9 | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ... | Mathlib_Analysis_MeanInequalities |
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q✝ : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
hpq : Real.IsConjugateExponent p q
hp₁ : 1 / p * p = 1
⊢ 1 / q * p = p / q | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | ring | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ... | Mathlib.Analysis.MeanInequalities.768_0.4hD1oATDjTWuML9 | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ... | Mathlib_Analysis_MeanInequalities |
case inr
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q✝ : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
hpq : Real.IsConjugateExponent p q
hp₁ : 1 / p * p = 1
hq : 1 / q * p = p - 1
⊢ (∑ i in s, f i) ^ p ≤ ↑(card s) ^ (p - 1) * ∑ i in s, f i ^ p | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using
ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ... | Mathlib.Analysis.MeanInequalities.768_0.4hD1oATDjTWuML9 | /-- For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions.
-/
theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i in s, f i ... | Mathlib_Analysis_MeanInequalities |
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
⊢ (∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | by_cases H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤ | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib_Analysis_MeanInequalities |
case pos
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
⊢ (∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | cases' H' with H' H' | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib_Analysis_MeanInequalities |
case pos.inl
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
H' : (∑ i in s, f i ^ p) ^ (1 / p) = ⊤
⊢ (∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | simp [H', -one_div] | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib_Analysis_MeanInequalities |
case pos.inr
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
H' : (∑ i in s, g i ^ p) ^ (1 / p) = ⊤
⊢ (∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | simp [H', -one_div] | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
H' : ¬((∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤)
⊢ (∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
H' : ¬((∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤)
pos : 0 < p
⊢ (∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤ | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib_Analysis_MeanInequalities |
case H'
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
H' : ¬((∑ i in s, f i ^ p) ^ (1 / p) = ⊤ ∨ (∑ i in s, g i ^ p) ^ (1 / p) = ⊤)
pos : 0 < p
⊢ (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤ | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H' | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
⊢ (∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in s, g i ^ p) ^ (1 / p) | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp) | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
↑((∑ i in s, ((fun i => ENNReal.toNNReal (f i)) i + (fun i => ENNReal.toNNReal (g i)) i) ^ p) ^ (1 / p)) ≤
↑((∑ i in s, (fun i => ENNReal.toNNReal (f i)) i ^ p) ^ (1 / p) +
(∑ i... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ p) ^ (1 ... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | convert this using 2 <;> [skip; congr 1; congr 1] | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib_Analysis_MeanInequalities |
case neg
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g x)) ^ p) ^ (1 ... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | convert this using 2 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_3.h.e'_5
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g x)) ... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | skip | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_4.h.e'_5
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g x)) ... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | congr 1 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_4.h.e'_6
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g x)) ... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | congr 1 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_3.h.e'_5
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g x)) ... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | refine Finset.sum_congr rfl fun i hi => ?_ | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_3.h.e'_5
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g x)) ... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | simp [H'.1 i hi, H'.2 i hi] | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_4.h.e'_5.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g ... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | refine Finset.sum_congr rfl fun i hi => ?_ | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_4.h.e'_5.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g ... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | simp [H'.1 i hi, H'.2 i hi] | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_4.h.e'_6.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g ... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | refine Finset.sum_congr rfl fun i hi => ?_ | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib_Analysis_MeanInequalities |
case h.e'_4.h.e'_6.e_a
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hp : 1 ≤ p
pos : 0 < p
H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤
this :
(∑ x in s, (↑(ENNReal.toNNReal (f x)) + ↑(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) ≤
(∑ x in s, ↑(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (∑ x in s, ↑(ENNReal.toNNReal (g ... | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | simp [H'.1 i hi, H'.2 i hi] | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib.Analysis.MeanInequalities.786_0.4hD1oATDjTWuML9 | /-- Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. -/
theorem Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, f i ^ p) ^ (1 / p) + (∑ i in... | Mathlib_Analysis_MeanInequalities |
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p✝ q✝ p q : R
hp : p ≠ 0
⊢ p / GCDMonoid.gcd p q ≠ 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Prin... | obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by
| Mathlib.RingTheory.EuclideanDomain.42_0.j84WZGwHDjQhSAS | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 | Mathlib_RingTheory_EuclideanDomain |
case intro
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p✝ q✝ p q : R
hp : p ≠ 0
r : R
hr : p = GCDMonoid.gcd p q * r
⊢ p / GCDMonoid.gcd p q ≠ 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Prin... | obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp) | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q
| Mathlib.RingTheory.EuclideanDomain.42_0.j84WZGwHDjQhSAS | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 | Mathlib_RingTheory_EuclideanDomain |
case intro.intro
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p✝ q✝ p q : R
hp : p ≠ 0
r : R
hr : p = GCDMonoid.gcd p q * r
pq0 : GCDMonoid.gcd p q ≠ 0
r0 : r ≠ 0
⊢ p / GCDMonoid.gcd p q ≠ 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Prin... | nth_rw 1 [hr] | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp)
| Mathlib.RingTheory.EuclideanDomain.42_0.j84WZGwHDjQhSAS | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 | Mathlib_RingTheory_EuclideanDomain |
case intro.intro
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p✝ q✝ p q : R
hp : p ≠ 0
r : R
hr : p = GCDMonoid.gcd p q * r
pq0 : GCDMonoid.gcd p q ≠ 0
r0 : r ≠ 0
⊢ GCDMonoid.gcd p q * r / GCDMonoid.gcd p q ≠ 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Prin... | rw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0] | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp)
nth_rw 1 [hr]
| Mathlib.RingTheory.EuclideanDomain.42_0.j84WZGwHDjQhSAS | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 | Mathlib_RingTheory_EuclideanDomain |
case intro.intro
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p✝ q✝ p q : R
hp : p ≠ 0
r : R
hr : p = GCDMonoid.gcd p q * r
pq0 : GCDMonoid.gcd p q ≠ 0
r0 : r ≠ 0
⊢ r ≠ 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Prin... | exact r0 | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp)
nth_rw 1 [hr]
rw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0]
| Mathlib.RingTheory.EuclideanDomain.42_0.j84WZGwHDjQhSAS | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 | Mathlib_RingTheory_EuclideanDomain |
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p✝ q✝ p q : R
hq : q ≠ 0
⊢ q / GCDMonoid.gcd p q ≠ 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Prin... | obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_right p q | theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 := by
| Mathlib.RingTheory.EuclideanDomain.50_0.j84WZGwHDjQhSAS | theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 | Mathlib_RingTheory_EuclideanDomain |
case intro
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p✝ q✝ p q : R
hq : q ≠ 0
r : R
hr : q = GCDMonoid.gcd p q * r
⊢ q / GCDMonoid.gcd p q ≠ 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Prin... | obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hq) | theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_right p q
| Mathlib.RingTheory.EuclideanDomain.50_0.j84WZGwHDjQhSAS | theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 | Mathlib_RingTheory_EuclideanDomain |
case intro.intro
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p✝ q✝ p q : R
hq : q ≠ 0
r : R
hr : q = GCDMonoid.gcd p q * r
pq0 : GCDMonoid.gcd p q ≠ 0
r0 : r ≠ 0
⊢ q / GCDMonoid.gcd p q ≠ 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Prin... | nth_rw 1 [hr] | theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_right p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hq)
| Mathlib.RingTheory.EuclideanDomain.50_0.j84WZGwHDjQhSAS | theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 | Mathlib_RingTheory_EuclideanDomain |
case intro.intro
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p✝ q✝ p q : R
hq : q ≠ 0
r : R
hr : q = GCDMonoid.gcd p q * r
pq0 : GCDMonoid.gcd p q ≠ 0
r0 : r ≠ 0
⊢ GCDMonoid.gcd p q * r / GCDMonoid.gcd p q ≠ 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Prin... | rw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0] | theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_right p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hq)
nth_rw 1 [hr]
| Mathlib.RingTheory.EuclideanDomain.50_0.j84WZGwHDjQhSAS | theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 | Mathlib_RingTheory_EuclideanDomain |
case intro.intro
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p✝ q✝ p q : R
hq : q ≠ 0
r : R
hr : q = GCDMonoid.gcd p q * r
pq0 : GCDMonoid.gcd p q ≠ 0
r0 : r ≠ 0
⊢ r ≠ 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Prin... | exact r0 | theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 := by
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_right p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hq)
nth_rw 1 [hr]
rw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0]
| Mathlib.RingTheory.EuclideanDomain.50_0.j84WZGwHDjQhSAS | theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 | Mathlib_RingTheory_EuclideanDomain |
R : Type ?u.6733
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
a b : R
⊢ Associated (gcd a b * lcm a b) (a * b) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Prin... | rw [EuclideanDomain.gcd_mul_lcm] | /-- Create a `GCDMonoid` whose `GCDMonoid.gcd` matches `EuclideanDomain.gcd`. -/
-- porting note: added `DecidableEq R`
def gcdMonoid (R) [EuclideanDomain R] [DecidableEq R] : GCDMonoid R where
gcd := gcd
lcm := lcm
gcd_dvd_left := gcd_dvd_left
gcd_dvd_right := gcd_dvd_right
dvd_gcd := dvd_gcd
gcd_mul_lcm a... | Mathlib.RingTheory.EuclideanDomain.71_0.j84WZGwHDjQhSAS | /-- Create a `GCDMonoid` whose `GCDMonoid.gcd` matches `EuclideanDomain.gcd`. -/
-- porting note: added `DecidableEq R`
def gcdMonoid (R) [EuclideanDomain R] [DecidableEq R] : GCDMonoid R where
gcd | Mathlib_RingTheory_EuclideanDomain |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
⊢ (aeval f) (charpoly f) = 0 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import line... | apply (LinearEquiv.map_eq_zero_iff (algEquivMatrix (chooseBasis R M)).toLinearEquiv).1 | /-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
| Mathlib.LinearAlgebra.Charpoly.Basic.64_0.6NA9VnT03sJgAKk | /-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
⊢ (AlgEquiv.toLinearEquiv (algEquivMatrix (chooseBasis R M))) ((aeval f) (charpoly f)) = 0 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import line... | rw [AlgEquiv.toLinearEquiv_apply, ← AlgEquiv.coe_algHom, ← Polynomial.aeval_algHom_apply _ _ _,
charpoly_def] | /-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
apply (LinearEquiv.map_eq_zero_iff (algEquivM... | Mathlib.LinearAlgebra.Charpoly.Basic.64_0.6NA9VnT03sJgAKk | /-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
⊢ (aeval (↑(algEquivMatrix (chooseBasis R M)) f)) (Matrix.charpoly ((toMatrix (chooseBasis R M) (chooseBasis R M)) f)) =
0 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import line... | exact Matrix.aeval_self_charpoly _ | /-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 := by
apply (LinearEquiv.map_eq_zero_iff (algEquivM... | Mathlib.LinearAlgebra.Charpoly.Basic.64_0.6NA9VnT03sJgAKk | /-- The **Cayley-Hamilton Theorem**, that the characteristic polynomial of a linear map, applied
to the linear map itself, is zero.
See `Matrix.aeval_self_charpoly` for the equivalent statement about matrices. -/
theorem aeval_self_charpoly : aeval f f.charpoly = 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
k : ℕ
⊢ f ^ k = (aeval f) (X ^ k %ₘ charpoly f) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import line... | rw [← aeval_eq_aeval_mod_charpoly, map_pow, aeval_X] | /-- Any endomorphism power can be computed as the sum of endomorphism powers less than the
dimension of the module. -/
theorem pow_eq_aeval_mod_charpoly (k : ℕ) : f ^ k = aeval f (X ^ k %ₘ f.charpoly) := by
| Mathlib.LinearAlgebra.Charpoly.Basic.90_0.6NA9VnT03sJgAKk | /-- Any endomorphism power can be computed as the sum of endomorphism powers less than the
dimension of the module. -/
theorem pow_eq_aeval_mod_charpoly (k : ℕ) : f ^ k = aeval f (X ^ k %ₘ f.charpoly) | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
⊢ coeff (minpoly R f) 0 ≠ 0 | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import line... | intro h | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
| Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
⊢ False | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import line... | obtain ⟨P, hP⟩ := X_dvd_iff.2 h | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
| Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
case intro
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
⊢ False | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import line... | have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_zero (isIntegral f) hP | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
| Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
⊢ degree P < degree (minpoly R f) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import line... | rw [hP, mul_comm] | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
| Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
⊢ degree P < degree (P * X) | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import line... | refine' degree_lt_degree_mul_X fun h => _ | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
| Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h✝ : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
h : P = 0
⊢ False | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import line... | rw [h, mul_zero] at hP | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
| Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h✝ : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = 0
h : P = 0
⊢ False | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import line... | exact minpoly.ne_zero (isIntegral f) hP | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
| Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
case intro
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
hdegP : degree P < degree (minpoly R f)
⊢ Fal... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import line... | have hPmonic : P.Monic := by
suffices (minpoly R f).Monic by
rwa [Monic.def, hP, mul_comm, leadingCoeff_mul_X, ← Monic.def] at this
exact minpoly.monic (isIntegral f) | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_z... | Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
hdegP : degree P < degree (minpoly R f)
⊢ Monic P | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import line... | suffices (minpoly R f).Monic by
rwa [Monic.def, hP, mul_comm, leadingCoeff_mul_X, ← Monic.def] at this | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_z... | Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
hdegP : degree P < degree (minpoly R f)
this : Monic (mi... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import line... | rwa [Monic.def, hP, mul_comm, leadingCoeff_mul_X, ← Monic.def] at this | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_z... | Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
hdegP : degree P < degree (minpoly R f)
⊢ Monic (minpoly... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import line... | exact minpoly.monic (isIntegral f) | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_z... | Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
case intro
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
hdegP : degree P < degree (minpoly R f)
hPmon... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import line... | have hzero : aeval f (minpoly R f) = 0 := minpoly.aeval _ _ | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_z... | Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
case intro
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
hdegP : degree P < degree (minpoly R f)
hPmon... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import line... | simp only [hP, mul_eq_comp, ext_iff, hf, aeval_X, map_eq_zero_iff, coe_comp, AlgHom.map_mul,
zero_apply, Function.comp_apply] at hzero | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_z... | Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
case intro
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : Nontrivial R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
f : M →ₗ[R] M
hf : Function.Injective ⇑f
h : coeff (minpoly R f) 0 = 0
P : R[X]
hP : minpoly R f = X * P
hdegP : degree P < degree (minpoly R f)
hPmon... | /-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import line... | exact not_le.2 hdegP (minpoly.min _ _ hPmonic (ext hzero)) | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 := by
intro h
obtain ⟨P, hP⟩ := X_dvd_iff.2 h
have hdegP : P.degree < (minpoly R f).degree := by
rw [hP, mul_comm]
refine' degree_lt_degree_mul_X fun h => _
rw [h, mul_zero] at hP
exact minpoly.ne_z... | Mathlib.LinearAlgebra.Charpoly.Basic.98_0.6NA9VnT03sJgAKk | theorem minpoly_coeff_zero_of_injective (hf : Function.Injective f) :
(minpoly R f).coeff 0 ≠ 0 | Mathlib_LinearAlgebra_Charpoly_Basic |
α : Type u
β : Type u_1
w x✝ y✝ z : α
inst✝ : GeneralizedBooleanAlgebra α
x y : α
⊢ x \ y ⊔ x ⊓ y = x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_comm, sup_inf_sdiff] | @[simp]
theorem sup_sdiff_inf (x y : α) : x \ y ⊔ x ⊓ y = x := by | Mathlib.Order.BooleanAlgebra.106_0.ewE75DLNneOU8G5 | @[simp]
theorem sup_sdiff_inf (x y : α) : x \ y ⊔ x ⊓ y = x | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x✝ y✝ z : α
inst✝ : GeneralizedBooleanAlgebra α
x y : α
⊢ x \ y ⊓ (x ⊓ y) = ⊥ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_comm, inf_inf_sdiff] | @[simp]
theorem inf_sdiff_inf (x y : α) : x \ y ⊓ (x ⊓ y) = ⊥ := by | Mathlib.Order.BooleanAlgebra.110_0.ewE75DLNneOU8G5 | @[simp]
theorem inf_sdiff_inf (x y : α) : x \ y ⊓ (x ⊓ y) = ⊥ | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
src✝ : Bot α := toBot
a : α
⊢ ⊥ ≤ a | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← inf_inf_sdiff a a, inf_assoc] | instance (priority := 100) GeneralizedBooleanAlgebra.toOrderBot : OrderBot α :=
{ GeneralizedBooleanAlgebra.toBot with
bot_le := fun a => by
| Mathlib.Order.BooleanAlgebra.115_0.ewE75DLNneOU8G5 | instance (priority | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
src✝ : Bot α := toBot
a : α
⊢ a ⊓ (a ⊓ a \ a) ≤ a | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | exact inf_le_left | instance (priority := 100) GeneralizedBooleanAlgebra.toOrderBot : OrderBot α :=
{ GeneralizedBooleanAlgebra.toBot with
bot_le := fun a => by
rw [← inf_inf_sdiff a a, inf_assoc]
| Mathlib.Order.BooleanAlgebra.115_0.ewE75DLNneOU8G5 | instance (priority | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : x ⊓ y ⊔ z = x
i : x ⊓ y ⊓ z = ⊥
⊢ x \ y = z | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm] | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
| Mathlib.Order.BooleanAlgebra.127_0.ewE75DLNneOU8G5 | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : x ⊓ y ⊔ z = x
i : x ⊓ y ⊓ z = ⊥
| x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← sup_inf_sdiff x y, sup_comm] | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => | Mathlib.Order.BooleanAlgebra.127_0.ewE75DLNneOU8G5 | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : x ⊓ y ⊔ z = x
i : x ⊓ y ⊓ z = ⊥
| x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← sup_inf_sdiff x y, sup_comm] | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => | Mathlib.Order.BooleanAlgebra.127_0.ewE75DLNneOU8G5 | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : x ⊓ y ⊔ z = x
i : x ⊓ y ⊓ z = ⊥
| x | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← sup_inf_sdiff x y, sup_comm] | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => | Mathlib.Order.BooleanAlgebra.127_0.ewE75DLNneOU8G5 | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : x ⊓ y ⊔ z = x \ y ⊔ x ⊓ y
i : x ⊓ y ⊓ z = ⊥
⊢ x \ y = z | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [sup_comm] at s | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]
| Mathlib.Order.BooleanAlgebra.127_0.ewE75DLNneOU8G5 | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : z ⊔ x ⊓ y = x \ y ⊔ x ⊓ y
i : x ⊓ y ⊓ z = ⊥
⊢ x \ y = z | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | conv_rhs at i => rw [← inf_inf_sdiff x y, inf_comm] | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]
rw [sup_comm] at s
| Mathlib.Order.BooleanAlgebra.127_0.ewE75DLNneOU8G5 | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : z ⊔ x ⊓ y = x \ y ⊔ x ⊓ y
i : x ⊓ y ⊓ z = ⊥
| ⊥ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← inf_inf_sdiff x y, inf_comm] | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]
rw [sup_comm] at s
conv_rhs at i => | Mathlib.Order.BooleanAlgebra.127_0.ewE75DLNneOU8G5 | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : z ⊔ x ⊓ y = x \ y ⊔ x ⊓ y
i : x ⊓ y ⊓ z = ⊥
| ⊥ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← inf_inf_sdiff x y, inf_comm] | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]
rw [sup_comm] at s
conv_rhs at i => | Mathlib.Order.BooleanAlgebra.127_0.ewE75DLNneOU8G5 | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : z ⊔ x ⊓ y = x \ y ⊔ x ⊓ y
i : x ⊓ y ⊓ z = ⊥
| ⊥ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [← inf_inf_sdiff x y, inf_comm] | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]
rw [sup_comm] at s
conv_rhs at i => | Mathlib.Order.BooleanAlgebra.127_0.ewE75DLNneOU8G5 | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z | Mathlib_Order_BooleanAlgebra |
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : z ⊔ x ⊓ y = x \ y ⊔ x ⊓ y
i : x ⊓ y ⊓ z = x \ y ⊓ (x ⊓ y)
⊢ x \ y = z | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | rw [inf_comm] at i | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z := by
conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]
rw [sup_comm] at s
conv_rhs at i => rw [← inf_inf_sdiff x y, inf_comm]
| Mathlib.Order.BooleanAlgebra.127_0.ewE75DLNneOU8G5 | theorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z | Mathlib_Order_BooleanAlgebra |
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