A Heuristic Prover for Real Inequalities

Abstract

We describe a general method for verifying inequalities between real-valued expressions, especially the kinds of straightforward inferences that arise in interactive theorem proving. In contrast to approaches that aim to be complete with respect to a particular language or class of formulas, our method establishes claims that require heterogeneous forms of reasoning, relying on a Nelson-Oppen-style architecture in which special-purpose modules collaborate and share information. The framework is thus modular and extensible. A prototype implementation shows that the method works well on a variety of examples, and complements techniques that are used by contemporary interactive provers.

Differential Equation Invariance Axiomatization

“…e (1) − 4υ 3 e) e (polynomial) identity for . e (2) in terms of . e (1) , e and their cofactors is obtained computationally by ideal membership checks for the polynomial ring R[x, 1 , 2 , 3 ] (the indeterminate i corresponds to υ i for i = 1, 2, 3), following Proposition 7.2. e next example illustrates how the extended term language allows e ective proofs of more invariants than possible with polynomial term languages.…”

“…Similarly, invariants of ODEs describe subsets of the state space from which solutions of the ODEs cannot escape. e three basic dL principles for reasoning about such invariants are: (1) di erential invariants, which enable local reasoning about trends of truth in di erential form, (2) di erential cuts, which accumulate knowledge about the evolution of an ODE from multiple proofs, and (3) di erential ghosts, which add di erential equations for new ghost variables to the existing system of di erential equations enabling reasoning about the historical evolution of ODE systems in integral form. ese reasoning principles relate to their discrete loop counterparts as follows: (1) corresponds to loop induction by analyzing the loop body, (2) corresponds to progressive re nement of the loop guards, and (3) corresponds to adding discrete ghost variables to remember intermediate program states.…”

“…(1) All analytic invariants, i.e., nite conjunctions and disjunctions of equations between extended terms, are provable using only the three ODE axioms outlined above. (2) With axioms internalizing the existence and uniqueness theorems for solutions of di erential equations, all local progress properties of ODEs are provable for all semianalytic formulas, i.e., propositional combinations of inequalities between extended terms. (3) With a real induction axiom that reduces invariance to local progress, the dL calculus is complete for proving all semianalytic invariants of di erential equations.…”

A formal proof of Hensel's lemma over the p-adic integers

“…It takes roughly 70 lines of Lean code to perform these computations, compared to roughly 10 in the informal presentation. Many of these computations fall under the scope of the tool Polya [2] developed by the author. In the future, such a tool could be used to significantly condense this portion of our proof.…”

“…The formalization described in this paper is incorporated into the Lean mathematical library, available on GitHub. 2 Since this library is regularly changing, we preserve a snapshot of its status at the time this paper was submitted. This snapshot, and a map between this paper and the formalization, can be found on the author's website.…”