Module S.Proof

A module to manipulate proofs.

Signature for a module handling proof by resolution from sat solving traces

Type declarations

exception Resolution_error of string

Raised when resolution failed.

type formula = formula
type atom = atom
type lemma = lemma
type clause = clause

Abstract types for atoms, clauses and theory-specific lemmas

type t = proof

Lazy type for proof trees. Proofs are persistent objects, and can be extended to proof nodes using functions defined later.

and proof_node = {
  1. conclusion : clause;
    (*

    The conclusion of the proof

    *)
  2. step : step;
    (*

    The reasoning step used to prove the conclusion

    *)
}

A proof can be expanded into a proof node, which show the first step of the proof.

and step =
  1. | Hypothesis of lemma
    (*

    The conclusion is a user-provided hypothesis

    *)
  2. | Assumption
    (*

    The conclusion has been locally assumed by the user

    *)
  3. | Lemma of lemma
    (*

    The conclusion is a tautology provided by the theory, with associated proof

    *)
  4. | Duplicate of t * atom list
    (*

    The conclusion is obtained by eliminating multiple occurences of the atom in the conclusion of the provided proof.

    *)
  5. | Hyper_res of hyper_res_step

The type of reasoning steps allowed in a proof.

and hyper_res_step = {
  1. hr_init : t;
  2. hr_steps : (atom * t) list;
}

Proof building functions

val prove : clause -> t

Given a clause, return a proof of that clause.

val prove_unsat : clause -> t

Given a conflict clause c, returns a proof of the empty clause.

val prove_atom : atom -> t option

Given an atom a, returns a proof of the clause [a] if a is true at level 0

val res_of_hyper_res : hyper_res_step -> t * t * atom

Turn an hyper resolution step into a resolution step. The conclusion can be deduced by performing a resolution between the conclusions of the two given proofs. The atom on which to perform the resolution is also given.

Proof Nodes

val parents : step -> t list

Returns the parents of a proof node.

val is_leaf : step -> bool

Returns wether the the proof node is a leaf, i.e. an hypothesis, an assumption, or a lemma. true if and only if parents returns the empty list.

val expl : step -> string

Returns a short string description for the proof step; for instance "hypothesis" for a Hypothesis (it currently returns the variant name in lowercase).

Proof Manipulation

val expand : t -> proof_node

Return the proof step at the root of a given proof.

val conclusion : t -> clause

What is proved at the root of the clause

val fold : ('a -> proof_node -> 'a) -> 'a -> t -> 'a

fold f acc p, fold f over the proof p and all its node. It is guaranteed that f is executed exactly once on each proof node in the tree, and that the execution of f on a proof node happens after the execution on the parents of the nodes.

val unsat_core : t -> clause list

Returns the unsat_core of the given proof, i.e the lists of conclusions of all leafs of the proof. More efficient than using the fold function since it has access to the internal representation of proofs

Misc

val check_empty_conclusion : t -> unit

Check that the proof's conclusion is the empty clause,

val check : t -> unit

Check the contents of a proof. Mainly for internal use.

module Tbl : Hashtbl.S with type key = t