`Fast.Int` Update by LorenzoMolena · Pull Request #1280 · agda/cubical

LorenzoMolena · 2025-12-16T14:44:43Z

See #1276.
I have decided to change the organization of the modules by introducing a common Fast directory, hosting the fast Int modules and, in the future, fast rationals as well. The same change has been made for instances of algebraic structures.
In my opinion, this organization seems conceptually clearer.
For example, in #1276 the instances of fast integers as AbGroup are in Cubical.Algebra.AbGroup.Instances.Int.Fast, with the file named Fast, whereas in this PR we have Cubical.Algebra.AbGroup.Instances.Fast.Int, with the file named Int.

Moreover, since in these PRs they are consistently referred to as "fast integers", having them under Fast.Int appears more natural than the current Int.Fast.

Finally, the only fast Int module already merged is the Base file, so this reorganization is minimal.

Update

After discussing with @marcinjangrzybowski, we agreed to adopt a different notion of order than the one currently used in the draft of #1276, and to update this PR accordingly, including the order module together with the related order instances.
In particular, I am not using Marcin’s solver for fast integers, since it is not required for this part of the development. There will still be a separate PR for the solver, which will be especially useful for the development of Fast Rationals, where it can help avoid long chains of equational reasoning.

This implementation of the order is inspired by Agda’s standard library and 1Lab, but relies on _<ᵗ_ from Cubical.Data.Nat.Order.Inductive rather than the order from Cubical.Data.Nat.Order (which is defined as a sigma type).

The main advantages are that implicit arguments are easier to infer and that the definitions reduce efficiently on concrete numerals, as can be seen by normalizing the proofs of decidability

I would also appreciate feedback on a few design choices in this PR. In particular, I experimented with the DISPLAY pragma to obtain a more informative printing of comparison results. I found it convenient, but I do not have strong feelings about keeping it, so I am happy to adjust if preferred.

* add fast implementation of integers and their properties * move properties of min and max over `ℕ` to the appropriate file * reorganize and format code * remove `--safe` flag * remove `--safe` flag * Fast int (#2) * direct solver application * direct solver application --------- Co-authored-by: Marcin Jan Turek-Grzybowski <marcinjangrzybowski@gmail.com> * add fast implementation of integers and their properties * move properties of min and max over `ℕ` to the appropriate file * reorganize and format code * remove `--safe` flag * remove `--safe` flag * remove `--safe` flag * Fast int (#2) * direct solver application * direct solver application --------- Co-authored-by: Marcin Jan Turek-Grzybowski <marcinjangrzybowski@gmail.com> * fix compatibility with the library * fix compatibility after rebase, start proving properties of fast integer min and max * add instances of `Nat` and `Int` as `Pseudolattice`, and of `Int` as `OrderedCommRing` * fix typo, add better make from Posets to Pseudolattices * update `makePseudolatticeFromPoset` and instances built using it * move `maxLUB` and `minGLB` to `Data.Nat.Order`, remove unnecessary imports * rewrite properties from `minAssoc` to `-max`, add temporary lemmas in `Nat`, add `UsingEq` into `min` and `max` over the Naturals * fix naming clash on import * rewrite properties from `pos+posLposMin` to `·DistNegsucLMax` * start work on Order: adapt properties from `isProp≤` to `≤SumLeftPos`, add boolean order, suggest alternative definition `_≤'_` * use more efficient implementation of quotient and remainder in `Int.Fast.Divisibility` * adapt properties from `pred-≤-pred` to `0≤o→≤-·o`, move import of `Bool.Base` together with the other imports * adapt properties from `<-·o` to `predℤ-≤-predℤ` * adapt properties from `¬m+posk<m` to `<-+pos-trans` * adapt properties from `<-pos+-trans` to `0<o→≤-·o-cancel` * adapt properties from `≤-o·-cancel` to `≤→min` ; add conversions between the order over the integers and the one over the naturals ; add `maxLUB`, `minGLB` , `≤→max` , `≤→min` to `Data.Nat.Order` * adapt properties from `≤MonotoneMin` to `_≟'_` ; add faster `_≟_` and tests. This finishes (a first but complete version) of `Data.Int.Fast` * edit `_≤'_` using recursive order, add `_≤ᵇ_` and `_≤''_` * update `minIdem` * clean comments, reorganize `Fast` modules, add more properties in `Int.Fast.Order`, add `Trichotomy` and `Ordering` in `BinaryRelation` * add fixity for `_≤_` in `PseudolatticeStr`, add `Properties` file * edit Fast using PL and OCR instances * remove `minIdem`, use PL instance of `Nat` in the `Int.Fast.Order` module * simplify some proofs, with additional converions between the order on Nat and the integer order between negsuc * Add order related instances of `Fast.Int` * revert addition of `Trichotomy` to `BinaryRelation`, add more tests to `Int.Fast.Order` * add missing `'` on commented test code * adapt instances of `Int` and `Int.Fast` as `OCR`, to the branch `ocr+pl-instances'` --------- Co-authored-by: Ettore Forigo <ettore.forigo@gmail.com> Co-authored-by: Marcin Jan Turek-Grzybowski <marcinjangrzybowski@gmail.com>

…seudolattices and simplify some proofs about min/max in `Int.Fast.Properties`

…ebraic instances

…er order)

anshwad10 · 2025-12-18T15:47:00Z

"Fast" doesn't feel like a word to me anymore, after reading that 😄

* Resolve naming inconsistency between Iso and Category.isIso (agda#1274) s/leftInv/ret; s/rightInv/sec --------- Co-authored-by: Madeleine Sydney Ślaga <msyds@deertopia.net> * isIso name change fix (agda#1281) * Update fin (agda#1279) * Exchange Fin in Cubical.Data.Fin for the deifinition from Cubical.Data.Fin.Inductive. WIP * Fixed finj, predFin, inject<, flast, ¬Fin0 to work with new definition of Fin. * Minor changes * Fix sumFinGen * fix any? * Everything type checks now. Important changes: - Implicit argument added to toN - Implicit arguments in toN-injective slightly changed - Argument changed from implicit to explicit in elim * some formatting * change ¬Fin0 * remove comments * Add <ᵗ-asym * make properties type check * Remove dependencies on Cubical.Data.Fin.Inductive.Base * Things that work * Properties type check * more working files * factorEquiv * comment fix * LehmerCode type checks * SumFin.Base type checks * some cleanup and Axiom.Choice type checks not reliant on old Fin * Remove dependence on Fin.Inductive * Remove dependence on Fin.Inductive * Remove dependency on Fin.Infuctive * Remove dependency on Fin.Inductive * Remove dependencies on Fin.Inductive * type check * fix FinSet * _<ᵗ_ is well founded * type checks with new Fin * Everything builds * Move everything from Fin.Inductive.Properties to Fin.Properties and remove Fin.Inductive, Fin.Inductive.Base, Fin.Inductive.Properties * some cleanup * more cleanup * fix trailing whitespaces * files end with newline --------- Co-authored-by: croos <76143n@gmail.com> * fix defintions of `min` and `max`, add comment on limitations of `UsingEq` (agda#1270) * Change implementation of Inductive Order on Nat (agda#1283) * use boolean comparison in `Nat.Order.Inductive`, add non-strict and flipped inductive order, make faster instance search in `Fin.Literals` * fix typo * Place holder for paper (agda#1286) * placeholder * whitespae * Cellpaper2 (agda#1287) * placeholder * whitespae * form * fix imports: `Fin.Inductive.Base/Properties` → `Fin.Base/Properties` --------- Co-authored-by: Madeleine Sydney Ślaga <65362461+msyds@users.noreply.github.com> Co-authored-by: Madeleine Sydney Ślaga <msyds@deertopia.net> Co-authored-by: Max S. New <maxsnew@gmail.com> Co-authored-by: Caroline Roos <croos.math@gmail.com> Co-authored-by: croos <76143n@gmail.com> Co-authored-by: Axel Ljungström <49276137+aljungstrom@users.noreply.github.com>

mortberg

Only a question, otherwise everything looks good. Is it ready to be merged?

mortberg · 2026-02-19T05:43:47Z

Cubical/Data/Fast/Int/Order.agda

+¬>ℕ→¬negsuc<negsuc : ∀ {x y} → ¬ (x ℕ.>ᵗ y) → ¬ negsuc x < negsuc y
+¬>ℕ→¬negsuc<negsuc ¬p (negsuc<negsuc p) = ¬p p
+
+{-# DISPLAY negsuc≤negsuc {x} {y} _ = negsuc x ≤ negsuc y #-}


What do these DISPLAY pragmas do?

They are used to show more informative outputs when running the properties.
I found it useful while writing the code, in particular for the proofs of decidability of order and of trichotomy:

running 123456 <? 123455 shows no (¬ pos 123456 < pos 123455) instead of no (¬<ℕ→¬pos<pos (λ x → x))

running -123455 ≟ -123456 shows gt (negsuc 123455 < negsuc 123454) instead of gt (negsuc<negsuc tt)

@flgrubm I wonder if this trick could be used to make category theory goals more readable as well

LorenzoMolena · 2026-02-19T06:39:56Z

Only a question, otherwise everything looks good. Is it ready to be merged?

Yes, it is ready to be merged

mortberg · 2026-02-20T09:02:17Z

Only a question, otherwise everything looks good. Is it ready to be merged?

Yes, it is ready to be merged

Thanks a lot for the contribution!

LorenzoMolena and others added 9 commits November 19, 2025 09:52

edit zero-≤pos, using an easier proof, add zero-<possuc

4fa577b

remove Int.Fast.Order and related instances ; merge properties of P…

0294656

…seudolattices and simplify some proofs about min/max in `Int.Fast.Properties`

move Fast Integer files to Fast folder, and similarly for the alg…

1d982c1

…ebraic instances

rename the algebraic instances of fast integers

37f7203

another rename

0218afb

fix imports, remove unnecessary code in Pseudolattice.Properties

d47295e

Merge branch 'agda:master' into fast-int-update-without-order

1b378c7

remove Ordering data type (I'll leave it for the PR with fast integ…

f2971e0

…er order)

LorenzoMolena and others added 6 commits January 24, 2026 14:33

Merge branch 'agda:master' into fast-int-update-without-order

fced844

add Fast integer order and the related instaces

f22fa64

Merge branch 'agda:master' into fast-int-update-without-order

c30d0b2

fix conflicts

8a4f28d

Merge branch 'master' into fast-int-update-without-order

6977b44

LorenzoMolena mentioned this pull request Feb 17, 2026

Instances for Pseudolattice and OrderedCommRing #1271

Merged

Merge branch 'agda:master' into fast-int-update-without-order

8e7872f

marcinjangrzybowski mentioned this pull request Feb 17, 2026

Comm ring solver improvement #1293

Draft

mortberg approved these changes Feb 19, 2026

View reviewed changes

mortberg merged commit 762f7a2 into agda:master Feb 20, 2026
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`Fast.Int` Update#1280

`Fast.Int` Update#1280
mortberg merged 16 commits intoagda:masterfrom
LorenzoMolena:fast-int-update-without-order

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