Proof assistants, such as Isabelle HOL, offer tools to facilitate inductive theorem proving. Isabelle experts know how to use these tools effectively; however they did not have a systematic way to encode their expertise. To address this problem, we present our domain-specific language, LiFtEr. LiFtEr allows experienced Isabelle users to encode their induction heuristics in a style independent of any problem domain. LiFtEr's interpreter mechanically checks if a given application of induction tool matches the heuristics, thus transferring experienced users' expertise to new Isabelle users.
The recent development in proof automation for higher-order logic takes the meta-tool approach. Gauthier al , for example, developed an automated tactic prover, TacticToe, on top of the HOL4 @cite_13 . TacticToe leans how human engineers used tactics and applies the knowledge to execute a tactic based Monte Carlo tree search. To automate proofs in Coq @cite_14 , Komendantskaya al developed ML4PG @cite_6 . ML4PG uses recurrent clustering to mine a proof database and attempts to find a tactic-based proof for a given proof goal. Both of them try to identify useful lemmas or hypotheses as arguments of a tactic; however, they do not identify promising terms as arguments of a tactic even though identify such terms is crucial to apply induction effectively.
@cite_14 @cite_13 @cite_6
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Techniques combining machine learning with translation to automated reasoning have recently become an important component of formal proof assistants. Such "hammer" tech- niques complement traditional proof assistant automation as implemented by tactics and decision procedures. In this paper we present a unified proof assistant automation approach which attempts to automate the selection of appropriate tactics and tactic-sequences com- bined with an optimized small-scale hammering approach. We implement the technique as a tactic-level automation for HOL4: TacticToe. It implements a modified A*-algorithm directly in HOL4 that explores different tactic-level proof paths, guiding their selection by learning from a large number of previous tactic-level proofs. Unlike the existing hammer methods, TacticToe avoids translation to FOL, working directly on the HOL level. By combining tactic prediction and premise selection, TacticToe is able to re-prove 39 percent of 7902 HOL4 theorems in 5 seconds whereas the best single HOL(y)Hammer strategy solves 32 percent in the same amount of time. Several approaches exist to data-mining big corpora of formal proofs. Some of these approaches are based on statistical machine learning, and some -- on theory exploration. However, most are developed for either untyped or simply-typed theorem provers. In this paper, we present a method that combines statistical data mining and theory exploration in order to analyse and automate proofs in dependently typed language of Coq.