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Module Internal.Make.Proof

module Proof: Res.S  with module St = St

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

module St: Solver_types.S

Module defining atom and clauses

Type declarations

exception Insuficient_hyps

Raised when a complete resolution derivation cannot be found using the current hypotheses.

type atom = St.atom 
type lemma = St.proof 
type clause = St.clause

Abstract types for atoms, clauses and theory-specific lemmas

type proof

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

type proof_node = {
   conclusion : clause; (*

The conclusion of the proof

*)
   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.

type step =
| Hypothesis (*

The conclusion is a user-provided hypothesis

*)
| Assumption (*

The conclusion has been locally assumed by the user

*)
| Lemma of lemma (*

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

*)
| Duplicate of proof * atom list (*

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

*)
| Resolution of proof * proof * atom (*

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.

*)

The type of reasoning steps allowed in a proof.

Resolution helpers

val to_list : clause -> atom list

Returns the sorted list of atoms of a clause.

val merge : atom list -> atom list -> atom list

Merge two sorted atom list using a suitable comparison function.

val resolve : atom list -> atom list * atom list

Performs a "resolution step" on a sorted list of atoms. resolve (List.merge l1 l2) where l1 and l2 are sorted atom lists should return the pair [a], l', where l' is the result of the resolution of l1 and l2 over a.

Proof building functions

val prove : clause -> proof

Given a clause, return a proof of that clause.

val prove_unsat : clause -> proof

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

val prove_atom : atom -> proof option

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

Proof Nodes

val parents : step -> proof 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 Res_intf.S.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 : proof -> proof_node

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

val fold : ('a -> proof_node -> 'a) -> 'a -> proof -> '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 : proof -> 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 : proof -> unit

Check the contents of a proof. Mainly for internal use

val print_clause : Format.formatter -> clause -> unit

A nice looking printer for clauses, which sort the atoms before printing.

Unsafe

module H: Hashtbl.S  with type key = clause

Hashtable over proofs.