Towards Smart Proof Search for Isabelle PSL and all that

Transcription

1 formerly known as NICTA Towards Smart Proof Search for Isabelle PSL and all that Yutaka Nagashima Trustworthy System Research Group March 2017 until last week

2 Example proof at Data61 Click to edit Master text styles Second level Third level Fourth level Fifth level taken from: 2 Presentation title Presenter name

3 PSL and try-hard for Isabelle/HOL The percentage of automatically proved obligations out of 1526 proof obligations (timeout = 300s) 100% 75% 50% 25% 73% Part 1 28% Part 2 20% 16% 57% Not specific to Isabelle! Other ITPs / Logic Programming 0% try_hard sledgehammer 3

4 Isabelle/HOL before PSL proof context tactic / sub-tool 4 error-message It's blatantly clear You stupid machine, subs that what I tell you is true (Michael Norrish) no sub-!

5 PSL (Proof Strategy Language) programming language extensible (Eisbach) extensive proof search low memory usage meta-tool approach parallel search PSL tactics sledgehammer almost no code clutter!! quickcheck runtime tactic generation efficient proof generation native Isabelle proof script 5

6 Isabelle/HOL with PSL context proof strategy proof context tactic / sub-tool PSL efficient tactic proved theorem / subs / message Much less interaction with Isabelle. 6

7 Tactics 1 preproces imp principle of explosion tactic False imp P Case 1 Case 2 new imp Case 3 sub 1 imp sub 2 imp imp 7 PSL and all that. Yutaka Nagashima

8 ,, [ ] Tactics 2 preproces Case 1 new Case 3 sub 1 imp imp tactic : thm Case 2 imp sub 2 imp imp 8 PSL and all that. Yutaka Nagashima

9 Tactics 2 preproces imp [ ] Case 4 (failure = empty list) tactic 9 PSL and all that. Yutaka Nagashima

10 Tactics 3 [,, ] :: thm tactic 1:: thm 2 :: thm fun tactic :: thm -> [ thm ] Lazy simp OR auto auto simp induct REPEAT simp induct THEN auto 10 PSL and all that. Yutaka Nagashima

11 Tactical (THEN) :: thm tactic1 THEN tactic2 tactic1 [,, ] 1 2 giant tactic? tactic2 tactic2 tactic2 [, ] , ]@[ 11

12 Giant tactic giant tactic? simp OR fast OR force OR auto problem 1: Default tactics are too weak! problem 2: Giant tactics are too slow! problem 3: Sledgehammer and quick-check are not tactics! 12

13 Thens [Dynamic(Induct), Auto, IsSolved] runtime interpretation (InductA ++ InductB ++ ) THEN auto THEN is_solved Dynamic ( Induct ) Auto sequential combination (THEN) IsSolved 13 Towards Smart Proof Search. non-determinism Yutaka Nagashima

14 Monadic interpretation type tactic = thm -> thm Seq.seq type a tactic = a -> a monad explicit tree construction? pointer? Dynamic ( Induct ) writer monad + non-deterministic monad Auto IsSolved efficient proof scripts as state 14

15 Sledgehammer as tactic problem 3: Sledgehammer and quick-check are not tactics! They work on Proof.state not on thm. type a tactic = 'a -> a nondet_state_monad type tactic = P.state -> P.state nondet_state_monad parallel persistant hammering PThenOne Thens [Dyn (Induct), Thens[Hammer+, IsSolved]] 15

16 try_hard: the default strategy strategy Basic = Ors [ Auto_Solve, Blast_Solve, FF_Solve, Thens [IntroClasses, Auto_Solve], Thens [Transfer, Auto_Solve], Thens [Normalization, IsSolved], Thens [DInduct, Auto_Solve], Thens [Hammer, IsSolved], Thens [DCases, Auto_Solve], Thens [DCoinduction, Auto_Solve], Thens [Auto, RepeatN(Hammer), IsSolved], Thens [DAuto, IsSolved]] strategy Try_Hard = Ors [Thens [Sub, Basic], Thens [DInductTac, Auto_Solve], Thens [DCaseTac, Auto_Solve], Thens [Sub, Advanced], Thens [DCaseTac, Solve_Many], Thens [DInductTac, Solve_Many] ] 16 PSL and all that. Yutaka Nagashima

17 PSL: Demo

18 PSL and try-hard for Isabelle/HOL The percentage of automatically proved obligations out of 1526 proof obligations (timeout = 300s) 100% 75% try_smart 73% 28% Part 2 50% Part 1 20% 16% 57% 25% 0% try_hard sledgehammer 18

19 PaMpeR: Proof Method Recommendation System huge and complex proof context strategy proof and context as a vector of boolean values context proof assertions? 19 proof method recommendation:: (proof method * double) list PaMpeR Regression Algorithm Type class mechanism? Recursively defined constant? Proof Data Base e.g. AFP & sel4

20 PaMpeR: Demo Affine_Arithmetic/Affine_Approximation

21 Future work: try-hard to try-smart PaMpeR proof context proof context small strategy proof context tactic / sub-tool try_smart PSL efficient tactic state monad transformer runtime tactic generation 21 Even better than PSL.

22 Isabelle/PSL on Github ( Leave a star if you like. I want you to use PSL / adopt the idea Isabelle/PaMpeR on Github (still work in progress) Lean/PSL coming soon(?) 22

23 Thank You TS/ProofEngineering Yutaka Nagashima Engineer