[Merged by Bors] - feat: fun x ↦ (x ⊓ z, x ⊔ z) is StrictMono in ModularLattice#17457
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[Merged by Bors] - feat: fun x ↦ (x ⊓ z, x ⊔ z) is StrictMono in ModularLattice#17457alreadydone wants to merge 1 commit intomasterfrom
fun x ↦ (x ⊓ z, x ⊔ z) is StrictMono in ModularLattice#17457alreadydone wants to merge 1 commit intomasterfrom
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PR summary 3d40d6db19Import changes for modified filesNo significant changes to the import graph Import changes for all files
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YaelDillies
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That's very nifty!
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🚀 Pull request has been placed on the maintainer queue by YaelDillies. |
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In the proof of `wellFounded_lt_exact_sequence`, it is implicitly shown that `fun x : α ↦ (x ⊓ z, x ⊔ z)` is StrictMono in a modular lattice, with the lexicographic order on `α × α`. Here we show the stronger result with `α × α` equipped with the product order, and golf the proof. If `G` is not an abelian group, then `α = Subgroup G` is not necessarily a modular lattice. However, if `z` is a normal subgroup, the result still holds true. We prove a closely related result `strictMono_comap_prod_map` which replaces `inf` with `comap` and `sup` with `map`, so it concerns a function to `Subgroup z × Subgroup (G ⧸ z)` instead. The same result for submodules immediately implies that Noetherian/Artinian are closed under extensions, which is proven in #17425 using `wellFounded_lt/gt_exact_sequence`. If `z` is not even normal, the function to `Subgroup z × Set (G ⧸ z)` is still StrictMono.
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fun x ↦ (x ⊓ z, x ⊔ z) is StrictMono in ModularLatticefun x ↦ (x ⊓ z, x ⊔ z) is StrictMono in ModularLattice
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In the proof of
wellFounded_lt_exact_sequence, it is implicitly shown thatfun x : α ↦ (x ⊓ z, x ⊔ z)is StrictMono in a modular lattice, with the lexicographic order onα × α. Here we show the stronger result withα × αequipped with the product order, and golf the proof.If
Gis not an abelian group, thenα = Subgroup Gis not necessarily a modular lattice. However, ifzis a normal subgroup, the result still holds true. We prove a closely related resultstrictMono_comap_prod_mapwhich replacesinfwithcomapandsupwithmap, so it concerns a function toSubgroup z × Subgroup (G ⧸ z)instead. The same result for submodules immediately implies that Noetherian/Artinian are closed under extensions, which is proven in #17425 usingwellFounded_lt/gt_exact_sequence.If
zis not even normal, the function toSubgroup z × Set (G ⧸ z)is still StrictMono.