Formal Mathematics and Proof Assistants - NMMB568

• Title: Formal Mathematics and Proof Assistants
• Guaranteed by: Department of Algebra (32-KA)
• Faculty: Faculty of Mathematics and Physics
• Actual: from 2023
• Semester: summer
• E-Credits: 4
• Hours per week, examination: summer s.:2/1, C+Ex [HT]
• Capacity: unlimited
• Min. number of students: unlimited
• 4EU+: no
• Virtual mobility / capacity: no
• State of the course: not taught
• Language: English
• Teaching methods: full-time
• Teaching methods: full-time
• Guarantor: Mgr. Josef Urban, Ph.D.; RNDr. Martin Suda, Ph.D.
• Class: M Mgr. MMIB; M Mgr. MMIB > Volitelné
• Classification: Mathematics > Algebra
• Incompatibility : NMMB566
• Interchangeability : NMMB566
• Is incompatible with: NMMB566
• Is interchangeable with: NMMB566

Annotation

English

Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (08.12.2022)

The topic of the course are formal (computer-understandable) mathematics and proof assistants (interactive theorem provers). A proof assistent is a tool to assist with the development of formal proofs by human-machine collaboration. The language of communication is some variant of formal logic and the main aim a high level of assurance that the proven actually holds.We will have a look at the proof assistent Lean, study its theoretical foundations and explore the ways in which it can be used to develop theories, model systems, and prove theorems in computer science as well as mathematics.

Course completion requirements

Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (08.12.2022)

Předmět je zakončen ústní zkouškou.

Literature

English

Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (08.12.2022)

Blanchette et at. The Hitchhiker's Guide to Logical Verification (available online)

Jeremy Avigad, Leonardo de Moura, and Soonho Kong. Theorem Proving in Lean (available online)

Requirements to the exam

English

Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (08.12.2022)

Will be specified later.

Czech

Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (08.12.2022)

Požadavky budou upřesněny.

Syllabus

English

Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (08.12.2022)

We will mostly follow the Hitchhiker's Guide:

Basics:

• Definitions and Statements,

• Backward Proofs, Forward Proofs

(Quantifiers, Connectives, Tactics, Propositions as Types / Proofs as Terms)

Functional–Logic Programming:

• Functional Programming

• Inductive Predicates

• Monads

(Basic Data Structures, Proofs by induction, Arithmetic tactics)

Program Semantics:

• Operational Semantics

• Hoare Logic

• Denotational Semantics

(Small-step/big-step, Partial vs Total correctness, Monotone functions, lattices and

fixpoints)

Mathematics:

• Logical Foundations of Mathematics

• Basic Mathematical Structures

• Rationals and Real Numbers

(Foundational principles, Subtypes, Quotient types, Type Classes)

Czech

Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (08.12.2022)

Většinou se budeme řídit textem The Hitchhiker's Guide to Logical Verification:

Základy:

• definice a výroky,

• Backward Proofs, Forward Proofs,

(kvantifikátory, spojky, taktiky, věty jako typy / důkazy jako termy).

Funkcionálně-logické programování:

• funkcionální programování,

• induktivní predikáty,

• monády,

(základní datové struktury, důkazy indukcí, aritmetické taktiky).

Sémantika programů:

• operační sémantika,

• Hoareova logika,

• denotační sémantika,

(small-step/big-step, částečná vs. úplná správnost, monotónní funkce, svazy a pevné body).

Matematika:

• logické základy matematiky,

• základní matematické struktury,

• racionální a reálná čísla,

(základní principy, podtypy, kvocientové typy, třídy typů).