Book cauchy reals by marcinjangrzybowski · Pull Request #1182 · agda/cubical

marcinjangrzybowski · 2025-01-22T23:36:42Z

This is an implementation of real numbers as defined in Chapter 11.3: Cauchy Reals of the HoTT Book.
(Note: results 11.3.11 and 11.3.50 are currently missing.)

Modules:

Cubical/HITs/CauchyReals/Base.agda
Cubical/HITs/CauchyReals/Closeness.agda
Cubical/HITs/CauchyReals/Continuous.agda
Cubical/HITs/CauchyReals/Inverse.agda
Cubical/HITs/CauchyReals/Lems.agda
Cubical/HITs/CauchyReals/Lipschitz.agda
Cubical/HITs/CauchyReals/Multiplication.agda
Cubical/HITs/CauchyReals/Order.agda

These modules closely follow the book. However, I had to adjust the order of definitions due to the issue discussed here: HoTT/book#899.

Additionally, the following modules:

Cubical/HITs/CauchyReals/Sequence.agda
Cubical/HITs/CauchyReals/Derivative.agda

are attempts to benchmark the definitions and evaluate their behavior. In Derivative.agda, I've managed to derive the Power rule (for natural exponents). In Sequence.agda, I defined exp (currently only for ℚ₊), allowing the definition of e.

To implement these properly (assuming we want to continue without countable choice or LEM), I plan to consult some books on constructive analysis. The HoTT Book references "Errett Bishop. Foundations of Constructive Analysis," but I welcome other suggestions.

While the development is not yet up to library standards, everything currently type checks. I will likely split this into smaller PRs. There is one remaining definition, invℝ, hidden behind abstract. However, I have a promising idea based on previous optimizations to expose it (or at least manage its evaluation more controllably by guarding it with lockUnit).

There is some redundancy, as I am still experimenting with different versions of definitions.

TODO:

Implement square root
Implement chain rule
Complete results 11.3.11 and 11.3.50 from the HoTT Book
Extend exp to all reals and provide a characterization
Properly generalize convergence tests and add a few more
Define logarithm
Parameterize definitions by implementation of Rationals
Figure out a more elegant way to employ CommRingSolver (temporary solution in Cubical/HITs/CauchyReals/Lems.agda)

felixwellen · 2025-01-23T07:41:04Z

Congrats! This is great :-)
Why don't you use opaque instead of abstract?

marcinjangrzybowski · 2025-01-23T07:59:24Z

I did not saw opaque utilised too much thru the library, did not considered it really :), will do now thanks!

anshwad10

oops accidentally started a review

Cubical/Algebra/Ring/Properties.agda

* 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 (agda#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 (agda#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>

* 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 (agda#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 (agda#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'` * add `Rationals.Fast` --------- Co-authored-by: Ettore Forigo <ettore.forigo@gmail.com> Co-authored-by: Marcin Jan Turek-Grzybowski <marcinjangrzybowski@gmail.com>

…updating PR

…ngEq` (agda#26)

… `OrderedCommRing` (agda#28)

* add fixity for `_≤_` in `PseudolatticeStr`, add `Properties` file * add `≤→∧≡Left` and `≥→∧≡Right` into the `Properties` module. Add `isPropIsPseudolattice` into the the `Base` file * add equivalences of Pseudolattices and other basic properties, following similar modules * remove implicit universe levels in the type of `PseudolatticeEquiv`

* Fast int properties with pl and ocr instances (agda#23) * 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 (agda#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 (agda#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> * edit `zero-≤pos`, using an easier proof, add `zero-<possuc` --------- Co-authored-by: Ettore Forigo <ettore.forigo@gmail.com> Co-authored-by: Marcin Jan Turek-Grzybowski <marcinjangrzybowski@gmail.com>

…Order`, add `0<` and its property into `Int.Fast.Order`, add Fast Rationals. The module `Rationals.Fast.Order.Properties` is still WIP, but the uncommented code currently typechecks

…loor-frac` typechecks

…y--rebased

marcinjangrzybowski · 2025-11-24T07:48:49Z

this is result of explorative development, code is messy but everything typechecks
It is intended to be moved piece-by-piece to smaller PRs, we are planning code sprint focused on cleaning this up and refactoring during AIM XLI

Nth roots are equivalences on non-negative real numbers
rational power of real numbers, extended to real power of positive real numbers
log/exp isomorphism , group isomorphism between additive and multiplicative group on reals
exp-log-group-hom : GroupHom +Groupℝ ·₊Groupℝ
trigonometric functions and their inverses (equivalences on restricted domain)
pi defined as smallest period of cos/sin
sin' = cos , lots of trigonometric identities
Rieman integral, both directions of Fundamental theorem of calculus
Circle on plane is equivalent to interval with welded ends (with arbitrary small overlap)
mean value theorem, etc...
uniform continuity, derivatives of power series
basic Convergence tests
bits and pieces of MetricSpaces

mortberg · 2025-11-24T08:24:22Z

Wow, sounds very cool! Are you relying on any classical axioms or is everything constructive?

marcinjangrzybowski · 2025-11-24T08:41:22Z

everything constructive

marcinjangrzybowski · 2025-11-24T08:42:45Z

after implementing HoTT chapter on cauchy reals I roughly followed Bishop (as book was advising :))

marcinjangrzybowski · 2025-11-24T08:46:23Z

and of course Auke Booij , Analysis in Univalent Type Theory

…-reals

…operties

… nth root of reals without laousy unification and using solvers

marcinjangrzybowski mentioned this pull request Jun 6, 2025

HIIT Cauchy real #1214

Closed

anshwad10 reviewed Jun 27, 2025

View reviewed changes

Cubical/Algebra/Ring/Properties.agda Show resolved Hide resolved

anshwad10 reviewed Jun 27, 2025

View reviewed changes

Cubical/Algebra/Ring/Properties.agda Show resolved Hide resolved

hexwell mentioned this pull request Aug 8, 2025

Integer operations using built-ins #1245

Merged

marcinjangrzybowski mentioned this pull request Nov 12, 2025

Fix to the definitions of min and max #1270

Merged

LorenzoMolena and others added 15 commits November 19, 2025 09:52

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

284358f

rebased, but I nedd to remove few more files with experiments before …

81a733c

…updating PR

fix defintions of min and max, add comment on limitations of `Usi…

7645332

…ngEq` (agda#26)

add instances of Nat and Int as Pseudolattices, and of Int as…

a316395

… `OrderedCommRing` (agda#28)

comment sign, sign·abs (they are duplicated), fix import on `Nat.…

1fefd78

…Order`, add `0<` and its property into `Int.Fast.Order`, add Fast Rationals. The module `Rationals.Fast.Order.Properties` is still WIP, but the uncommented code currently typechecks

Rational.Fast.Order.Properties: until equivInvℚ₊, every except `f…

6d3638d

…loor-frac` typechecks

removed underdeveloped files

32c9f0d

Merge remote-tracking branch 'lorenzo/cauchy--rebase-fast' into cauch…

63e9b01

…y--rebased

whitespace fix

87dba27

small fix

fbf0126

Merge branch 'agda:master' into cauchy--rebased

fa8a793

marcinjangrzybowski force-pushed the book-cauchy-reals branch from 24e5e8c to fa8a793 Compare November 24, 2025 07:36

remove underdeveloped file

feac02b

Merge branch 'agda:master' into fast-rationals-wip

0cb77e4

LorenzoMolena and others added 17 commits November 26, 2025 23:24

merge "minimal-ocr-properties"

f90b3d4

add Rationals.Fast instances as some algebraic and order structures

8c2aad4

wip

25176f1

wip

3140d66

Merge remote-tracking branch 'lorenzo/premetric-wip' into book-cauchy…

32d09a9

…-reals

fast rational reflection, no lousy *sic* unification in Fast.Order.Pr…

1477b5c

…operties

whitespace fix

a0f916e

Base, Closness and Lip[shits with fast integers, and no Lems

a62144a

small fix

4f24bd7

lipshitz and reals-order, without lousy unification, 2 goals short

dc8c665

reals - order, got rid of lossy unification, and aplied solver

862f81e

wip on fast solvers

92f9233

wip on fast solvers

44abd4c

reals multiplciaiton cleaned

953942c

reals multiplciaiton cleaned

0e2cfab

whitespace

e7f21a9

solver for reals, more robust solver for rationals, inverse real, and…

e9a29ba

… nth root of reals without laousy unification and using solvers

LorenzoMolena mentioned this pull request Dec 16, 2025

Update to Int.Fast #1276

Draft

Provide feedback

Saved searches

Use saved searches to filter your results more quickly

Book cauchy reals#1182

Book cauchy reals#1182
marcinjangrzybowski wants to merge 34 commits intoagda:masterfrom
marcinjangrzybowski:book-cauchy-reals

marcinjangrzybowski commented Jan 22, 2025 •

edited

Loading

Uh oh!

felixwellen commented Jan 23, 2025

Uh oh!

marcinjangrzybowski commented Jan 23, 2025

Uh oh!

anshwad10 left a comment

Uh oh!

Uh oh!

Uh oh!

marcinjangrzybowski commented Nov 24, 2025

Uh oh!

mortberg commented Nov 24, 2025

Uh oh!

marcinjangrzybowski commented Nov 24, 2025

Uh oh!

marcinjangrzybowski commented Nov 24, 2025

Uh oh!

marcinjangrzybowski commented Nov 24, 2025

Uh oh!

Reviewers

Assignees

Labels

Projects

Milestone

Development

Uh oh!

5 participants

Conversation

marcinjangrzybowski commented Jan 22, 2025 • edited Loading Uh oh! There was an error while loading. Please reload this page.

Uh oh!

TODO:

Uh oh!

felixwellen commented Jan 23, 2025

Uh oh!

marcinjangrzybowski commented Jan 23, 2025

Uh oh!

anshwad10 left a comment

Choose a reason for hiding this comment

Uh oh!

Uh oh!

Uh oh!

marcinjangrzybowski commented Nov 24, 2025

Uh oh!

mortberg commented Nov 24, 2025

Uh oh!

marcinjangrzybowski commented Nov 24, 2025

Uh oh!

marcinjangrzybowski commented Nov 24, 2025

Uh oh!

marcinjangrzybowski commented Nov 24, 2025

Uh oh!

Reviewers

Assignees

Labels

Projects

Milestone

Development

Uh oh!

5 participants

marcinjangrzybowski commented Jan 22, 2025 •

edited

Loading