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### Basic proof theory

Lev Gordeev
1998 Science of Computer Programming
Proof theory of arithmetic. Chapter 5. Refinements. 1.  ...  According to the reviewer's research, this advantage leads to a more general "nested" proof theory that continue to hold in finite-variable logic, which fails in Gentzen's theory.  ...

### Predicative proof theory of PDL and basic applications [article]

Lev Gordeev
2021 arXiv   pre-print
The whole proof can be formalized in PA extended by transfinite induction along φ_ω( 0) – actually in the corresponding primitive recursive weakening, 𝐏𝐑𝐀_φ_ω( 0).  ...  Furthermore we consider star-free formulas A in dual basic disjunctive normal form, and corresponding expansions S=⟨ P^∗⟩A∨ Z whose PDL-validity problem is known to be EXPTIME-complete.  ...  As usual in proof theory, our cutfree sequent calculi provide useful help in the verification of PDL-(un)provability of concrete formulas of simple shapes.  ...

### Alternate proofs of some basic theorems of finite group theory

Yakov Berkovich
2005 Glasnik Matematicki - Serija III
In this note alternate proofs of some basic results of finite group theory are presented. These notes contain a few new results.  ...  Our aim here is to give alternate and, as a rule, more short proofs of some basic results of finite group theory: theorems of Sylow, Hall, Carter, Kulakoff, Wielandt-Kegel and so on.  ...  Then, by Clifford theory, χ T = µ 1 + · · · + µ p , where µ 1 , . . . , µ p are pairwise distinct G-conjugate irreducible characters of M .  ...

### Short proofs of some basic characterization theorems of finite p-group theory

Yakov Berkovich
2006 Glasnik Matematicki - Serija III
We offer short proofs of such basic results of finite p-group theory as theorems of Blackburn,  ...  In Section 2 • we show that some basic results of p-group theory are easy consequences of Theorem 2.  ...  The main point of this section is to deduce from that theorem some basic results of p-group theory.  ...

### Towards a Proof Theory for Basic Logic [chapter]

Thomas Vetterlein
Advances in Soft Computing
However, for BL (Basic Logic), the logic of continuous t-norms and their residua, the situation is less satisfactory.  ...  Proof systems based on rules with the property that all formulas contained in the assumptions are contained as subformulas in the conclusion as well, are particularly suitable for automated proof search  ...  We will give an introduction to rHM L and compile its basic properties (Section 4) and in particular explain its proof search capabilities (Section 5). 2 Basic Logic -the usual Hilbert-style formulation  ...

### §36. Short proofs of some basic characterization theorems of finite p-group theory [chapter]

de Gruyter Expositions in Mathematics
We offer short proofs of such basic results of finite p-group theory as theorems of Blackburn,  ...  In Section 2 • we show that some basic results of p-group theory are easy consequences of Theorem 2.  ...  The main point of this section is to deduce from that theorem some basic results of p-group theory.  ...

### Comparing Constructive Arithmetical Theories Based on NP-PIND and coNP-PIND

M. Moniri
2003 Journal of Logic and Computation
In this note we show that the intuitionistic theory of polynomial induction on Π b+ 1 -formulas does not imply the intuitionistic theory IS 1 2 of polynomial induction on Σ b+ 1 -formulas.  ...  We also investigate the relation between various other intuitionistic first-order theories of bounded arithmetic.  ...  Proof See The theory IS 1 2 is the intuitionistic theory axiomatized by BASIC plus the scheme P IN D on positive Σ b 1 formulas (denoted Σ b+ 1 ), i.e. Σ b 1 -formulas which do not contain ¬ and →.  ...

### A Decision Method for the Intuitionistic Theory of successor

E.G.K. Lopez-Escobar
1968 Indagationes Mathematicae (Proceedings)
Proof is by induction on the length of A. If A is an atomic formula then the result is immediate. (e.g. a"= b"" is equivalent in TS to b" =a which is a basic formula).  ...  By the (elementary) intuitionistic theory of successor, TS, we understand the theory obtained from the formal theory given in IM () p. 82 by the following modifications: (1) omit from the list of formal  ...  TSc is the classical theory of successor). Then a consequence of the theorem and its proof is the following Corollary.  ...

### An Independence Result for Intuitionistic Bounded Arithmetic

Morteza Moniri
2006 Journal of Logic and Computation
This implies the unprovability of the scheme ¬¬PIND(Σ b+ 1 ) in the mentioned theory. However, this theory contains the sentence ∀x, y¬¬∃z ≤ y(x ≤ |y| → x = |z|).  ...  It is shown that the intuitionistic theory of polynomial induction on positive Π b 1 (coNP) formulas does not prove the sentence ¬¬∀x, y∃z ≤ y(x ≤ |y| → x = |z|).  ...  Proof See [K, Lemma 5.2.5]. IBASIC is the usual notation for the intuitionistic deductive closure of BASIC.  ...

### On the Hierarchy of Intuitionistic Bounded Arithmetic

M. Moniri
2007 Journal of Logic and Computation
In this paper we are concerned with cuts in models of Samuel Buss' theories of bounded arithmetic, i.e. theories like S i 2 and T i 2 .  ...  We study the basic properties of pcuts and small cuts. In particular, we prove some overspill and underspill properties for them.  ...  The theory S i 2 is axiomatized by adding the Σ b i − P IND axioms to BASIC, i.e.  ...

### Unprovability of consistency statements in fragments of bounded arithmetic

Samuel R. Buss, Aleksandar Ignjatović
1995 Annals of Pure and Applied Logic
Thus, the above sequence of lemmas clearly implies Theorem 1; namely that the following four theories are equivalent: (1) BASIC + Σ b i -PIND (2) BASIC + Σ b i -LIND (3) BASIC + Π b i -PIND (4) BASIC +  ...  Lemma 2 The following three functions can be Σ b 1 -defined in the theory BASIC + Σ b 0 -LIND, and basic properties of these functions are provable in this theory: (1) c = min(a, b). ( 2) c = LenP(a, b  ...  This finishes the proof of Lemma 22. 2  ...

### Automating Proofs in Category Theory [chapter]

Dexter Kozen, Christoph Kreitz, Eva Richter
2006 Lecture Notes in Computer Science
We introduce a semi-automated proof system for basic category-theoretic reasoning.  ...  It is based on a first-order sequent calculus that captures the basic properties of categories, functors and natural transformations as well as a small set of proof tactics that automate proof search in  ...  Proof Automation The implementation of the proof calculus described above enables us to create formal proofs for many theorems of basic category theory.  ...

### Closed Fragments of Provability Logics of Constructive Theories

Albert Visser
2008 Journal of Symbolic Logic (JSL)
In this paper we give a new proof of the characterization of the closed fragment of the provability logic of Heyting's Arithmetic.  ...  Proof The proof is by induction on the number of steps in the IQC-proof. P Fact 4.9 Let a finite tree for Basic be given.  ...  We have: HA P HA φ → P HA α Basic (φ). Proof In case α Basic (φ) = ∞, this is trivial. We assume that α Basic (φ) is finite.  ...

### Logic-based knowledge representation

1990 Discrete Applied Mathematics
Proof theories with either the Barcan formula or its converse (Proof theory with FB but without BF. Proof theory with BF but without FB. Correction: BF, FB and symmetry). Doxastic Reasoning.  ...  Proof theory for logics with the Barcan formula (Syntax. Kripke models. World indices. World-unification. M-unification. Proof theory. Examples. Soundness of w-unification. Soundness of proof theory.  ...

### Page 2073 of Mathematical Reviews Vol. , Issue 97D [page]

1997 Mathematical Reviews
Chapter 3, Set theorv, deals with the basic elements of basic set theory, unions, intersections, set products, etc. without attempting to delve too deeply into abstract set theory.  ...  Chapter |, Basic logic, introduces the basic connectives “and”, “or”, “not” and “if-then”, along with the quantifiers “for all” and “there exist”.  ...
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