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Formal Proof Framework for List Separation Logic and Inductive Reasoning in Coq

This document introduces a formal framework leveraging Separation Logic and Coq for verification of list operations such as allocation, deallocation, and insertion. It discusses generating proof terms from given assertions and demonstrates how to combine with the Simplex Prover for complex data structures. Key examples include handling inductive axioms within constructed proofs. The framework also addresses common difficulties like user-defined data structures and multi-assumption frame inferences, providing insights into semantics and proof generation methodologies.

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Formal Proof Framework for List Separation Logic and Inductive Reasoning in Coq

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  1. Separation Logic Prover

  2. Goal • Assertion Language • Assert :: Formula  Formula • Formula :: (pureF ‘|’ spaF) | Formula \/ Formula • pureF :: term /\ term | true • spaF :: sterm * sterm • term :: exp relop exp | id(explist) • sterm :: exp  exp | id(explist) | emp • Example • p=q | head -> v * head+4->p * lseg(q,NULL)  • true | list(head) * emp • Proof given assertion and generate Coq checkable proof term

  3. Status • Demo • list_alloclist_dealloclist_insert • function call • proof term of a small step in prover • Combine with Simplex Prover • Embedded “data structure” • lseg list clist • more… • Single assumption frame inference • Proof term generation • Coq library • Coding …

  4. Difficulties • User-defined data structure • hard to discover inductive axiom from definition • e.g. lseg(p,q)*lseg(q,r)  lseg(p,r) • Multi assumption frame inference • (A\/B) -> (C\/D) • what is a “frame” in this case?

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