13.3 Automated theorem proving and proof assistants

Automated theorem proving uses algorithms to prove mathematical statements. It combines techniques like resolution and unification with SAT and SMT solvers to tackle complex logical problems. These tools automate reasoning and find proofs faster than humans.

Proof assistants like Coq, Isabelle/HOL, and Lean help verify math and software. They offer interactive theorem proving, where users guide proofs with tactics. This blend of human insight and machine power tackles tricky proofs and boosts confidence in results.

Automated Theorem Proving

Resolution and Unification

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• Resolution is a rule of inference for propositional logic and first-order logic that produces a new clause from two clauses containing complementary literals
• Involves resolving two clauses by assuming one is true and the other is false, then eliminating complementary literals to produce a new clause
• Unification is a key concept in automated theorem proving that involves finding a substitution that makes two logical expressions identical
• Unification algorithms (syntactic unification, semantic unification) find the most general unifier between two expressions, allowing for variables to be replaced with terms

SAT and SMT Solvers

• SAT solvers are programs that determine the satisfiability of propositional logic formulas, checking if there exists an interpretation that makes the formula true
• Use algorithms like DPLL (Davis-Putnam-Logemann-Loveland) to efficiently search for satisfying assignments
• SMT (Satisfiability Modulo Theories) solvers build upon SAT solvers to decide the satisfiability of formulas in first-order logic with respect to background theories (arithmetic, arrays, uninterpreted functions)
• SMT solvers combine a SAT solver with specialized decision procedures for the background theories, allowing for reasoning about more expressive formulas

Proof Assistants

Popular Proof Assistants

• Coq is a proof assistant based on the Calculus of Inductive Constructions, a dependent type theory
• Supports writing formal specifications and proofs, allowing for the verification of complex mathematical theorems and programs
• Isabelle/HOL is a generic proof assistant that supports various logics, with higher-order logic (HOL) being the most widely used
• Offers a rich set of tools for interactive theorem proving and has been used to formalize substantial mathematical theories
• Lean is a newer proof assistant based on dependent type theory, designed for both mathematical reasoning and software verification
• Provides a powerful framework for writing formal proofs and has a growing library of mathematical results

Features and Applications

• Proof assistants offer expressive logics and type systems for specifying mathematical concepts and software systems
• Support the interactive development of formal proofs, with the ability to define custom tactics and automation
• Used in a wide range of applications, including verifying the correctness of algorithms, formalizing mathematical theories, and certifying the safety of critical systems
• Enable the construction of large, machine-checked proofs that can be independently verified, increasing confidence in the correctness of results

Interactive Theorem Proving

Proof Tactics and Interactivity

• Proof tactics are programs that automate common proof steps or strategies in interactive theorem proving
• Examples of tactics include

  intros

  for introducing hypotheses,

  apply

  for applying a lemma or hypothesis, and

  auto

  for applying a collection of standard tactics
• Interactive theorem proving involves a user guiding the construction of a proof by applying tactics and providing input when needed
• Allows for the development of complex proofs that may be difficult to fully automate, while still benefiting from the automation provided by tactics

Proof Scripts and Reproducibility

• Proof scripts are files that contain a sequence of proof commands and tactics used to construct a formal proof in a proof assistant
• Serve as a record of the proof development process and can be replayed to check the correctness of the proof
• Enable the reproducibility of proofs, as other users can examine the proof script to understand the reasoning behind each step
• Can be shared and built upon by the community, allowing for the collaborative development of large formalized theories
• Examples of proof scripts include

  .v

  files in Coq and

  .thy

  files in Isabelle/HOL, which contain the definitions, lemmas, and proofs of a formalization