Related papers: Performing Implicit Induction Reasoning with Certi…
An efficient intuitionistic first-order prover integrated into Coq is useful to replay proofs found by external automated theorem provers. We propose a two-phase approach: An intuitionistic prover generates a certificate based on the matrix…
Recently, we developed an automated theorem prover for projective incidence geometry. This prover, based on a combinatorial approach using matroids, proceeds by saturation using the matroid rules. It is designed as an independent tool,…
This paper describes a formal proof library, developed using the Coq proof assistant, designed to assist users in writing correct diagrammatic proofs, for 1-categories. This library proposes a deep-embedded, domain-specific formal language,…
Despite recent advances in automating theorem proving in full first-order theories, inductive reasoning still poses a serious challenge to state-of-the-art theorem provers. The reason for that is that in first-order logic induction requires…
Whereas proof assistants based on Higher-Order Logic benefit from external solvers' automation, those based on Type Theory resist automation and thus require more expertise. Indeed, the latter use a more expressive logic which is further…
We propose a procedure for automated implicit inductive theorem proving for equational specifications made of rewrite rules with conditions and constraints. The constraints are interpreted over constructor terms (representing data values),…
This paper describes SEPIA, a tool for automated proof generation in Coq. SEPIA combines model inference with interactive theorem proving. Existing proof corpora are modelled using state-based models inferred from tactic sequences. These…
Despite recent advances in automating theorem proving in full first-order theories, inductive reasoning still poses a serious challenge to state-of-the-art theorem provers. The reason for that is that in first-order logic induction requires…
We study induction on the program structure as a proof method for bisimulation-based compiler correctness. We consider a first-order language with mutually recursive function definitions, system calls, and an environment semantics. The…
In the context of interactive theorem provers based on a dependent type theory, automation tactics (dedicated decision procedures, call of automated solvers, ...) are often limited to goals which are exactly in some expected logical…
We propose a new cyclic proof system for automated, equational reasoning about the behaviour of pure functional programs. The key to the system is the way in which cyclic proof and equational reasoning are mediated by the use of contextual…
Mathematical theorems are human knowledge able to be accumulated in the form of symbolic representation, and proving theorems has been considered intelligent behavior. Based on the BHK interpretation and the Curry-Howard isomorphism, proof…
Humans prove theorems by relying on substantial high-level reasoning and problem-specific insights. Proof assistants offer a formalism that resembles human mathematical reasoning, representing theorems in higher-order logic and proofs as…
We consider the problem of explaining the predictions of an arbitrary blackbox model $f$: given query access to $f$ and an instance $x$, output a small set of $x$'s features that in conjunction essentially determines $f(x)$. We design an…
The ever-growing complexity of mathematical proofs makes their manual verification by mathematicians very cognitively demanding. Autoformalization seeks to address this by translating proofs written in natural language into a formal…
Automated theorem provers and formal proof assistants are general reasoning systems that are in theory capable of proving arbitrarily hard theorems, thus solving arbitrary problems reducible to mathematics and logical reasoning. In…
We describe a method for building composable and extensible verification procedures within the Coq proof assistant. Unlike traditional methods that rely on run-time generation and checking of proofs, we use verified-correct procedures with…
We have developed an alternative approach to teaching computer science students how to prove. First, students are taught how to prove theorems with the Coq proof assistant. In a second, more difficult, step students will transfer their…
In mathematics, it is common practice to have several constructions for the same objects. Mathematicians will identify them modulo isomorphism and will not worry later on which construction they use, as theorems proved for one construction…
Proof assistants like Coq are increasingly popular to help mathematicians carry out proofs of the results they conjecture. However, formal proofs remain highly technical and are especially difficult to reuse. In this paper, we present a…