[Merged by Bors] - feat(algebra/order/nonneg): properties about the nonnegative cone #9598
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| /-- If `Sup ∅ ≤ a` then `{x : α // a ≤ x}` is a `conditionally_complete_linear_order_bot`. -/ | ||
| @[reducible] | ||
| protected noncomputable def conditionally_complete_linear_order_bot | ||
| [conditionally_complete_linear_order α] {a : α} (h : Sup ∅ ≤ a) : | ||
| conditionally_complete_linear_order_bot {x : α // a ≤ x} := | ||
| { cSup_empty := (function.funext_iff.1 | ||
| (@subset_Sup_def α (set.Ici a) _ ⟨⟨a, le_rfl⟩⟩) ∅).trans $ subtype.eq $ | ||
| by { cases h.lt_or_eq with h2 h2, { simp [h2.not_le] }, simp [h2] }, | ||
| ..nonneg.order_bot, | ||
| .. @ord_connected_subset_conditionally_complete_linear_order α (set.Ici a) _ ⟨⟨a, le_rfl⟩⟩ _ } |
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Should this be defined on set.IcI like the semilattice_inf_bot isntance above is?
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I could, but order.lattice_intervals currently doesn't import order.conditionally_complete_lattice...
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) * Provide various classes on the type `{x : α // 0 ≤ x}` where `α` has some order (and algebraic) structure. * Use this to automatically derive the classes on `nnreal`. * We currently do not yet define `conditionally_complete_linear_order_bot nnreal` using nonneg, since that causes some errors (I think Lean then thinks that there are two orders that are not definitionally equal (unfolding only instances)). * We leave a bunch of `example` lines in `nnreal` to show that `nnreal` has all the old classes. These could also be removed, if desired. * We currently only give some instances and simp/norm_cast lemmas. This could be expanded in the future. Co-authored-by: Floris van Doorn <fpv@andrew.cmu.edu> Co-authored-by: Johan Commelin <johan@commelin.net>
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{x : α // 0 ≤ x}whereαhas some order (and algebraic) structure.nnreal.conditionally_complete_linear_order_bot nnrealusing nonneg, since that causes some errors (I think Lean then thinks that there are two orders that are not definitionally equal (unfolding only instances)).examplelines innnrealto show thatnnrealhas all the old classes. These could also be removed, if desired.