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(* Title *)
SML Library for Proof Representation and Manipulation.


(* Description *)

In the creation of both automated and interactive proof systems, the
legibility of generated proofs has long been a problem. Proofs
generated in such systems are notoriously low-level when compared to
the informal inscriptions of the practicing mathematician.  Modern
interactive proof environments like Isabelle strive to provide rich
higher-order languages for expressing proofs with many features to
improve legibility and reuse such as unix-pipe-like operators like
"hence" and "thus", as well as block structures to promote code
reuse. Such environments thus reach a balance between formal rigor and
ease of use. However, it remains a quite intensive process to formally
prove complicated theorems.

On the other hand, there exist numerous fully automated tools that use
different, often less expressive, logical formalisms but that are able
to produce difficult but highly illegible proofs. Utilities such as
Sledgehammer currently attempt to bridge these environments, allowing
the mathematician to consult, for example, automated theorem provers
for subgoals such that syntactically-valid HOL proofs are returned
upon success that can then be verified by Isabelle.

Yet, sledgehammer has come up against numerous difficulties. In
particular, it remains quite difficult to convert the output from, for
example, first-order theorem provers such as Vampire into HOL proofs,
especially when legibility is taken as a criteria of judgement.  The
aim of my summer project is to aid in precisely this issue: to create
an SML library to aid in the transition from low-level, machine
generated proofs to legible proofs containing justifications at a
higher level of abstraction. While in just this summer it is unlikely
that this library will be able to produce proofs of the same quality
as human users, the primary goal is to produce a standard data
structure atop which subsequent proof cleaning algorithms can be
implemented in years to come.

Following Karol Pak's work on legibility in proof, the following are
essential values to be considered:
i. maximization of the length of paths where every consecutive
   justification should refer to a previous line
ii. minimization of the quantity of introduced labels
iii. minimization of the total length of jumps to distant, previously
     justified facts iv. presentation using sub-proofs

Given the above, I propose to work on developing a proof
representation and manipulation library in SML for the purpose of:
 1) automatically improving the legibility of human and
    machine-generated proofs alike
 2) improving the performance of bridges between interactive and
    automatic theorem provers such as Sledgehammer.

This project will consist in several concrete stages:
(1) The development of an efficient graph data structure for proof
    representation in SML.
(2) Integration with TSTP
(3) Implementation of two proof manipulation algorithms to ensure the
    usability of proof data structure. Likely some refactoring.
    a) proof by contradiction into direct proof    
    b) factoring out of common proof components (avoid exponential 
       explosion)
(4) Integration with Isabelle
(5) Implementation of the rest of decided upon manipulation algorithms.
     a) linear reorganization of proofs (better legibility)
     b) introduction of structure in a proof using block abstraction 
        (subproofs)
     c) if time, implement Karol Pak's utilities
(6) Integration of this suite with Sledgehammer
(7) Integration with SMT solvers.

Parts (1) and (3) will require a significant amount of research into
existent proof representations and the difficulties in the current
representation within Sledgehammer. It thus seems that these first
three steps will take approximately the first half of the summer to
complete. Once the proof representation has been completed and the
usability of its API has been tested through the implementation of
several proof manipulation algorithms, it should be straightforward,
with the assistance of my mentor, Jasmin, and existent documentation
to integrate this tool with Isabelle and to implement the remaining
agreed upon proof manipulation algorithms. (6) is a research problem
that will probably not be completed during the summer, but that Jasmin
and I have discussed working on after its termination.