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Basic proof theory

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*. ...

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

2021
*
arXiv
*
pre-print

The whole

arXiv:1904.05131v2
fatcat:v6wuedrlyneqzp6youdraxryvq
*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. ...##
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Alternate proofs of some basic theorems of finite group theory

2005
*
Glasnik Matematicki - Serija III
*

In this note alternate

doi:10.3336/gm.40.2.03
fatcat:i5iwknf3sfdvjp23hnsofor56m
*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 . ...##
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Short proofs of some basic characterization theorems of finite p-group theory

2006
*
Glasnik Matematicki - Serija III
*

We offer short

doi:10.3336/gm.41.2.07
fatcat:bxnay6fqozdurgpyays6jalbfe
*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*. ...##
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Towards a Proof Theory for Basic Logic
[chapter]

*
Advances in Soft Computing
*

However, for BL (

doi:10.1007/978-3-540-72434-6_86
dblp:conf/ifsa/Vetterlein07
fatcat:kpk6uknmebg2vedooeqtumia3m
*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 ...##
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§36. Short proofs of some basic characterization theorems of finite p-group theory
[chapter]

*
de Gruyter Expositions in Mathematics
*

We offer short

doi:10.1515/9783110208221.333
fatcat:s2q7jiwjqrdx3glgdbnixsb6aa
*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*. ...##
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Comparing Constructive Arithmetical Theories Based on NP-PIND and coNP-PIND

2003
*
Journal of Logic and Computation
*

In this note we show that the intuitionistic

doi:10.1093/logcom/13.6.881
fatcat:six76ujd45ajhprhoq7t4trxfm
*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 →. ...##
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A Decision Method for the Intuitionistic Theory of successor

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 ([1]) 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. ...

##
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An Independence Result for Intuitionistic Bounded Arithmetic

2006
*
Journal of Logic and Computation
*

This implies the unprovability of the scheme ¬¬PIND(Σ b+ 1 ) in the mentioned

doi:10.1093/logcom/exi085
fatcat:gxdlsvxk6fgjbnegws5ua7ap74
*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*. ...##
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On the Hierarchy of Intuitionistic Bounded Arithmetic

2007
*
Journal of Logic and Computation
*

In this paper we are concerned with cuts in models of Samuel Buss'

doi:10.1093/logcom/exn003
fatcat:mwptrc4gtbhddjyyatd3zpkc5a
*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

1995
*
Annals of Pure and Applied Logic
*

Thus, the above sequence of lemmas clearly implies Theorem 1; namely that the following four

doi:10.1016/0168-0072(94)00049-9
fatcat:rjf74dqd7nejtfiqt67hxtvmx4
*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 ...##
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Automating Proofs in Category Theory
[chapter]

2006
*
Lecture Notes in Computer Science
*

We introduce a semi-automated

doi:10.1007/11814771_34
fatcat:uqvwhp6e4jfalayvsdphusfqoy
*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*. ...##
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Closed Fragments of Provability Logics of Constructive Theories

2008
*
Journal of Symbolic Logic (JSL)
*

In this paper we give a new

doi:10.2178/jsl/1230396766
fatcat:7lz7ddn5d5bo5bbqx6eqowbrmu
*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. ...##
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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*. ...

##
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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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