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chore: forward-port leanprover-community/mathlib#18854 (#4840)
This forward-ports all the files from leanprover-community/mathlib#18854 which have already been ported, and it also ports the new file `algebra.star.order`, which is a split from `algebra.star.basic` and was necessary to do at the same time. Co-authored-by: Chris Hughes <chrishughes24@gmail.com>
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| Original file line number | Diff line number | Diff line change |
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| /- | ||
| Copyright (c) 2023 Scott Morrison. All rights reserved. | ||
| Released under Apache 2.0 license as described in the file LICENSE. | ||
| Authors: Scott Morrison | ||
| ! This file was ported from Lean 3 source module algebra.star.order | ||
| ! leanprover-community/mathlib commit 31c24aa72e7b3e5ed97a8412470e904f82b81004 | ||
| ! Please do not edit these lines, except to modify the commit id | ||
| ! if you have ported upstream changes. | ||
| -/ | ||
| import Mathlib.Algebra.Star.Basic | ||
| import Mathlib.GroupTheory.Submonoid.Basic | ||
|
|
||
| /-! # Star ordered rings | ||
| We define the class `StarOrderedRing R`, which says that the order on `R` respects the | ||
| star operation, i.e. an element `r` is nonnegative iff it is in the `AddSubmonoid` generated by | ||
| elements of the form `star s * s`. In many cases, including all C⋆-algebras, this can be reduced to | ||
| `0 ≤ r ↔ ∃ s, r = star s * s`. However, this generality is slightly more convenient (e.g., it | ||
| allows us to register a `StarOrderedRing` instance for `ℚ`), and more closely resembles the | ||
| literature (see the seminal paper [*The positive cone in Banach algebras*][kelleyVaught1953]) | ||
| In order to accodomate `NonUnitalSemiring R`, we actually don't characterize nonnegativity, but | ||
| rather the entire `≤` relation with `StarOrderedRing.le_iff`. However, notice that when `R` is a | ||
| `NonUnitalRing`, these are equivalent (see `StarOrderedRing.nonneg_iff` and | ||
| `StarOrderedRing.ofNonnegIff`). | ||
| ## TODO | ||
| * In a Banach star algebra without a well-defined square root, the natural ordering is given by the | ||
| positive cone which is the _closure_ of the sums of elements `star r * r`. A weaker version of | ||
| `StarOrderedRing` could be defined for this case (again, see | ||
| [*The positive cone in Banach algebras*][kelleyVaught1953]). Note that the current definition has | ||
| the advantage of not requiring a topology. | ||
| -/ | ||
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| universe u | ||
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| variable {R : Type u} | ||
|
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||
| /-- An ordered `*`-ring is a ring which is both an `OrderedAddCommGroup` and a `*`-ring, | ||
| and the nonnegative elements constitute precisely the `AddSubmonoid` generated by | ||
| elements of the form `star s * s`. | ||
| If you are working with a `NonUnitalRing` and not a `NonUnitalSemiring`, it may be more | ||
| convenient to declare instances using `StarOrderedRing.ofNonnegIff'`. -/ | ||
| class StarOrderedRing (R : Type u) [NonUnitalSemiring R] [PartialOrder R] extends StarRing R where | ||
| /-- addition commutes with `≤` -/ | ||
| add_le_add_left : ∀ a b : R, a ≤ b → ∀ c : R, c + a ≤ c + b | ||
| /-- characterization of the order in terms of the `StarRing` structure. -/ | ||
| le_iff : | ||
| ∀ x y : R, x ≤ y ↔ ∃ p, p ∈ AddSubmonoid.closure (Set.range fun s => star s * s) ∧ y = x + p | ||
| #align star_ordered_ring StarOrderedRing | ||
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||
| namespace StarOrderedRing | ||
|
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||
| -- see note [lower instance priority] | ||
| instance (priority := 100) toOrderedAddCommMonoid [NonUnitalSemiring R] [PartialOrder R] | ||
| [StarOrderedRing R] : OrderedAddCommMonoid R := | ||
| { show NonUnitalSemiring R by infer_instance, show PartialOrder R by infer_instance, | ||
| show StarOrderedRing R by infer_instance with } | ||
| #align star_ordered_ring.to_ordered_add_comm_monoid StarOrderedRing.toOrderedAddCommMonoid | ||
|
|
||
| -- see note [lower instance priority] | ||
| instance (priority := 100) toExistsAddOfLE [NonUnitalSemiring R] [PartialOrder R] | ||
| [StarOrderedRing R] : ExistsAddOfLE R where | ||
| exists_add_of_le h := | ||
| match (le_iff _ _).mp h with | ||
| | ⟨p, _, hp⟩ => ⟨p, hp⟩ | ||
| #align star_ordered_ring.to_has_exists_add_of_le StarOrderedRing.toExistsAddOfLE | ||
|
|
||
| -- see note [lower instance priority] | ||
| instance (priority := 100) toOrderedAddCommGroup [NonUnitalRing R] [PartialOrder R] | ||
| [StarOrderedRing R] : OrderedAddCommGroup R := | ||
| { show NonUnitalRing R by infer_instance, show PartialOrder R by infer_instance, | ||
| show StarOrderedRing R by infer_instance with } | ||
| #align star_ordered_ring.to_ordered_add_comm_group StarOrderedRing.toOrderedAddCommGroup | ||
|
|
||
| -- set note [reducible non-instances] | ||
| /-- To construct a `StarOrderedRing` instance it suffices to show that `x ≤ y` if and only if | ||
| `y = x + star s * s` for some `s : R`. | ||
| This is provided for convenience because it holds in some common scenarios (e.g.,`ℝ≥0`, `C(X, ℝ≥0)`) | ||
| and obviates the hassle of `AddSubmonoid.closure_induction` when creating those instances. | ||
| If you are working with a `NonUnitalRing` and not a `NonUnitalSemiring`, see | ||
| `StarOrderedRing.ofNonnegIff` for a more convenient version. -/ | ||
| @[reducible] | ||
| def ofLeIff [NonUnitalSemiring R] [PartialOrder R] [StarRing R] | ||
| (h_add : ∀ {x y : R}, x ≤ y → ∀ z, z + x ≤ z + y) | ||
| (h_le_iff : ∀ x y : R, x ≤ y ↔ ∃ s, y = x + star s * s) : StarOrderedRing R := | ||
| { ‹StarRing R› with | ||
| add_le_add_left := @h_add | ||
| le_iff := fun x y => by | ||
| refine' ⟨fun h => _, _⟩ | ||
| · obtain ⟨p, hp⟩ := (h_le_iff x y).mp h | ||
| exact ⟨star p * p, AddSubmonoid.subset_closure ⟨p, rfl⟩, hp⟩ | ||
| · rintro ⟨p, hp, hpxy⟩ | ||
| revert x y hpxy | ||
| refine' AddSubmonoid.closure_induction hp _ (fun x y h => add_zero x ▸ h.ge) _ | ||
| · rintro _ ⟨s, rfl⟩ x y rfl | ||
| nth_rw 1 [← add_zero x] | ||
| refine' h_add _ x | ||
| exact (h_le_iff _ _).mpr ⟨s, by rw [zero_add]⟩ | ||
| · rintro a b ha hb x y rfl | ||
| nth_rw 1 [← add_zero x] | ||
| refine' h_add ((ha 0 _ (zero_add a).symm).trans (hb a _ rfl)) x } | ||
| #align star_ordered_ring.of_le_iff StarOrderedRing.ofLeIff | ||
|
|
||
| -- set note [reducible non-instances] | ||
| /-- When `R` is a non-unital ring, to construct a `StarOrderedRing` instance it suffices to | ||
| show that the nonnegative elements are precisely those elements in the `AddSubmonoid` generated | ||
| by `star s * s` for `s : R`. -/ | ||
| @[reducible] | ||
| def ofNonnegIff [NonUnitalRing R] [PartialOrder R] [StarRing R] | ||
| (h_add : ∀ {x y : R}, x ≤ y → ∀ z, z + x ≤ z + y) | ||
| (h_nonneg_iff : ∀ x : R, 0 ≤ x ↔ x ∈ AddSubmonoid.closure (Set.range fun s : R => star s * s)) : | ||
| StarOrderedRing R := | ||
| { ‹StarRing R› with | ||
| add_le_add_left := @h_add | ||
| le_iff := fun x y => by | ||
| haveI : CovariantClass R R (· + ·) (· ≤ ·) := ⟨fun _ _ _ h => h_add h _⟩ | ||
| simpa only [← sub_eq_iff_eq_add', sub_nonneg, exists_eq_right'] using h_nonneg_iff (y - x) } | ||
| #align star_ordered_ring.of_nonneg_iff StarOrderedRing.ofNonnegIff | ||
|
|
||
| -- set note [reducible non-instances] | ||
| /-- When `R` is a non-unital ring, to construct a `StarOrderedRing` instance it suffices to | ||
| show that the nonnegative elements are precisely those elements of the form `star s * s` | ||
| for `s : R`. | ||
| This is provided for convenience because it holds in many common scenarios (e.g.,`ℝ`, `ℂ`, or | ||
| any C⋆-algebra), and obviates the hassle of `AddSubmonoid.closure_induction` when creating those | ||
| instances. -/ | ||
| @[reducible] | ||
| def ofNonnegIff' [NonUnitalRing R] [PartialOrder R] [StarRing R] | ||
| (h_add : ∀ {x y : R}, x ≤ y → ∀ z, z + x ≤ z + y) | ||
| (h_nonneg_iff : ∀ x : R, 0 ≤ x ↔ ∃ s, x = star s * s) : StarOrderedRing R := | ||
| ofLeIff (@h_add) | ||
| (by | ||
| haveI : CovariantClass R R (· + ·) (· ≤ ·) := ⟨fun _ _ _ h => h_add h _⟩ | ||
| simpa [sub_eq_iff_eq_add', sub_nonneg] using fun x y => h_nonneg_iff (y - x)) | ||
| #align star_ordered_ring.of_nonneg_iff' StarOrderedRing.ofNonnegIff' | ||
|
|
||
| theorem nonneg_iff [NonUnitalSemiring R] [PartialOrder R] [StarOrderedRing R] {x : R} : | ||
| 0 ≤ x ↔ x ∈ AddSubmonoid.closure (Set.range fun s : R => star s * s) := by | ||
| simp only [le_iff, zero_add, exists_eq_right'] | ||
| #align star_ordered_ring.nonneg_iff StarOrderedRing.nonneg_iff | ||
|
|
||
| end StarOrderedRing | ||
|
|
||
| section NonUnitalSemiring | ||
|
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||
| variable [NonUnitalSemiring R] [PartialOrder R] [StarOrderedRing R] | ||
|
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||
| theorem star_mul_self_nonneg (r : R) : 0 ≤ star r * r := | ||
| StarOrderedRing.nonneg_iff.mpr <| AddSubmonoid.subset_closure ⟨r, rfl⟩ | ||
| #align star_mul_self_nonneg star_mul_self_nonneg | ||
|
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||
| theorem star_mul_self_nonneg' (r : R) : 0 ≤ r * star r := by | ||
| simpa only [star_star] using star_mul_self_nonneg (star r) | ||
| #align star_mul_self_nonneg' star_mul_self_nonneg' | ||
|
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||
| theorem conjugate_nonneg {a : R} (ha : 0 ≤ a) (c : R) : 0 ≤ star c * a * c := by | ||
| rw [StarOrderedRing.nonneg_iff] at ha | ||
| refine' AddSubmonoid.closure_induction ha (fun x hx => _) | ||
| (by rw [MulZeroClass.mul_zero, MulZeroClass.zero_mul]) fun x y hx hy => _ | ||
| · obtain ⟨x, rfl⟩ := hx | ||
| convert star_mul_self_nonneg (x * c) using 1 | ||
| rw [star_mul, ← mul_assoc, mul_assoc _ _ c] | ||
| · calc | ||
| 0 ≤ star c * x * c + 0 := by rw [add_zero]; exact hx | ||
| _ ≤ star c * x * c + star c * y * c := add_le_add_left hy _ | ||
| _ ≤ _ := by rw [mul_add, add_mul] | ||
| #align conjugate_nonneg conjugate_nonneg | ||
|
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| theorem conjugate_nonneg' {a : R} (ha : 0 ≤ a) (c : R) : 0 ≤ c * a * star c := by | ||
| simpa only [star_star] using conjugate_nonneg ha (star c) | ||
| #align conjugate_nonneg' conjugate_nonneg' | ||
|
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| theorem conjugate_le_conjugate {a b : R} (hab : a ≤ b) (c : R) : | ||
| star c * a * c ≤ star c * b * c := by | ||
| rw [StarOrderedRing.le_iff] at hab ⊢ | ||
| obtain ⟨p, hp, rfl⟩ := hab | ||
| simp_rw [← StarOrderedRing.nonneg_iff] at hp ⊢ | ||
| exact ⟨star c * p * c, conjugate_nonneg hp c, by simp only [add_mul, mul_add]⟩ | ||
| #align conjugate_le_conjugate conjugate_le_conjugate | ||
|
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| theorem conjugate_le_conjugate' {a b : R} (hab : a ≤ b) (c : R) : c * a * star c ≤ c * b * star c := | ||
| by simpa only [star_star] using conjugate_le_conjugate hab (star c) | ||
| #align conjugate_le_conjugate' conjugate_le_conjugate' | ||
|
|
||
| end NonUnitalSemiring |
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