feat: lemmas about sums of lists/arrays/vectors

[datokrat]

This PR provides more lemmas about sums of lists/arrays/vectors, especially sums of Nat or Int lists/arrays/vectors.

This change has been motivated by my experience solving human-eval-lean problems. Sums, minima and maxima are frequently required and the improvements provided in this PR make it easier to verify such programming tasks.

Changes:

• Added lemmas that sum equals foldl/foldr.
• Generalized sum_append_nat and sum_reverse_nat lemmas so that they are polymorphic, requiring only some type class instances about the list elements' type. The polymorphic lemmas aren't simp- or grind-annotated because I fear the instance synthesis overhead. However, the Nat and Int specializations are annotated (see below). Note that as {Array,Vector}.min do not exist, some lemmas can't be stated and were omitted.
• Added List.min_singleton and List.max_singleton lemmas as they were needed for some proofs.
• Nat-related:
  • Moved all {List,Array,Vector}.sum lemmas that are specific for Nat into their own module: Init.Data.List.Nat.Sum, Init.Data.Array.Nat and Int.Data.Vector.Nat.
  • Notably, moved Nat.sum_pos_iff_exists_pos and renamed it to List.sum_pos_iff_exists_pos_nat. This is more consistent and made it possible to add Array and Vector variants of this lemma.
  • Added lemmas proving that l.sum / l.length lies between the minimum and the maximum of a list.
• Added analogous lemmas for Int lists/arrays/vectors to parallel modules: Init.Data.List.Int.Sum, Init.Data.Array.Int and Int.Data.Vector.Int.
• Renamed sum_eq_sum_toList to sum_toList, which better represents the theorem's content.

[leanprover-bot]

Reference manual CI status:

• ❗ Reference manual CI can not be attempted yet, as the nightly-testing-2026-01-12 tag does not exist there yet. We will retry when you push more commits. If you rebase your branch onto nightly-with-manual, reference manual CI should run now. You can force reference manual CI using the force-manual-ci label. (2026-01-13 19:07:37)

[leanprover-community-bot]

Mathlib CI status (docs):

• ✅ Mathlib branch lean-pr-testing-11994 has successfully built against this PR. (2026-01-13 20:25:19) View Log
• 💥 Mathlib branch lean-pr-testing-11994 build failed against this PR. (2026-01-15 14:59:37) View Log
• 💥 Mathlib branch lean-pr-testing-11994 build failed against this PR. (2026-01-15 16:14:10) View Log
• 💥 Mathlib branch lean-pr-testing-11994 build failed against this PR. (2026-01-15 17:49:25) View Log
• ✅ Mathlib branch lean-pr-testing-11994 has successfully built against this PR. (2026-01-16 14:12:15) View Log
• ✅ Mathlib branch lean-pr-testing-11994 has successfully built against this PR. (2026-01-16 15:46:48) View Log

[datokrat]

!radar

[leanprover-radar]

Benchmark results for 75e46e1 against e56351d are in! @datokrat

Warnings (2)

These warnings may indicate that the benchmark results are not directly comparable, for example due to changes in the runner configuration or hardware.

• Runner for run build has different system configurations between commits.
• Runner for run other has different system configurations between commits.

Small changes (2✅, 10🟥)

• 🟥 build/module/Init.Data.Array.Lemmas//instructions: +239.0M (+0.3%)
• 🟥 build/module/Init.Data.Array.OfFn//instructions: +119.5M (+5.0%) (reduced significance based on *//lines)
• 🟥 build/module/Init.Data.List.Count//instructions: +68.7M (+1.8%) (reduced significance based on *//lines)
• 🟥 build/module/Init.Data.List.Lemmas//instructions: +657.7M (+1.3%) (reduced significance based on *//lines)
• 🟥 build/module/Init.Data.List.MinMax//instructions: +61.8M (+1.5%)
• 🟥 build/module/Init.Data.List.Nat.Find//instructions: +123.3M (+7.4%) (reduced significance based on *//lines)
• 🟥 build/module/Init.Data.Nat.Div.Lemmas//instructions: +420.3M (+8.1%) (reduced significance based on *//lines)
• 🟥 build/module/Init.Data.Vector.Lemmas//instructions: +405.1M (+0.7%) (reduced significance based on *//lines)
• 🟥 build/module/Init.Data.Vector.OfFn//instructions: +93.3M (+1.4%) (reduced significance based on *//lines)
• ✅ build/module/Lake.CLI//instructions: -16.4M (-2.1%)
• ✅ build/module/Lean.Elab.Tactic.Congr//instructions: -21.1M (-2.0%)
• 🟥 bv_decide_large_aig.lean//instructions: +194.4M (+0.4%)