A list of lists of SequentIndices denoting the auxiliary formula(s) of the rule.
A list of lists of SequentIndices denoting the auxiliary formula(s) of the rule. The first list contains the auxiliary formulas in the first premise and so on.
The conclusion of the rule.
The conclusion of the rule.
The immediate subproofs of this rule.
The immediate subproofs of this rule.
A list of SequentIndices denoting the main formula(s) of the rule.
A list of SequentIndices denoting the main formula(s) of the rule.
A list of occurrence connectors, one for each immediate subproof.
A list of occurrence connectors, one for each immediate subproof.
Assertions of the proof.
Assertions of the proof.
Attention: this is interpreted as assertions.toNegConjunction --> conclusion.toDisjunction
These assertions indicate the splitting assumptions this proof depends on.
A list of lists containing the auxiliary formulas of the rule.
A list of lists containing the auxiliary formulas of the rule. The first list constains the auxiliary formulas in the first premise and so on.
Operations that view the sub-proofs as a DAG, which ignore duplicate sub-proofs, see DagProof.DagLikeOps for a list.
Operations that view the sub-proofs as a DAG, which ignore duplicate sub-proofs, see DagProof.DagLikeOps for a list.
All definitions introduced by any subproof.
All definitions introduced by any subproof.
Exception
if inconsistent definitions are used
Depth of the proof, which is the maximum length of a path you can take via immediateSubProofs.
Depth of the proof, which is the maximum length of a path you can take via immediateSubProofs.
Definitions introduced by the bottom-most inference rule.
Is this a proof of the empty clause with empty assertions, and consistent definitions?
The name of this rule (in words).
The name of this rule (in words).
The list of main formulas of the rule.
The list of main formulas of the rule.
The name of this rule (in symbols).
The name of this rule (in symbols).
The upper sequents of the rule.
The upper sequents of the rule.
Set of all (transitive) sub-proofs including this.
Set of all (transitive) sub-proofs including this.
Operations that view the sub-proofs as a tree, see DagProof.TreeLikeOps for a list.
Operations that view the sub-proofs as a tree, see DagProof.TreeLikeOps for a list.
Proof using resolution.
Our resolution calculus integrates higher-order reasoning, structural clausification, and Avatar-style splitting as in [Vor14].
The judgments of this calculus are A-sequents. An A-sequent is a pair of a sequent of HOL formulas, and a conjunction of propositional literals---the assertion (that's where the "A" comes from).
Γ :- Δ <- A
We store the sequent as a HOLSequent in ResolutionProof#conclusion, and the negation of the assertion as a HOLClause in ResolutionProof#assertions (as the negation of a conjunction of literals is a clause). The judgment is interpreted as one of the following equivalent formulas:
assertions.toNegConjunction --> conclusion.toDisjunction (conclusion ++ assertions).toDisjunction // equivalent to the first one
Inferences such as Resolution or Paramod do not operate on the assertions. Unless specified otherwise, assertions are inherited by default:
There is no factoring on assertions, duplicate assertions are automatically removed.
Reading from top to bottom, a resolution proof is usually structured as follows:
Substitutions are not absorbed into resolution, factoring, and paramodulation; they are explicitly represented using Subst.
[Vor14] Andrei Voronkov, AVATAR: The Architecture for First-Order Theorem Provers. CAV 2014: pp. 696-710