[Merged by Bors] - feat(Topology/Algebra/InfiniteSum/NatInt): hasProd versions of ℕ+/ℕ transfer lemmas

[pepamontero]

The PNat section of Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean provides Multipliable and tprod lemmas for transferring infinite products between ℕ+ and ℕ, but not HasProd lemmas. This PR completes this gap with:

• hasProd_pnat_iff_hasProd_succ: HasProd (fun n : ℕ+ ↦ f n) m ↔ HasProd (fun n : ℕ ↦ f (n + 1)) m, the HasProd counterpart of the existing multipliable_pnat_iff_multipliable_succ
• hasProd_pnat_iff: in a topological group, HasProd (fun n : ℕ+ ↦ f n) g ↔ HasProd f (g * f 0), the counterpart of multipliable_pnat_iff_multipliable_nat
• tprod_pnat_eq_tprod_of_eq_one: if f 0 = 1, then ∏' n : ℕ+, f n = ∏' n, f n. Unlike tprod_zero_pnat_eq_tprod_nat, this requires no multipliability hypothesis (and no group structure or T2Space)

Motivation: the additive versions (hasSum_pnat_iff, tsum_pnat_eq_tsum_of_eq_zero, ...) replace
three private lemmas in the FLT project (hasSum_nat_of_pnat_add, hasSum_pnat_of_nat, tsum_pnat_of_zero), each of which becomes a one-line application, stated there for ℂ only.

AI Disclosure: original code in FLT was written by William Coram and Samuel Yin with the assistance of Claude. It was later cleaned up using Codex.

Co-authored-by: William Coram
Co-authored-by: Samuel Yin

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[github-actions]

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[github-actions]

PR summary 44608745a8

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ hasProd_pnat_iff
+ hasProd_pnat_iff_hasProd_succ
+ tprod_pnat_eq_tprod_of_eq_one

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 4460874).

• +6 new declarations
• −0 removed declarations

+hasProd_pnat_iff
+hasProd_pnat_iff_hasProd_succ
+hasSum_pnat_iff
+hasSum_pnat_iff_hasSum_succ
+tprod_pnat_eq_tprod_of_eq_one
+tsum_pnat_eq_tsum_of_eq_zero

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 44608745a8
Reference commit ed0d70fdcd

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/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).

[github-actions]

✅ PR Title Formatted Correctly

The title of this PR has been updated to match our commit style conventions.
Thank you!

[YaelDillies]

Thanks! 🚀

maintainer merge

[github-actions]

🚀 Pull request has been placed on the maintainer queue by YaelDillies.

[kbuzzard]

Thanks!

bors merge

[mathlib-bors]

Pull request successfully merged into master.

Build succeeded:

• ci (staging) / Post-CI job
• ci (staging) / Lint style
• ci (staging) / Build
• ci (staging) / Test and lint
• ci (staging) / Post-Build Step