Going Polymorphic - TH1 Reasoning for Leo-III

Abstract

While interactive proof assistants for higher-order logic (HOL) commonly admit reasoning within rich type systems, current theorem provers for HOL are mainly based on simply typed lambda-calculi and therefore do not allow such flexibility. In this paper, we present modifications to the higher-order automated theorem prover Leo-III for turning it into a reasoning system for rank-1 polymorphic HOL.To that end, a polymorphic version of HOL and a suitable paramodulation-based calculus are sketched. The implementation is evaluated using a set of polymorphic TPTP THF problems.

ProofBuddy: A Proof Assistant for Learning and Monitoring

“…• Proof assistants are useful for teaching mathematics [2,4,10,11,17] • Proof assistants are useful for teaching functional programming [4,5,17,18,21,26] • Proof assistants are useful for teaching logic [2,4,5,8,15,17,18,21,26,34,36,41] • Proof assistants are useful for teaching abstract thinking [4,11,34,36] • Proof assistants make the rules of formal reasoning clear [2,4,11] • Proof assistants help students learn how to structure proofs [4,5,8,10,11,17,19,21,31] • Students are helped by instant feedback on their proofs [2, 4, 8, 10-12, 15, 21, 26] • Proof competence gained using a proof assistant transfers to pen-and-paper [2,12,26,34] • Proof assistants help students fix their proof errors as early as possible [4,11] • Proof assistants make it easier for stude...…”

GRUNGE: A Grand Unified ATP Challenge

“…TH1-I TH0-I TH0-II TF1-I TF0-I TF0-II FOF-I FOF-II Union agsyHOL 1.0 [32] 1374 1187 1605 Beagle 0.9.47 [3] 2008 2047 2449 2498 3183 cocATP 0.2.0 899 599 1000 CSE E 1.0 [51] 4251 3102 4480 CVC4 1.6 [2] 4851 3991 5030 3746 5709 E 2.2 [41] 4277 3622 4618 3844 5118 HOLyHammer 0.21 [27] 5059 5059 iProver 2.8 [29] 2778 2894 3355 iProverModulo 2.5-0.1 [11] 2435 1639 1433 1263 2852 LEO-II 1.7.0 [4] 2579 1923 2119 1968 3702 Leo-III 1.3 [44,43] 6668 5018 3485 3458 4032 3421 3986 3185 7090 Metis 2.4 [24] 2353 474 2356 Princess 170717 [39,40] 3646 2138 3162 2086 4096 Prover9 1109a [33] 2894 1742 3128 Satallax 3.3 [10] 2207 1292 2494 SPASS 3.9 [50] 2850 3349 3821 Vampire 4.3 [30] 4837 4693 4008 4928 5929 ZenonModulo [15] 1071 1038 1041 1026 1198 1751 Zipperposition 1.4 [13] 2252 The TacticToe [17,18] prover built into HOL4 has been tested as a baseline comparison, and it (re)proves 5327 of 8855 chainy versions of the problems (60.2%). TacticToe is a machine-learning guided prover that searches for a tactical proof by selecting suitable tactics and theorems learned from human-written tactical proofs.…”

“…A method for encoding of polymorphic types as terms through type tags (as in our first translation) or type guards (as in our second translation) is described in [6]. Translations [21,44] from a polymorphic logic to a monomorphic poly-sorted logic without encoding typically rely on heuristic instantiations of type variables. However, heuristics may miss useful instantiations, and make the translation less modular (i.e., context dependent).…”

Extensional Higher-Order Paramodulation in Leo-III

“…Also, the data structures of Leo-III are already expressive enough to represent polymorphic formulas, cf. technical details in earlier work [SWB17].…”

“…Furthermore, Leo-III implements a locally nameless representation using de Bruijn indices [Bru72]. In the setting of polymorphism [SWB17], types may also contain variables. Consequently, the nameless representation of variables is extended to type variables [KRTU99].…”