Results 1 
7 of
7
Cut elimination for a logic with induction and coinduction
 JOURNAL OF APPLIED LOGIC
, 2012
"... ..."
A formal framework for specifying sequent calculus proof systems
, 2012
"... Intuitionistic logic and intuitionistic type systems are commonly used as frameworks for the specification of natural deduction proof systems. In this paper we show how to use classical linear logic as a logical framework to specify sequent calculus proof systems and to establish some simple consequ ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Intuitionistic logic and intuitionistic type systems are commonly used as frameworks for the specification of natural deduction proof systems. In this paper we show how to use classical linear logic as a logical framework to specify sequent calculus proof systems and to establish some simple consequences of the specified sequent calculus proof systems. In particular, derivability of an inference rule from a set of inference rules can be decided by bounded (linear) logic programming search on the specified rules. We also present two simple and decidable conditions that guarantee that the cut rule and nonatomic initial rules can be eliminated.
Reasoning about Computations Using TwoLevels of Logic
"... Abstract. We describe an approach to using one logic to reason about specifications written in a second logic. One level of logic, called the “reasoning logic”, is used to state theorems about computational specifications. This logic is classical or intuitionistic and should contain strong proof pri ..."
Abstract
 Add to MetaCart
Abstract. We describe an approach to using one logic to reason about specifications written in a second logic. One level of logic, called the “reasoning logic”, is used to state theorems about computational specifications. This logic is classical or intuitionistic and should contain strong proof principles such as induction and coinduction. The second level of logic, called the “specification logic”, is used to specify computation. While computation can be specified using a number of formal techniques—e.g., Petri nets, process calculus, and state machines—we shall illustrate the merits and challenges of using logic programminglike specifications of computation. 1
3.5. Deep Inference and Categorical Axiomatizations 4 3.6. Proof Nets and Combinatorial Characterization of Proofs 4 3.7. A Systematic Approach to Cut Elimination 5
"... c t i v it y e p o r t ..."
Author manuscript, published in "Principles and Practice of Declarative Programming (2013)" DOI: 10.1145/2505879.2505889 Reasoning About HigherOrder Relational Specifications
, 2013
"... The logic of hereditary Harrop formulas (HH) has proven useful for specifying a wide range of formal systems that are commonly presented via syntaxdirected rules that make use of contexts and sideconditions. The twolevel logic approach, as implemented in the Abella theorem prover, embeds the HH s ..."
Abstract
 Add to MetaCart
The logic of hereditary Harrop formulas (HH) has proven useful for specifying a wide range of formal systems that are commonly presented via syntaxdirected rules that make use of contexts and sideconditions. The twolevel logic approach, as implemented in the Abella theorem prover, embeds the HH specification logic within a rich reasoning logic that supports inductive and coinductive definitions, an equality predicate, and generic quantification. Properties of the encoded systems can then be proved through the embedding, with special benefit being extracted from the transparent correspondence between HH derivations and those in the encoded formal systems. The versatility of HH relies on the free use of nested implications, leading to dynamically changing assumption sets in derivations. Realizing an induction principle in this situation is nontrivial and the original Abella system uses only a subset of HH for this reason. We develop a method here for supporting inductive reasoning over all of HH. Our approach relies on the ability to characterize dynamically changing contexts through finite inductive definitions, and on a modified encoding of backchaining for HH that allows these finite characterizations to be used in inductive arguments. We demonstrate the effectiveness of our approach through examples of formal reasoning on specifications with nested implications in an extended version of Abella. 1
Author manuscript, published in "Second international conference on Certified Programs and Proofs (2012)" Proof Pearl: Abella Formalization of λCalculus Cube Property
, 2013
"... Abstract. In 1994 Gerard Huet formalized in Coq the cube property of λcalculus residuals. His development is based on a clever idea, a beautiful inductive definition of residuals. However, in his formalization there is a lot of noise concerning the representation of terms with binders. We reinterp ..."
Abstract
 Add to MetaCart
Abstract. In 1994 Gerard Huet formalized in Coq the cube property of λcalculus residuals. His development is based on a clever idea, a beautiful inductive definition of residuals. However, in his formalization there is a lot of noise concerning the representation of terms with binders. We reinterpret his work in Abella, a recent proof assistant based on higherorder abstract syntax and provided with a nominal quantifier. By revisiting Huet’s approach and exploiting the features of Abella, we get a strikingly compact and natural development, which makes Huet’s idea really shine. 1