Ahmed Ryan, Md Erfan, Akond Ashfaque Ur Rahman +1cs.LO cs.LG
Large language models (LLMs) can generate text that resembles a mathematical proof, but resemblance does not establish correctness. A formal proof checker verifies whether each proof step follows established logical rules. Coq bases its rules on the Calculus of Inductive Constructions, a logical framework that defines which proof steps the system may accept. This pilot study evaluated six open-weight LLMs on the same 100 theorems from CoqStoq, a benchmark derived from real Coq projects. Each LLM received one attempt per theorem with the temperature set to 0, and Coq checked every proposed proof in the theorem's original project environment. We counted a proof as successful only if the Coq kernel accepted it. Gemma 4 verified 12 of 100 theorems, Llama 3.3 verified 8, and DeepSeek Coder V2 Lite verified 1. Qwen 3.5, Mistral Small 3.1, and GPT-OSS verified none. The 21 successful model-theorem results covered 15 distinct theorems, 11 of which were not solved by a baseline of standard Coq tactics. All verified theorems had short or medium human-written reference proofs; no model verified a theorem with a long reference proof. Because the proof-length analysis was exploratory, this pattern does not establish that proof length caused the difference. For the three models with at least one success, the total generation cost per verified proof ranged from 741 to 36,193 output tokens, 14.9 to 178.0 seconds, and 0.0167 to 0.2000 aggregate GPU hours. We could not calculate these ratios for models with no verified proofs. Across 600 attempts, the models produced 21 kernel-verified proofs, giving an overall success rate of 3.5%. The study reports descriptive differences among the models but does not statistically test whether one model outperforms another. Therefore, the results do not establish a universal ranking of the six models.
We propose ZX-Calculus (Knowledge Evolution Calculus), a conservative extension of Martin-Lof Dependent Type Theory (MLTT) integrating trace-indexed types, presheaf non-monotone semantics, and constructive AGM belief revision. A Coq mechanisation accompanies the paper (34 complete proofs; zero admits for the two central results). (I) Trace types. FinTrace(s0,sn) is an inductive family of typed execution traces. FinTrace and Star(Step) are isomorphic as path types but not judgementally equal; TraceElim exposes the event label e:Event explicitly, giving a more ergonomic interface for event-driven induction. We prove the Trace-Reachability Correspondence, Deterministic Replay, and a canonicity framework via reducibility candidates with a Transport Lemma (RC-elim deferred; all other Core results are Coq-verified). (II) Sheaf semantics. Trace-indexed propositions are contravariant sheaves over the free trace partial-order category Tf. A Separation Theorem (explicit countermodel) distinguishes proof-theoretic monotonicity from semantic non-monotonicity. The term model is an initial CwF (syntactic universal property, not classical completeness). (III) AGM belief revision. We give an explicit constructive partial meet contraction algorithm verified against (C1)-(C4). All eight AGM postulates (R1)-(R8) are theorems. Proofs of R7 and R8 use the Disjunctive Entrenchment Lemma, given a self-contained constructive derivation. (IV) Integration. B^AGM fails the sheaf composition law BP-comp for sequential revision (explicit countermodel, Coq-verified). We introduce Single-Step Revision Systems (SSRS), prove B^AGM is a valid SSRS (Coq-verified), and show this suffices for trace morphisms, retraction characterisation, and revision witnesses. The BP-comp failure reveals a fundamental tension between path-dependent belief revision and functor consistency, not previously identified.