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feat(Data/Real): add embedding of archimedean groups into Real#25157

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wwylele wants to merge 6 commits intomasterfrom
wwylele-hahn-real
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feat(Data/Real): add embedding of archimedean groups into Real#25157
wwylele wants to merge 6 commits intomasterfrom
wwylele-hahn-real

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@wwylele wwylele commented May 24, 2025

This is part of #25140, and is the special case of Hahn embedding theorem applied to archimedean group. The construction is rather unpleasant as it uses explicit Cauchy sequence with essentially the binary representation of numbers, but I haven't found a better way to do this. Rewritten with Dedekind cut


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@github-actions github-actions bot added the t-data Data (lists, quotients, numbers, etc) label May 24, 2025
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github-actions bot commented May 24, 2025

PR summary 8744850cc1

Import changes for modified files

No significant changes to the import graph

Import changes for all files
Files Import difference
Mathlib.Data.Real.Embedding (new file) 700

Declarations diff

+ embed_real
+ embed_real_add
+ embed_real_strictMono
+ embed_real_zero
+ exists_orderAddMonoidHom_real_injective
+ mkRat_add_mkRat_of_den
+ mkRat_mem_ratLt
+ orderAddMonoidHom_real
+ orderAddMonoidHom_real_apply
+ orderAddMonoidHom_real_injective
+ orderAddMonoidHom_real_one
+ ratLt
+ ratLt'
+ ratLt'_add
+ ratLt'_bddAbove
+ ratLt'_nonempty
+ ratLt_add
+ ratLt_bddAbove
+ ratLt_nonempty

You can run this locally as follows
## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.


No changes to technical debt.

You can run this locally as

./scripts/technical-debt-metrics.sh pr_summary
  • The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
  • The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

@wwylele wwylele force-pushed the wwylele-hahn-real branch from 05bcde9 to abfc2e8 Compare May 24, 2025 20:38
@wwylele wwylele mentioned this pull request May 25, 2025
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@ScottCarnahan
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I don't really know how difficult this is to implement, but in informal mathematics, I would start attacking this problem by attaching to each element m the set of rational numbers a/b such that a/b • one < m, interpreted as a • one < b • m. By the archimedean property, this set has an upper bound, and we can assign the least upper bound as a real number. This approach may produce more reusable intermediate results (or have parts that are already implemented).

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wwylele commented Jun 12, 2025

That sounds good. Let me try and see if I can formalize it this way

@wwylele wwylele added the WIP Work in progress label Jun 12, 2025
@wwylele wwylele force-pushed the wwylele-hahn-real branch from 6f8e3b1 to 5467a6a Compare June 16, 2025 02:13
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wwylele commented Jun 16, 2025

I got most of the proof with the new approach, except for one sorry left. It looks like something obvious but it might take me a while to figure out figured it out, though that particular part is not pretty

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wwylele commented Jun 18, 2025

This PR has been migrated to a fork-based workflow: #26114

@wwylele wwylele closed this Jun 18, 2025
@YaelDillies YaelDillies deleted the wwylele-hahn-real branch August 18, 2025 07:31
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3 participants