Lemma discovery in automating induction

Dynamic Rippling, Middle-Out Reasoning and Lemma Discovery

“…Another approach to finding lemmas is interactive lemma speculation [16], which can provide the user with some useful feedback, even when the lemma cannot be fully instantiated. A lemma discovery algorithm similar to speculation has been proposed in [18]. Like lemma speculation, subterms in the goal are equated with the hypothesis, and meta-variables used for unknown term-structure.…”

“…It has long been considered a promising technique to improve rippling based proofs [6,5,15], and a similar approach has also been proposed in [18]. Lemma speculation constructs a schematic lemma which preserves parts of the goal that are similar to the inductive hypothesis and introduces meta-variables to stand for unknown term-structure.…”

Using an induction prover for verifying arithmetic circuits

“…Another heuristic found especially useful for proving properties of tail-recursive definitions (which is indeed the case for arithmetic circuits including adders and multipliers) is that of generating bridge lemmas which facilitate the use of induction hypotheses in a proof attempt of an attempted valid conjecture. A constraint-based approach is developed to speculate intermediate conjectures as well as guess instantiations for non-induction variables in a conjecture [30,46]. In verifying properties of arithmetic circuits, some of the intermediate lemmas needed can be generated from the circuit structure and component specifications, as illustrated later for multiplier circuits.…”

“…Instantiations for non-induction variables in an induction hypothesis may have to be guessed; if a conjecture does not have any non-induction variables, they may have to be introduced by attempting a generalization. A constraint-based approach for speculating conjectures for this was proposed in [30]. Constraints are generated from the simplified conclusion and the induction hypothesis, and aided by the definitions and known properties of function symbols appearing in the conjecture.…”

“…This can be determined by analyzing the definitions of the function symbols rca, bton as discussed in [30,46]. Two equations are generated corresponding to these potential applications of the hypothesis: 1: bton(rca(half(x, y1, 0), y, cons(z1, z))) == bton(rca(half(x', y1, z1), y, cons(0, z))) if {len(y) = add1(len(z)), len(rca(half(x',y1,z1), y, z)) = len(app(z, cons(0, nil)))}.…”