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6f03bb937ef6ae23e118d69d74294f7b64a3c8af | subsection | 68 | 84 | Summary of results | In this paper we have computed the couplings of NS-NS and R-R fluxes to
fermionic bilinears living on general brane intersections (including
instantonic ones). The couplings have been extracted from disk amplitudes
among two open string vertex operators and one closed string vertex
representing the background fluxes. T... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1854,
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"raw": "M. Grana, D3-brane action in a supergravity background: The fermionic story, Phys. Rev. D66 (2002) 045014, arXiv:hep-th/0202118.",
"source_ref_id": "ad0470c84bc2a48b1dd9b1e88b0022420453c81f",... | 10.1088/1126-6708/2008/10/112 | 0807.1666 | Flux interactions on D-branes and instantons | [
"Marco Billo'",
"Livia Ferro",
"Marialuisa Frau",
"Francesco Fucito",
"Alberto Lerda",
"Jose F. Morales"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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9301f5947b7deccb1ad1ad4cd2d0988a5df7b456 | subsection | 69 | 84 | Summary of results | For D3-branes in flat space one obtains{\mathcal {A}}_{\mathrm {D}3} = \frac{2\pi \mathrm {i}}{3!}\,c_F(\Lambda )\,{\mathrm {Tr}}
\Big [\, \Lambda ^{\alpha A}
\Lambda _{\alpha }^{{\phantom{\alpha }}B}
\big (\overline{\Sigma }^{mnp}\big )_{AB}\,G_{mnp}^{\mathrm {IASD}} -
\bar{\Lambda }_{\dot{\alpha }A}\bar{\Lambda }^{\d... | {
"cite_spans": []
} | 10.1088/1126-6708/2008/10/112 | 0807.1666 | Flux interactions on D-branes and instantons | [
"Marco Billo'",
"Livia Ferro",
"Marialuisa Frau",
"Francesco Fucito",
"Alberto Lerda",
"Jose F. Morales"
] | [
"hep-th"
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06ecf9f3afd6b6cc6f1776a2e0c495258112e347 | subsection | 70 | 84 | Summary of results | In particular for pure {\cal N}=1 SYM,
the flux couplings for both gauge and exotic instantons follow from
(REF ,REF ,REF ) by restricting the
spinor components to A=B=0. The only contributions to fermionic
mass terms come in this case from the components G_{(3,0)} and
G_{(0,3)} related to the soft symmmetry breaking g... | {
"cite_spans": []
} | 10.1088/1126-6708/2008/10/112 | 0807.1666 | Flux interactions on D-branes and instantons | [
"Marco Billo'",
"Livia Ferro",
"Marialuisa Frau",
"Francesco Fucito",
"Alberto Lerda",
"Jose F. Morales"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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f42906105dc35e1efeb13cac49d41a991a2b4333 | subsection | 71 | 84 | Summary of results | The presence
of the \lambda ^2-term in (REF ) suggests that
R-R and NS-NS fluxes can provide a valid alternative mechanism in the
case of oriented gauge theories.
A precise study of the low energy couplings generated by instantons in presence
of such fluxes will be presented in the companion paper .AcknowledgmentsWe th... | {
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{
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"raw": "M. Billo', L. Ferro, M. Frau, F. Fucito, A. Lerda and J. F. Morales, Non-perturbative effective interactions from fluxes, arXiv:0807.4098 [hep-th].",
"source_ref_id": "ce676f22f8766b2fb447e007... | 10.1088/1126-6708/2008/10/112 | 0807.1666 | Flux interactions on D-branes and instantons | [
"Marco Billo'",
"Livia Ferro",
"Marialuisa Frau",
"Francesco Fucito",
"Alberto Lerda",
"Jose F. Morales"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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5499fa955268806a9f159519a920a131ccb6fd59 | subsection | 72 | 84 | Notations and conventions | We use the following notations for space-time indices in the real basis:d=10 vector indices: M,N, ... \in \lbrace 0, ..., 9\rbrace ;
d=4 vector indices: \mu ,\nu , ... \in \lbrace 0, ..., 3\rbrace ;
d=6 vector indices: m,n, ... \in \lbrace 4, ..., 9\rbrace .The corresponding complex indices are denoted by I, J=1, ...... | {
"cite_spans": []
} | 10.1088/1126-6708/2008/10/112 | 0807.1666 | Flux interactions on D-branes and instantons | [
"Marco Billo'",
"Livia Ferro",
"Marialuisa Frau",
"Francesco Fucito",
"Alberto Lerda",
"Jose F. Morales"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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61eb3a3b1d15bf92c44db9b4c966b8d8fd00908b | subsection | 73 | 84 | Body | In a d=10 Euclidean space the \Gamma matrices which satisfy
\left\lbrace \Gamma ^M,\Gamma ^N\right\rbrace = 2\delta ^{MN},
can be given the following explicit representation in terms of the Pauli matrices \tau ^c:\begin{aligned}\Gamma ^0 & = \tau ^1 \otimes ~\mathbf {1} \otimes ~\mathbf {1} \otimes ~\mathbf {1} \otimes... | {
"cite_spans": []
} | 10.1088/1126-6708/2008/10/112 | 0807.1666 | Flux interactions on D-branes and instantons | [
"Marco Billo'",
"Livia Ferro",
"Marialuisa Frau",
"Francesco Fucito",
"Alberto Lerda",
"Jose F. Morales"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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0.01... |
b8722dc93a41ed57793036000c745888a9128621 | subsection | 74 | 84 | Body | In fact, by writing\Gamma ^\mu =\gamma ^\mu \otimes \mathbf {1}~~,~~
\Gamma ^m =\gamma _{(5)} \otimes \ \gamma ^m~~,~~
\Gamma _{(11)}^{\mathrm {E}} =\gamma _{(5)} \otimes \gamma _{(7)}~~,~~
C =C_4\otimes C_6~,we can read off the explicit representation of the Dirac matrices \gamma ^\mu and
\gamma ^m for d=4 and d=6, re... | {
"cite_spans": []
} | 10.1088/1126-6708/2008/10/112 | 0807.1666 | Flux interactions on D-branes and instantons | [
"Marco Billo'",
"Livia Ferro",
"Marialuisa Frau",
"Francesco Fucito",
"Alberto Lerda",
"Jose F. Morales"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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0.... |
2c126556ecfc05c7e18e11fc565a3f72e02f136b | subsection | 75 | 84 | Body | The matrices \big (\sigma ^\mu \big )_{\alpha \dot{\beta }} and
\big (\overline{\sigma }^\mu \big )^{\dot{\alpha }\beta } act on spinors of definite chirality \psi _{\alpha } and \psi ^{\dot{\alpha }} as\big (\sigma ^\mu \big )_{\alpha \dot{\beta }} \,\psi ^{\dot{\beta }}\quad \mbox{and}\quad \big (\overline{\sigma }^\... | {
"cite_spans": []
} | 10.1088/1126-6708/2008/10/112 | 0807.1666 | Flux interactions on D-branes and instantons | [
"Marco Billo'",
"Livia Ferro",
"Marialuisa Frau",
"Francesco Fucito",
"Alberto Lerda",
"Jose F. Morales"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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0.... |
1db36c7e0a07cc8594af1af79dbab88c58584598 | subsection | 76 | 84 | Body | In this basis the
matrices \gamma ^m\,C_6^{-1} may be written in the block diagonal form\gamma ^m \, C_6^{-1}=
\begin{pmatrix}
\Sigma ^m & 0 \\
0 & \overline{\Sigma }^{\,m}
\end{pmatrix}where (\Sigma ^m)^{AB} and (\overline{\Sigma }^{\,m})_{AB} are 4\times 4 anti-symmetric matrices.If we order the four chiral indices a... | {
"cite_spans": []
} | 10.1088/1126-6708/2008/10/112 | 0807.1666 | Flux interactions on D-branes and instantons | [
"Marco Billo'",
"Livia Ferro",
"Marialuisa Frau",
"Francesco Fucito",
"Alberto Lerda",
"Jose F. Morales"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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4b17d0f5e76ca80ab79277bd1d4b47d13c995e4c | subsection | 77 | 84 | Useful formulas | The previous formulas allow us to obtain the explicit expressions for the fermion bilinears which have been discussed in Sections and .
In this respect we point out that in writing a fermion bilinear, like the one appearing for instance
in Eq.(REF ), we always understand the inverse charge conjugation matrix C^{-1}.
Th... | {
"cite_spans": []
} | 10.1088/1126-6708/2008/10/112 | 0807.1666 | Flux interactions on D-branes and instantons | [
"Marco Billo'",
"Livia Ferro",
"Marialuisa Frau",
"Francesco Fucito",
"Alberto Lerda",
"Jose F. Morales"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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14d21dca55c5bb2d6aa6560d138d4b612930afc7 | subsection | 78 | 84 | Useful formulas | Indeed, computing the holomorphic
and anti-holomorphic products \gamma ^{123} and \gamma ^{\bar{1} \bar{2} \bar{3}} and combining
them with the charge conjugation matrix we find\begin{aligned}\gamma ^{123} \,C_6^{-1} &= -\,\begin{pmatrix}
0\phantom{0}& 0 \\
0\phantom{0} & 1
\end{pmatrix}
\otimes \begin{pmatrix}
0\phant... | {
"cite_spans": []
} | 10.1088/1126-6708/2008/10/112 | 0807.1666 | Flux interactions on D-branes and instantons | [
"Marco Billo'",
"Livia Ferro",
"Marialuisa Frau",
"Francesco Fucito",
"Alberto Lerda",
"Jose F. Morales"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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b3576c3cf39d4b2d8ce5135ce883d20c9a3cbcb6 | subsection | 79 | 84 | The orbifold | The orbifold group
\mathbb {Z}_2\times \mathbb {Z}_2 acting on the orthonormal complex
coordinates Z^i of \mathcal {T}_6 as in Table REF is
a discrete subgroup of \mathrm {SO}(6) that contains 4 elements
h_I (I=0,1,2,3), with h^0\equiv e being the identity element,
andh^1 = \mathrm {e}^{\mathrm {i}\pi (J_3 - J_2)}~,~~~... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1909,
"openalex_id": "",
"raw": "D. Friedan, E. J. Martinec, and S. H. Shenker, Conformal Invariance, Supersymmetry and String Theory, Nucl. Phys. B271 (1986) 93.",
"source_ref_id": "96271dd998cb0396c24f487cab615e5468aeb35d"... | 10.1088/1126-6708/2008/10/112 | 0807.1666 | Flux interactions on D-branes and instantons | [
"Marco Billo'",
"Livia Ferro",
"Marialuisa Frau",
"Francesco Fucito",
"Alberto Lerda",
"Jose F. Morales"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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5e8bb099bfc58ca703408b62b4f3d319a0d5116c | subsection | 80 | 84 | The orbifold | Using this information, we easily deduce from (REF ) the
transformation properties of the various spin fields, which are
summarized in the following table\begin{}{c|c|c}
irrep R_A & S^A & S_A \\
\hline R_0 & S^0 \equiv S^{+++}
& \Big .S_{0} \equiv S_{---}
\\
R_1 & S^1 \equiv S^{+--}
& \Big .S_{1} \equiv S_{-++}
\\
R_2 ... | {
"cite_spans": []
} | 10.1088/1126-6708/2008/10/112 | 0807.1666 | Flux interactions on D-branes and instantons | [
"Marco Billo'",
"Livia Ferro",
"Marialuisa Frau",
"Francesco Fucito",
"Alberto Lerda",
"Jose F. Morales"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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b9825fe126f1b85d3cf1ce402f87f6e2d4b10e89 | subsection | 81 | 84 | Soft supersymmetry breaking on fractional D9 branes | In the orbifold \mathcal {T}_6/{\mathbb {Z}_2}\times {\mathbb {Z}_2}, we can realize an \mathcal {N}=1 d=4 gauge theory using fractional D9 branes that completely wrap the internal compact space.
Such brane configuration preserves a different \mathcal {N}=1 supersymmetry
with respect to the fractional D3 branes conside... | {
"cite_spans": []
} | 10.1088/1126-6708/2008/10/112 | 0807.1666 | Flux interactions on D-branes and instantons | [
"Marco Billo'",
"Livia Ferro",
"Marialuisa Frau",
"Francesco Fucito",
"Alberto Lerda",
"Jose F. Morales"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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405b11b642a91d965dbdf690ec4dede1404450f7 | subsection | 82 | 84 | Soft supersymmetry breaking on fractional D9 branes | Notice also that
\int _{\mathcal {T}_6/(\mathbb {Z}_2\times \mathbb {Z}_2)}\!\!\! d^6y \sqrt{-g_{(6)}} =
\frac{1}{4}\, (2\pi \sqrt{\alpha ^{\prime }})^6\, \mathrm {e}^{3\varphi /2}\mathcal {V}
since it corresponds to the internal volume in the string frame, which is related by the factor of \mathrm {e}^{3\phi /2}
to... | {
"cite_spans": []
} | 10.1088/1126-6708/2008/10/112 | 0807.1666 | Flux interactions on D-branes and instantons | [
"Marco Billo'",
"Livia Ferro",
"Marialuisa Frau",
"Francesco Fucito",
"Alberto Lerda",
"Jose F. Morales"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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d0982b66e5250b988b37cb2347bc594cbba4a6a2 | subsection | 83 | 84 | Soft supersymmetry breaking on fractional D9 branes | (REF ) in the D3 case,
since now the normalization c_F contains contains the topological factor \mathcal {C}_{(10)}
suitable for D9 disk amplitude, namely \mathcal {C}_{(10)} = \frac{\mathcal {C}_{(4)}}{(2\pi \sqrt{\alpha ^{\prime }})^6}~.On the other hand to obtain the four dimensional couplings, we have to
dimensiona... | {
"cite_spans": []
} | 10.1088/1126-6708/2008/10/112 | 0807.1666 | Flux interactions on D-branes and instantons | [
"Marco Billo'",
"Livia Ferro",
"Marialuisa Frau",
"Francesco Fucito",
"Alberto Lerda",
"Jose F. Morales"
] | [
"hep-th"
] | 2,008 | en | Physics | [
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badedf8786f2c3fe1676ddbb76db79acc4101f76 | abstract | 0 | 1 | Abstract | We consider perturbations of the complex quadratic map $ z \to z^2 +c$ and
corresponding changes in their quasi-Mandelbrot sets. Depending on particular
perturbation, visual forms of quasi-Mandelbrot set changes either sharply (when
the perturbation reaches some critical value) or continuously. In the latter
case we ha... | {
"cite_spans": []
} | 0807.1667 | Quasi-Mandelbrot sets for perturbed complex analytic maps: visual
patterns | [
"A. V. Toporensky"
] | [
"cs.GR"
] | 2,008 | en | Computer Science | [
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4d982b6c34420e94b99ad20d5022c3b2f7ec0ae6 | abstract | 0 | 71 | Abstract | In this article we present a method for formally proving the correctness of
the lazy algorithms for computing homographic and quadratic transformations --
of which field operations are special cases-- on a representation of real
numbers by coinductive streams. The algorithms work on coinductive stream of
M\"{o}bius map... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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fbd78a13f32755b2cdd24acb820774678b236452 | subsection | 1 | 71 | Introduction | Exact real numbers constitute one of the prime examples of infinite objects in
computer science. The ubiquity and theoretical importance of real numbers as
well as recent safety-critical applications of exact arithmetic makes them an
important candidate for applying various approaches to formal verification.
Among such... | {
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{
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b5eb682e444ea0f6880b51a95cabb766e22f7f18 | subsection | 2 | 71 | Introduction | All the lemmas in this
article are formalised and proven in Coq, a list of their machine checked
counterparts is given in Appendix . Most of the
proofs of lemmas are omitted in the paper, however in some cases the proof is
given in human language. In any case all the machine checked version of the
statements and proofs... | {
"cite_spans": [
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"raw": "M. Niqui. http://homepages.cwi.nl/ milad/ETrees/coinductive-field/ and http://coq.inria.fr/contribs/CoinductiveReals.html [cited 9 July 2008], June 2007. Files for Coq 8.1.",
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596be224ae375ea5dc0fdd54e976bcc55ec76425 | subsection | 3 | 71 | Related Work | The stream representation of exact real numbers have been recently formalised in
a coinductive setting by Ciaffaglione and Di
Gianantonio , , Bertot ,
Hou and Gibbons . Ciaffaglione and Di
Gianantonio use the Coq proof assistant to formalise a representation of real
numbers in [-1,1] as ternary streams and to prove th... | {
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6160beb651512457f5bc19be12eba5fb19e533d3 | subsection | 4 | 71 | Related Work | The big picture that
we are working on, i.e., the formalisation of the Edalat–Potts normalisation
algorithm is related to the work in , ,
that reconcile the coalgebraic structure of real numbers with algebraic
operations on them.The general issue of formalising functions from streams to streams within
logical framewor... | {
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{
"arxiv_id": "",
"doi": "",
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"raw": "D. Pavlović and M. H. Escardó. Calculus in coinductive form. In Proc. of the 13th Annual IEEE Symposium on Logic in Computer Science, pages 408–417, 1998.",
"source_ref_id": "9a22e285308aef35c... | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
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cf6b918facd6964798f3a6a91199e23b16a506b9 | subsection | 5 | 71 | Type Theoretic Coinduction | The Coq proof assistant is an implementation of Calculus of
Inductive Constructions (CIC) extended with coinductive types. This is is an
extension of Martin-Löf intensional type theory. Coinductive types are intended
for accommodating infinite objects such as streams and infinite trees in type
theory . This is in cont... | {
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c3275c442c119cefb7f17e5f872786ece16c5a36 | subsection | 6 | 71 | Type Theoretic Coinduction | When given a well-typed definition that satisfies a
guardedness condition, this operator will introduce an infinite object
that inhabits the coinductive type.The typing rule for this operator is given by the following judgement (here, let
I be a coinductive type with parameters P_0,\dots ,P_i).\frac{\displaystyle \stru... | {
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"raw": "E. Giménez. Un Calcul de Constructions Infinies et son Application a la Verification des Systemes Communicants. PhD thesis PhD 96-11, Laboratoire de l'Informatique du Parallélisme, Ecole Normale Su... | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
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883a0a0aa649b4b953544299d145a569d1d2323f | subsection | 7 | 71 | Type Theoretic Coinduction | From a
coalgebraic point of view this treatment of coinductive types by means of
constructors and cofixed point operator might seem unnatural: final coalgebras
are about observations and not constructions; final coalgebra should be given
using its destructor. Nevertheless, presenting the coinductive types in the
Coq wa... | {
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"raw": "M. Abbott, T. Altenkirch, and N. Ghani. Containers - constructing strictly positive types. Theoret. Comput. Sci., 342:3–27, September 2005.",
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bdc2a1c3f7b3224eae93464326c630e92dc436de | subsection | 8 | 71 | Type Theoretic Coinduction | For instance, for the
case of filter on prime numbers, one has to (1) consider a number theoretic
constructive proof of the infinitude of primes, (2) from this proof extract a
function \kappa that returns the nth prime number, (3) use \kappa to
rewrite \mathsf {filter} in a way that it passes syntactic tests of product... | {
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"raw": "M. Niqui. Productivity of Edalat–Potts exact arithmetic in constructive type theory. Theory Comput. Syst., 41(1):127–154, July 2007.",
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34364660d0d6238778c329ef2c40d345df05e3a5 | subsection | 9 | 71 | Type Theoretic Coinduction | This
leads us to the following definition for a coinductive extensional equality on
streams which we denote by \mathop {\cong }.CoInductive \mathop {\cong } \colon A^{\omega } \mathord {\rightarrow } A^{\omega } \mathord {\rightarrow } \textsf {Prop}:=
| \mathop {\cong } _{c} \colon \forall (\alpha _{1} \alpha _{2}\col... | {
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7c85cf11eb2137ea37e33c362967818aa27b20e3 | subsection | 10 | 71 | Homographic and Quadratic Algorithms | The homographic and quadratic algorithms are similar to Gosper's
algorithm for addition and multiplication on continued
fractions and form the basis of the Edalat–Potts approach to lazy exact real
arithmetic , .Here we use a representation which is much simpler than the continued fractions
but it is redundant enough t... | {
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"raw": "R. W. Gosper. HAKMEM, Item 101 B. http://www.inwap.com/pdp10/hbaker/hakmem/cf.htmlitem101b, [cited 9 July 2008], February 29 1972. MIT AI Laboratory Memo No. 239.",
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301e6d75749d6e022b7c46513862dbde7f7a0028 | subsection | 11 | 71 | Homographic and Quadratic Algorithms | We interpret each digit by a
refining Möbius map as follows.\mathbf {L} =\left[\begin{}
\frac{1}{2}&\frac{-1}{2}\\
\frac{1}{2}&\frac{3}{2}\\
\end{}\right]\hspace{5.0pt},\quad \mathbf {R} =\left[\begin{}
\frac{1}{2}&\frac{1}{2}\\
\frac{-1}{2}&\frac{3}{2}\\
\end{}\right]\hspace{5.0pt},\quad \mathbf {M} =\left[\begin{}
1&... | {
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15c5e669d82c3380a3519ec1bf7a66f2892d1bfb | subsection | 12 | 71 | Homographic and Quadratic Algorithms | This enables us to state the
homographic algorithm:\mathsf {homographic} \quad \!\mu \quad \!(x\colon \!\!\!\colon xs) :=
\\
{\left\lbrace \begin{array}{ll}
\mathbf {L} \colon \!\!\!\colon \mathsf {homographic} \quad \!(\mathbf {L} ^{-1} \mathop {\circ }\mu )\quad \!(x\colon \!\!\!\colon xs)&\qquad \textbf {\upshape { ... | {
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c5f9c2b1cb68e758862335ac666b8ad19ab777f1 | subsection | 13 | 71 | Homographic and Quadratic Algorithms | We start
pushing \mu towards the infinity by absorbing digits (hence obtaining a new
refining Möbius map) and emitting digits whenever the emission condition holds,
i.e., whenever the range of Möbius map applied to the interval [-1,1] fits inside
the range of a digit.\mu \mathop {\circ }\phi _0\mathop {\circ }\phi _1 \... | {
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"raw": "P. J. Potts, A. Edalat, and M. H. Escardó. Semantics of exact real number arithmetic. In Proc. of the 12th Annual IEEE Symposium on Logic In Computer Science, pages 248–257, 1997.",
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55e327f7b73205e245e2d6de583e1f4e3907f68d | subsection | 14 | 71 | Homographic and Quadratic Algorithms | With this notation we can present the
quadratic algorithm:\mathsf {quadratic} \quad \!\xi \quad \!(x\colon \!\!\!\colon xs)\quad \!(y\colon \!\!\!\colon ys):=
\\
{\left\lbrace \begin{array}{ll}
\mathbf {L} \colon \!\!\!\colon \mathsf {quadratic} \quad \!(\mathbf {L} ^{-1} \mathop {\circ } \xi )\quad \!(x\colon \!\!\!\c... | {
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5b2b903e1c6ddb1a53205c1b6e7dfa74350bb09f | subsection | 15 | 71 | Homographic and Quadratic Algorithms | However, we mention that transferring the
computation to the whole real line is possible. A possibility would be to first
move to [0,+\infty ] via the inverse of the above conjugacy map. Form here we
can follow where a redundant sign bit is added by considering a
fourth order elliptic Möbius map that leads to a cyclic... | {
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fe6eb318eeb87a8fe143f95c2ce040199e2fb091 | subsection | 16 | 71 | General Corecursive Version of the algorithms | Algorithms of the previous section specify partial functions into the
coinductive type of streams. This partiality is problematic for us. Translating
these specifications into the language of Coq means that we should ensure that
the returned value is provably an infinite stream, which is obviously not always
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74ba43a9600162a3672ea0656f5a30fd82a5e070 | subsection | 17 | 71 | Homographic Algorithm | Let
\mathbb {M} (resp. \mathbb {T}) be the set of Möbius maps (resp. quadratic maps) in
Coq They can be considered as \mathbb {Q} ^{4} and \mathbb {Q} ^{8} respectively,
forgetting about the refining and nonsingular properties. Those properties
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"raw": "C. Dubois and V. V. Donzeau-Gouge. A step towards the mechanization of partial functions: domains as inductive predicates. In M. Kerber, editor, Proc. Workshop on Mechanization of Partial Functions... | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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0... |
5d3a04c85c4ac792c618258837c1b565ebebd534 | subsection | 18 | 71 | Homographic Algorithm | According to this method we need
to define an inductively defined predicate E_h(\mu ,\alpha ) with the intended
meaning that \mu and \alpha are in the domain of m_h which in turn means
that the homographic algorithm should emit at least one digit when applied on
\mu and \alpha . Thus, as a first step in the definition ... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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0.011179659515619278,
-0.0... |
bb478512aaeadb4a0a1b20c32107eafd730ee36c | subsection | 19 | 71 | Homographic Algorithm | E_{h}(\mu ,\alpha )\longrightarrow \mathbf {DIG} \times \mathbb {M} \times \mathbf {DIG}^{\omega }as follows.Fixpoint \overline{m}_{h}(\mu \colon \mathbb {M} )(\alpha \colon \mathbf {DIG}^{\omega } )(t\colon \!E_{h} \mu \alpha ){struct t}\colon \mathbf {DIG} *(\mathbb {M} *\mathbf {DIG}^{\omega }):=
match \operatorname... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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... |
f33188a29eb295213c100cd515494ef3fd6c57b2 | subsection | 20 | 71 | Homographic Algorithm | Thus we have to use
\mathop {\oplus } \colon \textsf {Prop} {\times }\textsf {Prop} \longrightarrow \textsf {Set} — with left and
right its coprojections— to transfer propositions into a boolean sum on
which we can pattern match. Hence the need for the above lemma is inevitable,
although its proof is quite trivial.The ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1179,
"openalex_id": "",
"raw": "Y. Bertot and P. Castéran. Interactive Theorem Proving and Program Development. Coq'Art: The Calculus of Inductive Constructions. EATCS Series. Springer-Verlag, 2004.",
"source_ref_id": "78c9... | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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0.015151513740420341,
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... |
736d7cba1c501b8a26cb3e26888b4754db10b3b1 | subsection | 21 | 71 | Homographic Algorithm | E_{h}(\mu ,\alpha )\mathord {\rightarrow } \textsf {Prop} \hspace{5.0pt}.The Lemma REF enables us to prove the fixed point
equations of the \overline{m}_{h} function. These are in fact unfolding
of the body of the definition of \overline{m}_{h}; they are crucial for
proving similar results for the homographic algorithm... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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0.0... |
2e184f4ac50e86095ce5e50d177f46ff981ab4f4 | subsection | 22 | 71 | Homographic Algorithm | If \lnot \operatorname{\mathbf {Incl}(\mu ,\mathbf {L})} , \lnot \operatorname{\mathbf {Incl}(\mu ,\mathbf {R})} and
\lnot \operatorname{\mathbf {Incl}(\mu ,\mathbf {M})} holds then for all t^{\prime } a proof of property
E_{h}\big (\mu \mathop {\circ }(\mathsf {hd} (\alpha )),\mathsf {tl} (\alpha )\big ) we have
\ove... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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0.00405813381075859... |
067d167b8f9b02daee3748f73dcdb2c009074ef7 | subsection | 23 | 71 | Homographic Algorithm | This predicate will be an inductive
type with one constructor.Inductive P_{h}\colon \mathbb {M} \mathord {\rightarrow } \mathbf {DIG}^{\omega } \mathord {\rightarrow } \textsf {Prop}:=
|P_{hab}\colon \forall (\mu \colon \mathbb {M} )(\alpha \colon \mathbf {DIG}^{\omega } ),(\forall (n\colon \mathbb {N} ), \Psi _{h} (n{... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
-0.031115161255002022,
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1ab935cbec6bba71b71deef01125bcce2b572243 | subsection | 24 | 71 | Homographic Algorithm | Then if \Psi _{h}(n+1,\mu ,\alpha ) holds then
\Psi _{h}\big (n,\pi _{23}(\overline{m}_{h}(\mu ,\alpha ,t)),\pi _{33}(\overline{m}_{h}(\mu ,\alpha ,t))\big )\hspace{5.0pt}.\qquad \mathrm {{}}Finally we are ready to define the homographic algorithm as a function\bar{h}\colon \Pi (\mu \colon \mathbb {M})(\alpha \colon \m... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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... |
92f92e0d17a600bcddbd69adb7d5e2f20ad220b2 | subsection | 25 | 71 | Homographic Algorithm | Note the resemblance with the definition of the modulus of
productivity \overline{m}_{h}.Fixpoint \overline{\delta }_{h} (\mu \colon \mathbb {M} )(\alpha \colon \mathbf {DIG}^{\omega } )(t\colon \!E_{h} \mu \alpha ){struct t}\colon \mathbb {N}:=
match \operatorname{\mathbf {Incl}_{dec}(\mu ,\mathbf {L})} with
| left \R... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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0.01... |
227368a72dad45029ffa7f3c1d8c3f3dc02d17b7 | subsection | 26 | 71 | Homographic Algorithm | Then for all n if
\overline{\delta }_{h}(\mu ,\alpha ,t)=n then one of the following three
cases always holds.(a)
\;\operatorname{\mathbf {Incl}(\mu {\mathop {\circ }}\alpha _{0}{\mathop {\circ }}\ldots {\mathop {\circ }}\alpha _{n-1},\mathbf {L})} \wedge \pi _{13}\big (\overline{m}_{h}(\mu ,\alpha ,t)\big )=\mathbf {L... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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... |
c9b00d46310e2dedd8d4a65b1654e505f1ae7004 | subsection | 27 | 71 | Homographic Algorithm | Then there exist
n\in \mathbb {N} and \phi \in \mathbf {DIG} that satisfy the following three conditions.P_{h}(\phi ^{-1}{\mathop {\circ }}\mu {\mathop {\circ }}\alpha _{0}{\mathop {\circ }}\ldots {\mathop {\circ }}\alpha _{n-1},\alpha |_{n})\hspace{5.0pt};
\operatorname{\mathbf {Incl}(\mu {\mathop {\circ }}\alpha _{0... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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adbac373e248f3bcd357274f896445a0298b8a67 | subsection | 28 | 71 | Homographic Algorithm | If \textsf {\upshape rep} (\alpha ,r) holds then\textsf {\upshape rep} (\bar{h}(\mu ,\alpha ,p), \mu (r))\hspace{5.0pt}.By Lemma REF there exist n, \phi and p^{\prime }
such that\operatorname{\mathbf {Incl}(\mu {\mathop {\circ }}\alpha _{0}{\mathop {\circ }}\ldots {\mathop {\circ }}\alpha _{n-1},\phi )} \hspace{5.0pt},... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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0.001... |
5139349b2a64670b86442490fc6953d911d9eedf | subsection | 29 | 71 | Homographic Algorithm | According to Lemma REF , from (REF ) it
follows that \mu _{c} is refining. | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
-0.036020584404468536,
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87eb3214c3716faec521a7c0513fc8d02865914b | subsection | 30 | 71 | Homographic Algorithm | Note that by Properties (REF ) and
(REF ) of rep we have\alpha _{n-1}^{-1}{\mathop {\circ }}\ldots \alpha _{0}^{-1}(r)\in [-1,1] \hspace{5.0pt};and thus according to the refining property r_1\in [-1,1] .From here and (REF ), according to the statement of the
constructor \textsf {\upshape rep} _{\phi } of rep applied to... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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... |
33682a4df5ce18cfe0da79b1e34ad11f9268cec4 | subsection | 31 | 71 | Homographic Algorithm | By
examining these predicates we observe that the only topological notion appears
in the type of the constructors of E_{h}, in the form of the emission
condition. We should follow this link to obtain the productivity predicate for
a refining Möbius map.First we state some elementary properties of the interval predicate... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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0.0... |
2c87d90adc6fe189452980637568fcce0b3fc3f3 | subsection | 32 | 71 | Homographic Algorithm | The intended meaning is that for two digit sets \Phi _1 and \Phi _2,
with the same number of elements, if \operatorname{\mathsf {red}}(\Phi _1) > \operatorname{\mathsf {red}}(\Phi _2) then
\Phi _1 has more redundancy. Note that this intended meaning does not work for
adding extra digits (which decreases \operatorname{\... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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0... |
186321c72d3945112334d85e8b50759ac871ca0d | subsection | 33 | 71 | Homographic Algorithm | The
identity follows by straightforward calculation (See ).
Since all the digits are refining, we can apply
part (REF ) with
\mu _2:=\alpha _{0}{\mathop {\circ }}\ldots {\mathop {\circ }}\alpha _{n-1}.Next we can prove that after finitely many absorption steps the emitting
condition holds and thus the algorithm is `in... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 59,
"openalex_id": "",
"raw": "R. Heckmann. Contractivity of linear fractional transformations. Theoret. Comput. Sci., 279(1–2):65–82, May 2002.",
"source_ref_id": "7e77509b89932f7c95b99dd29b6db33d1c58cce8",
"start": 0... | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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d1282095adca72d3c20be627230dff355f2d7d5e | subsection | 34 | 71 | Homographic Algorithm | This is possible because we are dealing with a decidable predicate \operatorname{\mathbf {Incl}} (see \operatorname{\mathbf {Incl}_{dec}}), on the well-founded set \mathbb {N} \times \mathbf {DIG} . The idea is
that once we have a witness we can perform a search bounded by this witness to
obtain the smallest witness. T... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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... |
22f1ccb1093a4aaad3b7907b6aa99542f88db8db | subsection | 35 | 71 | Homographic Algorithm | Therefore by induction
hypothesis we haveE_{h}(\mu \mathop {\circ }\alpha _{0},\alpha |_{1})\hspace{5.0pt}.Since 0<k we know by
Lemma REF .REF that
\operatorname{\mathbf {Incl}(\mu ,\phi )} does not hold for any \phi , i.e.\lnot \operatorname{\mathbf {Incl}(\mu ,\mathbf {L})} \hspace{5.0pt},\quad \lnot \operatorname{\m... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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-0.0... |
0fa3a4b221b399e4b9133428b4be5df5968e12c6 | subsection | 36 | 71 | Homographic Algorithm | REF\lnot \operatorname{\mathbf {Incl}(\mu ,\mathbf {L})} \hspace{5.0pt},\quad \lnot \operatorname{\mathbf {Incl}(\mu ,\mathbf {R})} \hspace{5.0pt},\quad \lnot \operatorname{\mathbf {Incl}(\mu ,\mathbf {M})} \hspace{5.0pt}.From here together with the definition of \overline{\delta }_{h} and
(REF ) we obtain\overline{\de... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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0.000... |
9684c35340191f71d85df51912c15e37cfe61c26 | subsection | 37 | 71 | Homographic Algorithm | Then \Psi _{h}(n,\mu ,\alpha ) holds.
By induction on n. If n=0 then by Lemma REF we
know that E_{h}(\mu ,\alpha ) holds, and hence we can apply the constructor
\Psi _{h0} to obtain the conclusion. Assume the conclusion holds for n=k
and arbitrary refining Möbius maps. Let t_0 be the specific proof
given by Lemma REF ... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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62502a80606c1bc0b740f1ef580c84805ebfa62d | subsection | 38 | 71 | Cofixed Point Equations | Next we show that \bar{h} satisfies the specification of the homographic
algorithm. At this point we need to use the extensional equality \mathop {\cong } on
streams to prove an extensional proof irrelevance for \bar{h}. The proof of
this lemma uses Lemma REF .Let \mu \in \mathbb {M},\alpha \in \mathbf {DIG}^{\omega } ... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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956cc427933183c86b6d47e44685476c8b99bb46 | subsection | 39 | 71 | Cofixed Point Equations | If \lnot \operatorname{\mathbf {Incl}(\mu ,\mathbf {L})} , \lnot \operatorname{\mathbf {Incl}(\mu ,\mathbf {R})} and
\lnot \operatorname{\mathbf {Incl}(\mu ,\mathbf {M})} holds then for all p^{\prime } a proof of property
P_{h}\big (\mu \mathop {\circ }(\mathsf {hd} (\alpha )),\mathsf {tl} (\alpha )\big ) we have
\bar... | {
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"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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047efc8dd1b0c7a0cb70a6289e3bbed8fa0869dc | subsection | 40 | 71 | Quadratic Algorithm | In the case of the quadratic algorithm we follow the same method that we used
for the homographic algorithm. We start by defining the inductive type for the
domain of the modulus function.Inductive E_{q} \colon \mathbb {T} \mathord {\rightarrow } \mathbf {DIG}^{\omega } \mathord {\rightarrow } \mathbf {DIG}^{\omega } \... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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48f349a2fdc5c6b08862943b03d1e97935738aae | subsection | 41 | 71 | Quadratic Algorithm | Note that in this case the modulus function
\overline{m}_{q} returns a quadruple
\langle \phi ,\langle \xi ^{\prime },\langle \alpha ^{\prime },\beta ^{\prime }\rangle \rangle \rangle consisting of the emitted
digit, the new quadratic map passed to the continuation of the quadratic
algorithm and the remainder (unabsorb... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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368544b5186fd8b0f91859e3401018a1aad91b06 | subsection | 42 | 71 | Quadratic Algorithm | For brevity we do not mention them
here but their statement and proofs can be found in .Next we define the inductive predicate \Psi _{q} that ensures the validity
of E_{q} for finite iterations of \overline{m}_{q}:Inductive \Psi _{q} \colon \mathbb {N} \mathord {\rightarrow } \mathbb {T} \mathord {\rightarrow } \mathbf... | {
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"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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87ccff3019e8be61a591b777033cc5bf1fae9ec8 | subsection | 43 | 71 | Quadratic Algorithm | P_q(\xi ,\alpha ,\beta )\longrightarrow \mathbf {DIG}^{\omega }using the cofixed point operator of Coq:CoFixpoint \bar{q} (\xi \colon \mathbb {T} ) (\alpha \beta \colon \mathbf {DIG}^{\omega } ) (p\colon \!P_{q} \xi \alpha \beta ) \colon \mathbf {DIG}^{\omega }:=
Cons \pi _{14}(\overline{m}_{q} \xi \alpha \beta (P_{q}_... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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5af45c69673415a40ea0514cdfbb712f4a207a10 | subsection | 44 | 71 | Quadratic Algorithm | If \lnot \operatorname{\mathbf {Incl}(\xi ,\mathbf {L})} but \operatorname{\mathbf {Incl}(\xi ,\mathbf {R})} holds then
\bar{q}(\xi ,\alpha ,\beta ,p) \;\mathop {\cong } \; \textsf {\upshape Cons} \quad \!\mathbf {R} \quad \!\bar{q}(\mathbf {R} ^{-1}\mathop {\circ }\xi ,\alpha ,\beta )\hspace{5.0pt}.
If \lnot \ope... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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b9336dd2a3599828bcdde26550d55b07da4ee975 | subsection | 45 | 71 | Quadratic Algorithm | We can prove the
properties similar to those of \overline{\delta }_{h}.The main auxiliary lemma in this case is the following.Let \xi \in \mathbb {T},
\alpha ,\beta \in \mathbf {DIG}^{\omega } and p be a proof that P_{q}(\xi ,\alpha ,\beta )
holds. Then there exist n\in \mathbb {N} and \phi \in \mathbf {DIG} that satis... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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69293a1b88bf432b24e016548bf1bdc8813bc069 | subsection | 46 | 71 | Quadratic Algorithm | If \operatorname{\mathbf {Incl}(\xi ,\phi )} then \phi ^{-1}\mathop {\circ } \xi is a
refining tensor.From these we can prove the correctness of the quadratic algorithm. In
particular we do not need any additional property of rep apart from
those that were used for the homographic algorithm. The proof is quite similar
... | {
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"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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4d22adfa639e3a8f28d50ed12ab73a9bd26bbea1 | subsection | 47 | 71 | Quadratic Algorithm | If \xi is bounded and for each r_1,r_2\in [-1,1] ,
\xi (r1,r2)\in [\frac{-1}{3},\frac{1}{3}] then
\operatorname{\mathbf {Incl}(\xi ,\mathbf {M})}.Recall that there were two ways of composing a quadratic map and a Möbius map.
Using the lemma above we can derive the following about these two products.Let \xi be a
quadrat... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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1c6197dc64a431ae3be21d76b248588ff5cdac48 | subsection | 48 | 71 | Quadratic Algorithm | If \xi ([-1,1],[-1,1])=[x,y], we consider the following three cases.\text{If } y\le 0:&\qquad \qquad \operatorname{\mathbf {Incl}(\xi ,\mathbf {L})} &\quad \text{by Lemma~\ref {lemma:incl_properties_quadratic}.\ref {item:incl_properties_quadratic_3}}\hspace{5.0pt},\\
\text{else if } 0\le x:&\qquad \qquad \operatorname{... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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733431c801f61cb1177c24ddd7b0bb957938dbad | subsection | 49 | 71 | Quadratic Algorithm | Then for all 0<\varepsilon \in \mathbb {Q} ^{+} there exist
0<\vartheta _0,\vartheta _1\in \mathbb {Q} ^{+} such that for all x_0,x_1,y_0,y_1\in [-1,1]
if |x_0-x_1|<\vartheta _0 and |y_0-y_1|<\vartheta _1 then\operatorname{\mathbf {diam}_2(\xi ,[x_0,y_0],[x_1,y_1])} <\varepsilon \hspace{5.0pt}.At this point we are rea... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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aa7f9b8063bd3d012d890e3d883c9047114b33ed | subsection | 50 | 71 | Quadratic Algorithm | Taken:=\max (\lceil \frac{2}{\vartheta _0}\rceil ,\lceil \frac{2}{\vartheta _1}\rceil )\hspace{5.0pt}.Letx_0:=l_n(\alpha )\hspace{5.0pt},\quad y_0:=l_n(\beta )\hspace{5.0pt},\quad x_1:=u_n(\alpha )\hspace{5.0pt},\quad y_1:=u_n(\beta )\hspace{5.0pt},Note that due to (REF ) we have|x_0-x_1|\le \frac{2}{n+1}\hspace{5.0pt}... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1717,
"openalex_id": "",
"raw": "R. Heckmann. Big integers and complexity issues in exact real arithmetic. In A. Edalat, A. Jung, K. Keimel, and M. Z. Kwiatkowska, editors, Third Workshop on Computation and Approximation (Comprox ... | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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ca7bc2b9cfc851ae18ccd353f39e1798ce0d8796 | subsection | 51 | 71 | Quadratic Algorithm | Then
E_{q}(\xi ,\alpha ,\beta ) holds.Let n\in \mathbb {N} , \xi be
a refining quadratic map and \alpha ,\beta \in \mathbf {DIG}^{\omega } . Then
\Psi _{q}(n,\xi ,\alpha ,\beta ) holds.Let \xi be a
refining quadratic map and \alpha ,\beta \in \mathbf {DIG}^{\omega } . Then
P_{q}(\xi ,\alpha ,\beta ) holds.As expected t... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 829,
"openalex_id": "",
"raw": "P. J. Potts. Exact Real Arithmetic using Möbius Transformations. Ph.D. thesis, University of London, Imperial College, July 1998.",
"source_ref_id": "3776b425021ed0fec6518ae80deeb289f3ebc737",... | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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0.... |
81e56a3abfb10de8e98f53e503abefeb90389596 | subsection | 52 | 71 | General Corecursion? | Evidently the method for formalising the quadratic algorithm mimics precisely
the one used for the homographic algorithm. This suggests that one can
generalise this method to obtain a scheme in style of for
formalising specification of partial functions on coinductive types. Such a
method would be the dual of the Bove... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 276,
"openalex_id": "",
"raw": "D. Cancila, F. Honsell, and M. Lenisa. Generalized coiteration schemata. In P. Gumm, editor, Proc. of 6th Workshop on Coalgebraic Methods in Computer Science, CMCS'03, Warsaw, 5-6 Apr. 2003, volume ... | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
-0.03910055756568909,
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f6973c408e0f6a85e8b919dd6ecb87b227e6258d | subsection | 53 | 71 | General Corecursion? | There, this function is employed to provide a universal
quantifier for total predicates on streams and is used for obtaining higher
order functions such as the numerical integration. However, this function which
is the basis for defining other functions in , itself does not
satisfy the Coq guardedness condition and hen... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 428,
"openalex_id": "",
"raw": "A. K. Simpson. Lazy functional algorithms for exact real functionals. In L. Brim, J. Gruska, and J. Zlatuska, editors, Mathematical Foundations of Computer Science 1998, 23rd International Symposium... | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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70459d2892fa6a06ae6848bb1f65beead96724c5 | subsection | 54 | 71 | Representation | As it is the case with all algorithms, `to prove the correctness' of the
homographic and quadratic algorithms can point to different concepts:(i)
To prove that the algorithms satisfy their Haskell-like specification.
(ii)
To prove that the algorithms turn the set \mathbf {DIG}^{\omega } to a partial field
and behave as... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1928,
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"raw": "M. Niqui. Formalising Exact Arithmetic: Representations, Algorithms and Proofs. Ph.D. thesis, Radboud Universiteit Nijmegen, September 2004.",
"source_ref_id": "0523616ca7874699aec53dd55466d4... | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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551d9be124b7849d6d1a19629e109efba9e92e5b | subsection | 55 | 71 | Representation | This leads to the following
definition for a binary predicate \textsf {\upshape rep} \colon \mathbf {DIG}^{\omega } \times [-1,1] \longrightarrow \textsf {Prop} with the intended meaning that
\textsf {\upshape rep} (\alpha ,r) holds if \rho (\alpha )=\lbrace r\rbrace .CoInductive rep \colon \mathbf {DIG}^{\omega } \mat... | {
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"raw": "Y. Bertot. CoInduction in Coq. In Lecture Notes of TYPES Summer School 2005, August 1526 2005, Göteborg, Sweden. vol II, 2005. http://www.cs.chalmers.se/Cs/Research/Logic/TypesSS05/Extra/lectnotes... | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
"cs.LO"
] | 2,008 | en | Computer Science | [
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45becefa28edf6e7245c230868792b22cba818a6 | subsection | 56 | 71 | Representation | The reason is due to the
guardedness condition of Coq: even without the \mathop {\cong } clause in the
constructors of rep we could find a proof X, by coinduction, for the property
that\forall \alpha \beta r,\;\;\textsf {\upshape rep} (\alpha ,r) \rightarrow \alpha \mathop {\cong } \beta \rightarrow \textsf {\upshape r... | {
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57a170b12beedb76de4063b1c2f377a3c83f6241 | subsection | 57 | 71 | Representation | In fact this
function calculates the limit of converging sequence of shrinking intervals that
is obtained by successive application of the digits starting from the base
interval. To be able to define {\mathopen {[\![}{\_}\mathclose {]\!]}} we should show this converging
property. This proof is directly dependent on the... | {
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f1fc6cf1d27054a44c8ba2948501d4769da5a626 | subsection | 58 | 71 | Representation | Hence if we define l_k(\alpha ) (resp. u_k(\alpha )) to be
the lower bound (resp. upper bound) of the interval \alpha _0\mathop {\circ }\alpha _1\mathop {\circ }\ldots \alpha _{k-1}([-1,1]) we can defineNote that we
could have equivalently used the upper bounds.{\mathopen {[\![}{\alpha }\mathclose {]\!]}} =\lim \limits... | {
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149a6efb77c038a3ee80cafaec4930dbf875a8c0 | subsection | 59 | 71 | Representation | However, the \mathop {\cong } equality
does not know anything about this redundancy and it distinguishes the two
streams representing the same real number. Therefore for two different
representations \alpha _1,\alpha _2 of a real number r, there are two
different proofs \textsf {\upshape rep} (\alpha _1,r) and \textsf ... | {
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} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
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304d28a7e55414d0e52ad4afd894da0512a9446f | subsection | 60 | 71 | Coinductive Correctness | We are going to prove that the homographic and quadratic algorithms correspond
to Möbius and quadratic maps on [-1,1] as a subset of the standard model of
\mathbb {R}. We base our correctness proofs on the coinductive predicate rep and we
prove that for the functions \bar{h} and \bar{q} of
Section we have&\forall \mu ... | {
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} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
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6473a03809d373cedbc9821818d7fd9f0804b85f | subsection | 61 | 71 | Final step: Refining, Productivity and Topological Correctness | So far we have shown that the homographic and quadratic algorithms are `correct'
modulo the existence of the productivity predicates P_h and P_q. In this
section we will prove that if a Möbius map (resp. quadratic map) is refining
then irrespective of the used input stream(s) the property P_h (resp. P_q)
is always sati... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
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c06bb34336acf93a83f0e62c28ccddc6f6a46535 | subsection | 62 | 71 | Reexamining The Method | We can outline the path that we followed in this article in the following steps.Implementing algorithms in type theoretic language.
Proving that they satisfy their Haskell-like
specification.
Proving that the algorithms correspond to partial Möbius and
quadratic maps on [-1,1]:
They are total on those subsets of \ma... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
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8ef96f829a0b0e00f75adbf5688585dd90f6fa97 | subsection | 63 | 71 | Dependence on Representation | None of the steps above depend on the representation, although the
specific proofs about this representation, as well as our choice of the base
interval appeared frequently in our reasoning. In we
show that as long as a representation satisfies a few properties with respect to
the effect of its digits on the chosen bas... | {
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30a8ea4bc5d2c7706f9d20d8262a67238ed13d05 | subsection | 64 | 71 | Type Theoretic vs. Topological Properties | We pointed out this correlation throughout the article. Here we summarise it for
all the above steps. Step REF above is purely type theoretic. It
simply consist of writing a function parametrised by a proof obligation that
passes the type checking in the functional programming language of Coq. The
proofs (proof irrelev... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
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050bb499ab55e55ce150b5584e05a94def137429 | subsection | 65 | 71 | Statistics on Formalisation | Finally we present some of the statistics pertaining to the formalised
algorithms. They indicate the size (in kilobytes) and length (in number of
linesNumber of lines is obtained using the command coqwc
which disregards the commented and blank lines.) of the ASCII code of the
formalisation.In Table REF we present the r... | {
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] | [
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10d7c9c6c768c1f79d070e5ebc5cc6cc44c0606d | subsection | 66 | 71 | Conclusions and Further Work | We have shown the correctness of the homographic and quadratic algorithms on a
stream representation of real numbers in [-1,1]. Following the general set-up of
the method is easily extensible to any admissible digit
set for any compact proper subinterval of the extended real numbers
[-\infty ,+\infty ]. Our correctnes... | {
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9f6957e4fa4be31b840f52e330cb5b812b428328 | subsection | 67 | 71 | Interval Predicates for Quadratic Algorithm | Let \xi =\left[ \begin{}
a&b&c&d\\
e&f&g&h\\
\end{} \right] be a quadratic map and \phi =\left[\begin{}
\phi _{00}&\phi _{01}\\
\phi _{00}&\phi _{01}\\
\end{}\right] \in \mathbf {DIG} . | {
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} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
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9fd85f7009b7d7787be46f14c097c0931de04c23 | subsection | 68 | 71 | Interval Predicates for Quadratic Algorithm | Then:\operatorname{\mathbf {Bounded}(\xi )} &:= (0{<}e{+}f{+}g{+}h \wedge 0{<}e{-}f{-}g{+}h \wedge 0{<}{-}e{-}f{+}g{+}h \wedge 0{<}{-}e{+}f{-}g{+}h \bigvee \\
&\quad \qquad e{+}f{+}g{+}h{<}0\wedge e{-}f{-}g{+}h{<}0\wedge {-}e{-}f{+}g{+}h{<}0\wedge {-}e{+}f{-}g{+}h{<}0)\hspace{5.0pt};\\\operatorname{\mathbf {Ref}(\xi )}... | {
"cite_spans": []
} | 10.2168/LMCS-4(3:6)2008 | 0807.1669 | Coinductive Formal Reasoning in Exact Real Arithmetic | [
"Milad Niqui"
] | [
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7385ccffdbf61cb2ffa8943f2413f9d3b514d75f | subsection | 69 | 71 | Correspondence with the formalised | In the following table we present the correspondence between the terms and
lemmas in the article and their formalised version in . In the
second column foo.bar refers to the Coq term bar in file
foo.v which is available for public download at . Note
that for notations that are overloaded between the homographic and qua... | {
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d397e2b19eca0780de57bdb73032714c00d262bd | subsection | 70 | 71 | Correspondence with the formalised | \operatorname{\mathbf {Bounded}(\mu )} is in fact formalised as
digits.Bounded_M mu but we ignore mu.
[c]|l|l|
Item in article Formalised Version\mathop {\cong } digits.bisim\operatorname{\mathbf {Bounded}(\mu )} digits.Bounded_M\operatorname{\mathbf {Bounded}(\xi )} digits.Bounded_T\operatorname{\mathbf {Ref}(\mu )} d... | {
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034bdd5f4a9a947d58f98d7ec3ae4cffdb486f69 | abstract | 0 | 15 | Abstract | The LHCb collaboration has studied various promising ways to determine the
Unitarity Triangle angle $\gamma$. Three complementary methods will be
considered. The potential of the $B \to DK^{(*)}$ decays has been studied by
employing the combined Gronau-London-Wyler (GLW) and the Atwood-Dunietz-Soni
(ADS) methods, makin... | {
"cite_spans": []
} | 0807.1670 | $gamma$ measurements at LHCb | [
"Jacopo Nardulli"
] | [
"hep-ex"
] | 2,008 | en | Physics | [
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0886d9f4316994b5cab5b86a623a65448e667e12 | subsection | 1 | 15 | Introduction | LHCb aims to study CP violation and rare B-meson decays with high precision, using
the Large Hadron Collider (LHC), where all species of B-mesons are produced in
14 \mathrm {TeV} pp collisions , .
In these events the b\bar{b} pairs
are frequently produced in the same forward (or backward) direction.
The LHCb detector i... | {
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2130b22f45b04eaddc2a6256801359d9b99977a0 | subsection | 2 | 15 | Event Selection | LHCb will collect large samples of all types of B mesons and baryons.
These samples will allow the precise measurement of all the three (\alpha ,
\beta and \gamma ) angles of the Unitarity triangle and of the B_s mixing phase.
Here, three complementary methods to extract the Unitarity Triangle angle
\gamma will be cons... | {
"cite_spans": []
} | 0807.1670 | $gamma$ measurements at LHCb | [
"Jacopo Nardulli"
] | [
"hep-ex"
] | 2,008 | en | Physics | [
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086df525747013092e677f9fb29d175c86daefd4 | subsection | 3 | 15 | Extracting | Interfering tree diagrams in the B^{\pm } \rightarrow \tilde{D} K^{\pm } decays allow the determination of
the Unitarity Triangle angle \gamma . Here, \tilde{D} can be a D^0 or a
\overline{D^0} and the D^0 and the \overline{D^0} are reconstructed in a
common final state.In LHCb this is done by employing the combined Gr... | {
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... | 0807.1670 | $gamma$ measurements at LHCb | [
"Jacopo Nardulli"
] | [
"hep-ex"
] | 2,008 | en | Physics | [
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0... | |
3c61039d8f8007bb5fb1e2d7366ac96e479ab26f | subsection | 4 | 15 | Extracting | Using the ADS
method the following equations can be written for the decay rates for the
neutral B^0 \rightarrow D^0 (\rightarrow K\pi ) K^{*0} channels (very similar
equation can be written for the charged decays, see also ) :\Gamma (B^0 \rightarrow (K^+\pi ^-)_D K^{*0}) &=& N_{K\pi } (1+(r_B r_D)^2 +
2r_Br_D \cdot \\ ... | {
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... | 0807.1670 | $gamma$ measurements at LHCb | [
"Jacopo Nardulli"
] | [
"hep-ex"
] | 2,008 | en | Physics | [
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0.03... | |
1c202f860e377c2cdf99087e3842524939804441 | subsection | 5 | 15 | Extracting | Further information can be added
by including the decays to CP-eigenstates, such as \pi ^+\pi ^- or K^+K^- and so:\Gamma (B^0 \rightarrow D_{CP} K^{*0}) &=& N_{CP} (1+r_B^2 + \\ & & 2r_B\cos (\delta _B+\gamma ))\;\;\;,\\
\Gamma (\overline{B^0} \rightarrow D_{CP} \overline{K^{*0}}) &=& N_{CP} (1+r_B^2 +
\\ & & 2r_B\cos ... | {
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... | 0807.1670 | $gamma$ measurements at LHCb | [
"Jacopo Nardulli"
] | [
"hep-ex"
] | 2,008 | en | Physics | [
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0... | |
bbbaa7cebf8403436ba7404a50ddc2eba7d8b157 | subsection | 6 | 15 | Extracting | The time evolution of the flavour eigenstates |B^0\rangle and
|\overline{B^{0}}\rangle is then given by the four decay equations:\Gamma _{B \rightarrow f}(t) &=&
\left|A_f\right|^2 (1+|\lambda _f|^2) \frac{e^{-\Gamma t}}{2} \cdot \\ & &
( \cosh {\frac{\Delta \Gamma t}{2}} + D_f \sinh {\frac{\Delta \Gamma t}{2}} +
\\ & ... | {
"cite_spans": []
} | 0807.1670 | $gamma$ measurements at LHCb | [
"Jacopo Nardulli"
] | [
"hep-ex"
] | 2,008 | en | Physics | [
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... | |
086eb57841d3291d0fb249ed52bb858b82f16f18 | subsection | 7 | 15 | Extracting | A_{f} =
\langle f |T| B^0\rangle ) and\lambda _f = \frac{q}{p}\frac{\overline{A}_f}{A_f} \;\;\; , \;\;\;
\lambda _{\overline{f}} = \frac{q}{p}\frac{\overline{A}_{\overline{f}}}{A_{\overline{f}}} \;\;\; .For the B_{s}^0 \rightarrow D_s^{\mp } K^{\pm } decay channels (see Feynman diagrams in
Fig. REF ) a B^0_s, as well a... | {
"cite_spans": []
} | 0807.1670 | $gamma$ measurements at LHCb | [
"Jacopo Nardulli"
] | [
"hep-ex"
] | 2,008 | en | Physics | [
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e6f108f96f9cf4f389c151ebccc6834b6e9bdd56 | subsection | 8 | 15 | Extracting | The terms \lambda _f and \overline{\lambda }_{\overline{f}}
are calculated as\lambda _{D^{-}_s K^{+}} &=& \left(\frac{q}{p}\right)_{B_s}
\frac{\overline{A}_{D^{-}_s K^{+}}}{A_{D^{-}_s K^{+}}}
= \left(\frac{V_{tb}^*V_{ts}}{V_{tb}V_{ts}^*}\right)
\left(\frac{V_{ub}V_{cs}^*}{V_{cb}^*V_{us}}\right) \cdot \\
& & \left|\frac... | {
"cite_spans": [
{
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"end": 1109,
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... | 0807.1670 | $gamma$ measurements at LHCb | [
"Jacopo Nardulli"
] | [
"hep-ex"
] | 2,008 | en | Physics | [
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f80979b3e26440635a7ff0c0b66bc325e5ff8232 | subsection | 9 | 15 | Extracting | REF the B_{(s)}^0 \rightarrow h^+h^{\prime -} tree diagrams are shown.
[Figure: The B_{(s)}^0 \rightarrow h^+h^{\prime -} tree diagram.]For a neutral B-meson decaying into a CP eigenstate f, the time-dependent CP asymmetry
is given by:{\cal A}_{CP}(t) &=& \frac{\Gamma (\overline{B^0}_{d/s}(t) \rightarrow f)-\Gamma (B^0... | {
"cite_spans": [
{
"arxiv_id": "",
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"end": 2013,
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"en... | 0807.1670 | $gamma$ measurements at LHCb | [
"Jacopo Nardulli"
] | [
"hep-ex"
] | 2,008 | en | Physics | [
0.006796203553676605,
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0.017... | |
47264145e063e4a298637d781e3337b21d8e1820 | subsection | 10 | 15 | Extracting | This system of four equations
and five unknowns (d,d^{\prime },\theta ,\theta ^{\prime } and \gamma ) can be solved with the help of
the U-spin symmetry,
as a consequence d = d^{\prime } and \theta = \theta ^{\prime } . This results in an over-constrained
system of three unknowns and four equations. | {
"cite_spans": []
} | 0807.1670 | $gamma$ measurements at LHCb | [
"Jacopo Nardulli"
] | [
"hep-ex"
] | 2,008 | en | Physics | [
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-... | |
6069012a6a8592e0b2c2a4cfec8d8592ebdc134f | subsection | 11 | 15 | Sensitivity to | As a result of the event selection, the annual yields
together with the total efficiencies and with the background to signal
ratios are listed in Table REF , .
[Table: Summary of event yield (S),experimental efficiency (\epsilon _{tot}), and backgroundto signal ratios.]Both for the charged and neutral B-meson decays a... | {
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... | 0807.1670 | $gamma$ measurements at LHCb | [
"Jacopo Nardulli"
] | [
"hep-ex"
] | 2,008 | en | Physics | [
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0.0... |
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