[Merged by Bors] - feat(order/galois_connection): formula for upper_bounds (l '' s)
#8478
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* upgrade `galois_connection.upper_bounds_l_image_subset` and `galois_connection.lower_bounds_u_image` to equalities; * prove `bdd_above (l '' s) ↔ bdd_above s` and `bdd_below (u '' s) ↔ bdd_below s`; * move `galois_connection.dual` to the top and use it in some proofs; * use `order_iso.to_galois_connection` to transfer some of these results to `order_iso`s; * rename `rel_iso.to_galois_insertion` to `order_iso.to_galois_insertion`.
eric-wieser
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Jul 31, 2021
src/order/galois_connection.lean
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| @@ -67,6 +67,11 @@ lemma monotone_intro (hu : monotone u) (hl : monotone l) | |||
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| include gc | |||
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| protected lemma dual [pα : preorder α] [pβ : preorder β] | |||
| {l : α → β} {u : β → α} (gc : galois_connection l u) : | |||
| @galois_connection (order_dual β) (order_dual α) _ _ u l := | |||
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Shouldn't l and u be composed with to_dual?
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I moved this lemma without touching the code. I mostly use order_dual for @dual_lemma (order_dual α) _ _ proofs, so I don't care about to_dual. Feel free to add it if you want to. I guess, something like
Suggested change
| @galois_connection (order_dual β) (order_dual α) _ _ u l := | |
| galois_connection (order_dual.to_dual ∘ u ∘ order_dual.of_dual) | |
| (order_dual.to_dual ∘ l ∘ order_dual.of_dual) := |
should work.
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Oh, I missed that this lemma was moved. Yeah, that's basically what I was thinking . You'll need to import order.order_dual, but that seems like a good change to make to me.
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…8478) * upgrade `galois_connection.upper_bounds_l_image_subset` and `galois_connection.lower_bounds_u_image` to equalities; * prove `bdd_above (l '' s) ↔ bdd_above s` and `bdd_below (u '' s) ↔ bdd_below s`; * move `galois_connection.dual` to the top and use it in some proofs; * use `order_iso.to_galois_connection` to transfer some of these results to `order_iso`s; * rename `rel_iso.to_galois_insertion` to `order_iso.to_galois_insertion`.
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upper_bounds (l '' s)upper_bounds (l '' s)
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galois_connection.upper_bounds_l_image_subsetandgalois_connection.lower_bounds_u_imageto equalities;bdd_above (l '' s) ↔ bdd_above sandbdd_below (u '' s) ↔ bdd_below s;galois_connection.dualto the top and use it in some proofs;order_iso.to_galois_connectionto transfer some of theseresults to
order_isos;rel_iso.to_galois_insertiontoorder_iso.to_galois_insertion.