feat(Query): query complexity framework with sorting examples by kim-em · Pull Request #401 · leanprover/cslib

kim-em · 2026-03-05T05:57:57Z

This PR implements Sebastian Graf's unified approach to query complexity (discussed in the CSLib Algorithm frameworks thread), combining the strengths of #372 (explicit Prog/FreeM query types) and #376 (monad-parametric approach).

Programs are Prog Q α (free monad over query type Q), and the oracle is supplied after the program produces its query plan — giving anti-cheating guarantees for both upper and lower bounds. No WP/Hoare triple machinery is needed: correctness is just equations about Prog.eval oracle, and cost is just equations about Prog.queriesOn oracle.

This provides an alternative to the TimeM-based cost analysis already in the repo: here query counting is structural (derived from the Prog tree) rather than annotation-based.

New files

File	Contents
`Query/Prog.lean`	Core `Prog` type, `eval`, `queriesOn`, simp lemmas
`Query/Bounds.lean`	`UpperBound` and `LowerBound` definitions
`Query/QueryTree.lean`	Decision trees with fixed response type, for lower bound proofs
`Query/Sort/LEQuery.lean`	Comparison query type for sorting
`Query/Sort/IsSort.lean`	`IsSort` correctness specification
`Query/Sort/Insertion/{Defs,Lemmas}.lean`	Insertion sort: correctness + O(n²) upper bound
`Query/Sort/Merge/{Defs,Lemmas}.lean`	Merge sort: correctness + n·⌈log₂ n⌉ upper bound
`Query/Sort/QueryTree.lean`	`Prog`-to-`QueryTree` bridge + pigeonhole depth lemma
`Query/Sort/LowerBound.lean`	Any correct comparison sort needs ≥ ⌈log₂(n!)⌉ queries

Results

Insertion sort: correctness (permutation + sortedness), n² upper bound, IsSort instance
Merge sort: correctness, n·⌈log₂ n⌉ upper bound, IsSort instance
Lower bound: any IsSort on an infinite type makes ≥ ⌈log₂(n!)⌉ queries. The proof constructs n! distinct total orders via permutations of embedded elements, shows they force distinct sorted outputs, then applies an adversarial pigeonhole argument on QueryTree depth.

🤖 Prepared with Claude Code

Shreyas4991

Please check author lists.

I didn't intend to GitHub-approve it. I wanted to click "Comment"

Shreyas4991 · 2026-03-05T08:44:02Z

Cslib/Algorithms/Lean/Query/Prog.lean

+/-
+Copyright (c) 2026 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Sorrachai Yingchareonthawornchai, Kim Morrison, Sebastian Graf, Shreyas Srinivas


Sorrachai is not an author on this file, only on the TimeM files. Further on all prog files I am first author

Add a framework for proving upper and lower bounds on query complexity of comparison-based algorithms, using `Prog` (free monad over query types) with oracle-parametric evaluation and structural query counting. Results: - Insertion sort: correctness + O(n²) upper bound - Merge sort: correctness + n·⌈log₂ n⌉ upper bound - Lower bound: any correct comparison sort on an infinite type needs ≥ ⌈log₂(n!)⌉ queries (via adversarial pigeonhole on QueryTree depth) Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

Add `Prog.cost`, a weighted generalization of `Prog.queriesOn` where each query type can have a different cost. Demonstrate this with complex multiplication: naive (4 muls + 2 adds) vs Gauss's trick (3 muls + 5 adds), proving correctness, exact parametric costs, and the crossover condition. Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

Shreyas4991 · 2026-03-05T09:03:03Z

Cslib/Algorithms/Lean/Query/Arith/Defs.lean

+Authors: Kim Morrison
+-/
+module
+


This is the Arith Prog from my CslibTests files?

Shreyas4991

Thus arith query is similar to mine in the the CslibTests folder

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

Shreyas4991

A non-comprehensive comparison with previous PR and authorship issues that arise therein.

Shreyas4991 · 2026-03-05T11:00:32Z

Cslib/Algorithms/Lean/Query/Sort/Insertion/Defs.lean

+namespace Cslib.Query
+
+/-- Insert `x` into a sorted list using comparison queries. -/
+@[expose] def orderedInsert (x : α) : List α → Prog (LEQuery α) (List α)


This is identical to the Prog version upto query type.

Shreyas4991 · 2026-03-05T11:02:01Z

Cslib/Algorithms/Lean/Query/Sort/Insertion/Lemmas.lean

+
+-- ## Query count proofs
+
+theorem orderedInsert_queriesOn_le (oracle : {ι : Type} → LEQuery α ι → ι)


You oracle is just my Model.evalQuery. Likewise for time.

Shreyas4991 · 2026-03-05T11:03:13Z

Cslib/Algorithms/Lean/Query/Sort/LowerBound.lean

+    toQTOracle (fromQTOracle f) = f := rfl
+
+/-- Convert a `Prog (LEQuery α)` program to a `QueryTree (α × α) Bool` decision tree. -/
+@[expose] def Prog.toQueryTree : Prog (LEQuery α) β → QueryTree (α × α) Bool β


This is essentially traceSort from cslib#372

Shreyas4991 · 2026-03-05T11:06:37Z

Cslib/Algorithms/Lean/Query/Sort/Merge/Defs.lean

+/-
+Copyright (c) 2026 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Sorrachai Yingchareonthawornchai, Kim Morrison


The authorship claims on this go to an equal extent for me and Sorrachai. We both use a different recursive structure and I provide the Prog version of his code. Since the recursion structure is different here (and more akin to the upstream version), potentially neither of us are co-authors or both of us are.

Shreyas4991 · 2026-03-05T11:07:48Z

Cslib/Algorithms/Lean/Query/Sort/QueryTree.lean

@@ -0,0 +1,107 @@
+/-
+Copyright (c) 2026 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.


I published a similar proof of model wide lower bound a few hours before cslib#376 got there.

eric-wieser · 2026-03-05T12:49:50Z

Cslib/Algorithms/Lean/Query/Prog.lean

+/-- Evaluate a program by answering each query using `oracle`. -/
+@[expose] def eval (oracle : {ι : Type} → Q ι → ι) : Prog Q α → α
+  | .pure a => a
+  | .liftBind op cont => eval oracle (cont (oracle op))


I claim that pattern matching on the free monad is exploiting an implementation detail, and that everything should really go through the universal property, FreeM.liftM. (this is what #372 does)

kim-em force-pushed the combined-query-complexity branch 2 times, most recently from dc724df to 8371c12 Compare March 5, 2026 08:34

Shreyas4991 approved these changes Mar 5, 2026

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kim-em force-pushed the combined-query-complexity branch from 8371c12 to dbb9e41 Compare March 5, 2026 08:46

kim-em force-pushed the combined-query-complexity branch from dbb9e41 to 9acdbbd Compare March 5, 2026 08:50

Shreyas4991 reviewed Mar 5, 2026

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Cslib/Algorithms/Lean/Query/Arith/Defs.lean

Authors: Kim Morrison

-/

module

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This is the Arith Prog from my CslibTests files?

Shreyas4991 reviewed Mar 5, 2026

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docs(Query/Arith): clarify these are toy examples of parametrized costs

83d95c4

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

Shreyas4991 reviewed Mar 5, 2026

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feat(Query): query complexity framework with sorting examples#401

feat(Query): query complexity framework with sorting examples#401
kim-em wants to merge 3 commits intoleanprover:mainfrom
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		-- ## Query count proofs

		theorem orderedInsert_queriesOn_le (oracle : {ι : Type} → LEQuery α ι → ι)

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