A computer-checked proof of the Four Colour Theorem Georges Gonthier Microsoft Research Cambridge
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1 A computer-checked proof of the Four Colour Theorem Georges Gonthier Microsoft Research Cambridge This report gives an account of a successful formalization of the proof of the Four Colour Theorem, which was fully checked by the Coq v7.3.1 proof assistant [13]. This proof is largely based on the mixed mathematics/computer proof [26] of Robertson et al, but contains original contributions as well. This document is organized as follows: section 1 gives a historical introduction to the problem and positions our work in this setting; section 2 defines more precisely what was proved; section 3 explains the broad outline of the proof; section 4 explains how we exploited the features of the Coq assistant to conduct the proof, and gives a brief description of the tactic shell that we used to write our proof scripts; section 5 is a detailed account of the formal proof (for even more details the actual scripts can be consulted); section 6 is a chronological account of how the formal proof was developed; finally, we draw some general conclusions in section 7. 1 The story The Four Colour Theorem is famous for being the first long-standing mathematical problem to be resolved using a computer program. The theorem was first conjectured in 1852 by Francis Guthrie, and after over a century of work by many famous mathematicians [36,28] (including De Morgan, Peirce, Hamilton, Cayley, Birkhoff, and Lebesgue), and many incorrect “proofs”, it was finally proved by Appel and Haken in 1976 [3]. This proof broke new ground because it involved using computer programs to carry out a gigantic case analysis, which could not, in any real sense, be performed by hand: it covered, quite literally, a billion cases. The Appel and Haken proof attracted a fair amount of criticism. Part of it concerned the proof style: the statement of the Four Colour Theorem is simple and elegant so many mathematicians expected a simple and elegant proof that would explain, at least informally, why the theorem was true — not opaque IBM 370 assembly language programs [5]. Another part, however, was more rational skepticism: computer programming is known to be error-prone, and difficult to relate precisely to the formal statement of a mathematical theorem. The fact that the proof also involved an initial manual case analysis [4] that was large (10,000 cases), difficult to verify, and in which several small errors were detected, also contributed to the uncertainty. In 1995, Robertson, Sanders, Seymour, and Thomas published a more streamlined proof of the theorem [26], using an argument quite similar to the one used by Appel and Haken, but employing C programs to check both the giant and the large case analyses. This second proof, along with the edition of a definitive monograph for the original proof [6], cleared up most doubts concerning the truth of the theorem. Futhermore, Robertson et al. used the larger computational resources available to them to search for and find a significantly (four times) smaller set of cases for the second analysis, by systematically exploring variants in the initial case analysis. 2 Our work can be seen as an ultimate step in this clarification effort, completely removing the two weakest links of the proof: the manual verification of combinatorial arguments, and the manual verification that custom computer programs correctly fill in parts of those arguments. To achieve this, we have written a formal proof script that covers both the mathematical and computational parts of the proof. We have run this script through the Coq proof checking system [13,9], which mechanically verified its correctness in all respects. Hence, even though the correctness of our proof still depends on the correct operation of several computer hardware and software components (the processor, its operating system, the Coq proof checker, and the Ocaml compiler that compiled it), none of these components are specific to the proof of the Four Colour Theorem. All of them come off-the-shelf, fulfill a more general purpose, and can be (and are) tested extensively on numerous other jobs, probably much more than the mind of an individual mathematician reviewing a proof manuscript could ever be. In addition, the most specific component we use — the Coq system, to which the script is tuned — can output a proof witness, i.e., a longhand detailed description of the chain of formal logical steps that was used in the proof. This witness can, in principle, be checked independently (technically, it is a term in a higher-order lambda calculus). Because this witness records only logical steps, and not computation steps, its size remains reasonable, despite the large amount of computation needed for actually checking the proof. Although this work is purportedly about using computer programming to help doing mathematics, we expect that most of its fallout will be in the reverse direction — using mathematics to help programming computers. The approach that proved successful for this proof was to turn almost every mathematical concept into a data structure or a program, thereby converting the entire enterprise into one of program verification. Most of our work consisted in experimenting with and developing proof techniques for dealing with such “higher-level” programs, leading to a proof style that departs significantly from common practice in machine-assisted theorem proving. For instance, we have almost no use for decision procedures or complex proof tactics, and insert relatively few explicit intermediate statements. In fact, many of our proofs look more like debugger or testing scripts than mathematical arguments. We believe that the fact that the “highbrow” mathematics involved in the Four Colour Theorem can be dealt with effectively in this fashion indicates that there is considerable scope for using more effectively mathematical abstraction in software engineering. 2 The statement of the theorem Part of the attractiveness of the Four Colour Theorem is its nutshell-sized statement: Four Colour Theorem: Any planar map can be coloured with only four colours. Of course, this is somewhat of a cheat, and we should be more explicit about what we mean by “coloured map”: Four Colour Theorem: The regions of any simple planar map can be coloured with only four colours, in such a way that any two adjacent regions have different colours. 3 This still leaves the terms simple planar map, region, and adjacent open; in formal mathematics, even such intuitive graphical notions must be precisely defined. It is posible to give a quite general definition, using some basic notions in abstract topology: Definition: A planar map is a set of pairwise disjoint subsets of the plane, called regions. A simple map is one whose regions are connected open sets. Definition: Two regions of a map are adjacent if their respective closures have a common point that is not a corner of the map. Definition: A point is a corner of a map iff it belongs to the closures of at least three regions. Without the restriction to simple maps in the first definition we would have the Empire Colouring Problem, where regions can consist of two or more separate pieces; such disjointed maps can require more than four colours (up to 6r+1, to be exact, when regions are allowed to have up to r disjoint open parts each). The second definition is just a formal way of saying that regions are adjacent when they “touch” at points other than corners. For example, consider a map consisting of four regions a, b, c, d that are the interior of the four squares below Then the closure of a is the union of a with its framing square, similarly for b, c, and d. The closures of a and b have a common point P that is not in the closure of c or d, so a and b are adjacent. On the other hand, a and d are not adjacent, since the only common point of their respective closures is Q, and Q also belongs to the closures of b and d. Note again that without this restriction more than four colours could be needed, e.g., if a, b, c, and d were all mutually adjacent, the the map obtained by adding the outside of the large square would require five colours. Note also that “corners” are not necessarily isolated points: replace the horizontal line segments in the map above with two arcs of the curve y = sin 1 /x, and there’s an entire line segment [Q1Q2] of “corners” all touching each of a, b, c, and d. The other notions we used in the above – open sets, connected sets, closures – are all standard basic undergraduate material. Fully formalizing these notions along with the definitions above takes up about 30 lines of our formal proof, in the file realmap.v. But, precisely because this is a formal proof, these few lines, along with the formal definition of the real numbers in file real.v (about 200 lines in total) are all that one needs to read and understand in order to ascertain that the statement below in file fourcolor.v is indeed a proof of the Four Colour Theorem: d Q1 c Q2 a b a b c d Q P 4 Variable R : real_model. Theorem four_color : (m : (map R)) (simple_map m) -> (map_colorable (4) m). Proof. Exact (compactness_extension four_color_finite). Qed. The other 60,000 or so lines of the proof can be read for insight or even entertainment, but need not be reviewed for correctness. That is the job of the the Coq proof assistant, a job for computers. The definition we use for the real numbers is both axiomatic and constructive, following the approach used in the C-Corn library for intuitionistic real numbers [17]. We define a real_model as a type of objects with operations that satisfy a fairly standard set of axioms1 for the real numbers and prove the Four Colour Theorem for an abitrary real_model. However, we also provide an explicit construction of a model (file dedekind.v) and a proof that this model is unique up to isomorphism (file realcategorical.v). Thus, it is not necessary to read through the somewhat involved Dedekind cut construction of the real model to ascertain that the R in the statement of four_color indeed stands for the usual real numbers. Nevertheless, the existence of the construction ensures that the real axioms we use are consistent with the Coq type theory2 [13,2,35,8,24]. Although the statement of the Four Colour Theorem uses notions from Analysis, the Four Colour Theorem is essentially a result in Combinatorics: the essence of the theorem is a statement of properties of certain finite arrangements of finite objects. Indeed, most mathematical papers on the subject pay only lip service to the continuous statement and quickly (and informally) rephrase the problem in graph theory: colouring a map is trivially equivalent to colouring the graph obtained by taking the regions of the map as nodes, and linking every pair of adjacent regions. In principle, it is even possible to completely forget that regions are sets of points in the graph problem, but this is never done in practice, because the purely graph theoretic characterizations of graphs generated by planar maps, such as the Kuratowski minor exclusion [23] or the Hopcroft-Tarjan planarity test [20], are not intuitively compelling and are difficult to use in proofs. Instead, the planarity of a graph is characterized by means of a planar embedding that maps nodes to points and edges to pairwise disjoint continuous arcs (the justification for this mapping is often informal). It is then possible to apply the Jordan Curve Theorem: The complement of any Jordan (i.e., closed continuous) curve is the disjoint union of two components, each of whose frontier is the whole curve. 1 There are a few twists due to the fact that Coq is based on intuitionistic type theory rather than classical set theory. The main consequence is that we have to state all the arithmetic axioms, such as the commutativity of +, using the extensional equality of the structure (defined by x=y ≤y x ≥ y), rather than Coq’s primitive intensional equality; we would need additional extensionality axioms to construct the quotient type in Coq. Also, we had to pick a form of the supremum axioms that implies the excluded middle ¬P 2 The axiomatization of the reals in the Coq standard library includes a ceiling function that maps the reals to the inductive integers, and whose consistency with the impredicative type theory of Coq v7 is questionable. Our construction relies only on an excluded middle assumption, whose consistency has been well studied 5 to make use of the planarity asssumption, as every cycle in the graph is mapped to a Jordan curve by the embedding. However, this approach results in proofs that, while convincing for humans, are difficult to formalize because they contain an informal mix of combinatorics and topology. We therefore decided to use a different kind of combinatorial structure, known as hypermaps[15,31,32,16], to carry out the combinatorial core of the proof (hypermaps will be described in detail in section 5.1). This design decision had several advantages: • We could formulate a self-contained combinatorial theorem, and prove it under weaker logical assumptions (using only intuitionistic logic). • The use of Analysis was circumscribed to the discretization of the problem, and even this could be carried out directly by elementary methods, without appealing to the Jordan Curve Theorem (in fact, the discretization does not follow from that Theorem, so it is somewhat off topic). The second point is important because the Jordan Curve Theorem is a rather difficult result3 , whose known proofs rely heavily on either complex analysis and/or homology. The approach we have outlined above allows us to avoid the Jordan Curve Theorem, although we do use a combinatorial analog (see Section 5.1). 3 A historical overview of the proof Aside from the hypermap formulation and the preliminary discretization lemma, our proof follows very closely the 1994 proof by Robertson, Sanders, Seymour and Thomas [26,27]; in turn, that proof follows the same outline as the 1976 proof by Appel and Haken [4,5], whose general principles go back to the original (incorrect) 1879 proof by Kempe[22], with important contributions by Birkhoff[10], Heesch[18], and many others (see [36] for a full history). In presenting the general ides of the proof, we can only pay tribute to the ingenuity of our predecessors; our actual work was to find a systematic way of getting the details right – all of them. Our (combinatorial) hypermap colouring theorem can be interpreted as a special case of the general Four Colour Theorem map colouring theorem, so we shall continue to use the map colouring formulation here, even though most of the works we reference use the graph formulation introduced in Section 2. We shall make the combinatorial interpretation evident by focusing on a particular class of maps: Definition: A polygonal outline is the pairwise disjoint union of a finite number of open line segments, called edges, with a set of nodes that contains the endpoints of the edges. The regions of a finite polygonal planar map, called faces, are the connected components of the complement of a polygonal outline. 3 The machine formalisation of the Jordan Curve Theorem was the subject of several projects; T. Hales has just announced that he had completed the proof using HOL Light. There seemed to be little point in trying to replicate these works. 6 Finite polygonal maps correspond quite closely to hypermaps, and indeed the construction use to prove our discretization lemma can also be used to show that the restriction to finite polygonal maps entails no loss of generality. On the contrary, the definition of polygonal maps is slightly too broad. It allows isolated nodes, disconnected outlines, and extraneous edges that are bordered by the same face on either side, called bridges. The discretization construction avoids such degeneracies; it generates polygonal maps that are bridgeless and connected. These properties play an important rôle in the combinatorial proof; indeed, the exact statement of our combinatorial theorem is Theorem four_color_hypermap : (g : hypermap) (planar_bridgeless g) -> (four_colorable g). The faces of a finite bridgeless connected polygonal map are the interior of simple polygons (and are thus bounded), except one, which is the exterior of a simple polygon (and is thus unbounded). The edges and nodes of such a map are the sides and vertices of these polygons, respectively. Thus the map is the planar projection of a polyhedron, hence the following Definition: A polyhedral map is a finite bridgeless connected polygonal map. The number N of nodes, F of faces, and E of edges of a polyhedral map are related by the well-known Euler formula N + F – E = 2 This formula characterizes planarity. We use it to define planar hypermaps; in the discretization construction, it is established by counting points, lines and squares on a rectangular grid. In the rest of this section we give an informal outline of the combinatorial proof, transposed to polyhedral map. To avoid spurrious computational geometry considerations, we will allow edges to be broken (as opposed to straight) line segments; this is only apparent in the “digon” case in c) below. The basic proof outline of the Four Colour Theorem, oddly, has not changed since the initial, incorrect proof attempt[22] by Kempe in 1879. Kempe’s proof went as follows: a) It is enough to consider only cubic maps, where exactly three edges meet at each node: just cover each node with a new polygonal face; if the resulting cubic map can be coloured, then simply deleting the faces that were added yields a colouring of the original map. b) In a cubic map, we have 3N = 2E edge endpoints, so the Euler formula can be rewritten 2E = 6F – 12. Since 2E is also the total number of sides of all faces, this states that every face has on average almost six sides, but that globally bridge bridge isolated node 7 there are 12 missing sides (this is why it takes 12 pentagons, along with the hexagons, to stitch a football together). c) Consider a minimal cubic counter example to the Four Colour Theorem: assuming there are polyhedral maps that require at least five colours, pick a cubic one with the smallest number of faces. d) By b) this minimal counter example map must have a face with five sides or less, so the neighborhood around that face must look like one of these If we erase one or two well chosen sides of the central face, we get a smaller cubic polyhedral map, which must therefore be four colourable. (For the square and pentagon, we erase two sides x and y, chosen so that the faces on the other side of x and y, respectively, are neither equal nor adjacent.) e) If the central face is not a pentagon, we can immediately find a colour for it (recalling that for a square, we erase two opposite sides). f) If the central face is a pentagon, we may need to modify the map colouring so that a colour is free for the pentagon. Roughly,4 this is done by locally interchanging two colours in any group of faces that is surrounded by the other two colours. Such two-toned groups (called “Kempe chains”) cannot cross each other, since the map is planar. Kempe enumerated all the possible ways in which chains could connect the faces in the ring surrounding the central pentagon to show that recolouring was always possible. The error that took ten years to spot and almost a century to fix occurred in the last step — some cases were missing in the enumeration (Kempe failed to note that interchanging colours in one chain may scramble other chains). The correct proof follows the same principle, but considers larger map fragments, called configurations. A configuration consists of a connected group of (whole) faces, called its kernel, surrounded by a ring of partial faces. The first major steps towards a correct proof were made by Birkhoff in 1913[10]. He showed that g) For some reducible configurations, such as the one below whose kernel is a group of four pentagons (known as the Birkhoff diamond), the Kempe argument f) that failed for single pentagons could be carried out soundly. 4 See section 5.2 for a precise description of the version of this operation that is used in the formal proof. 8 h) The Kempe argument is sound for any configuration surrounded by a ring of at most five faces—except the one containing a single pentagon. Let us say that a ring is trivial when its interior is either empty or consists of a single pentagon; then it follows from the above that a minimal counter example can contain no non-trivial rings of length 5 or less: it must be internally 5- connected. We refer to this result as the “Birkhoff lemma” in the sequel. i) As a consequence of h), in a minimal counter example, every face is surrounded by two concentric simple rings, i.e., its second neighborhood must have a shape similar to This excludes, for instance, maps where two non-consecutive neighbors of the central face are adjacent, as in the figure below, because such maps contain a configuration with a three-face ring (indicated by the heavy mixed dash). We refer to this result as the “Birkhoff theorem” in the sequel. These results suggest the following strategy for proving the Four Colour Theorem: Find an explicit list R of configurations such that one can establish I. reducibility: every configuration in R is reducible, using some variant of g). II. unavoidability: every minimal counter example must contain at least one of the configurations in R, using b), e), h), and i). The Four Colour Theorem then follows by contradiction, since no reducible configuration should occur in a minimal counter example. 9 Although much of the work that followed focused on finding a suitable R, for us it was a given: we just used the list of 633 configurations devised by Robertson et al[26]. The methods used in g), with some generalizations and minor improvements, are sufficient for part I, reducibility. It was only in 1969, however, that a systematic method for part II, unavoidability, was published by Heesch5 [18]. The method, called discharging, can be outlined as follows 1) Find a formula for computing the “average” arity (number of sides) of the polygons in the second neighborhood of a face, such that the sum of these averages over all faces is equal to 2E, the total number of sides in the graph. 2) Show that in a minimal counter example, every second neighborhood whose average arity, computed using the formula of 1), is strictly less than 6, contains one of the reducible configurations in R. Unavoidability follows from 2), since by b) and 1) any cubic map must contain some second neighborhood whose average arity is less than 6. By i) step 2) can be carried out by enumerating the possible arities of the faces in a second neighborhood. The average arity constraint usually provides an upper bound for this enumeration, and this bound is quite tight, since by e) all arities are at least 5. Thus, the total size of the enumeration is relatively small (in the 10,000 range); furthermore, the neighborhoods produced by the enumeration will have a high density of pentagons and hexagons, and thus stand a good chance of being reducible. Heesch also gave a method for constructing suitable averaging formulas for step 1). To insure that the sum-of-averages constraint in 1) is always met, the averaging process is described in terms explicit transfers of (fractions of) arity between neighboring faces: the average arity δ(a) of the neighborhood a face a is computed as δ(a) = δ(a) + ∑b Tba – ∑b Tab where δ(a) is the arity of a, and Tba is the fraction of arity transferred to a from one of its neighbors b. Heesch defined the value of Tba with a set of explicit patterns, called discharge rules. Each rule specifies that a fraction of arity should be tranferred from b to a if their common neighborhoood fits its pattern, and Tba is the sum of all these fractions. For instance (following Robertson et al.) we use the following rule This transfers to a pentagon one fifth of a side from any adjacent face that has more than five sides. Therefore, the neighborhood of any pentagon that is not adjacent to another pentagon will have an average arity of exactly 6. Robertson et al. showed [26] that with a handful of similar rules, this property can be extended to all pentagons and hexagons: unless they are part of a reducible clump of pentagons and hexagons, their average arity is exactly 6. In other terms, the “missing side” of each pentagon is exactly borrowed from the heptagons, octagons, and polygons of higher arity in its second neighborhood. Since each rule transfers only a fraction of a side, the second neighborhood of, say, an octagon with an average arity of less than 6 must contain 5 Heesch had been developing and using the methods since the 1940s, however. 1 a /5 b 10 many pentagons, which increases the likelyhood that it will contain a reducible configuration. Unlike a weighted arithmetic average, the average arity obtained by discharging might not depend on the arities of all the faces in the neighborhood, so it may not be possible to enumerate the arities of all the faces in a neighborhood with an average arity less than 6. This is not a problem in practice, since the enumeration can be stopped as soon as a reducible configuration occurs, and the set of discharge rules can be adjusted so that this is always the case. Discharging also gives bounds on the size of neighborhoods that need to be enumerated. The rules proposed by Robertson et al. never transfer more than half a side between two faces, so there is no need to enumerate the neighborhoods of dodecagons and larger polygons, since their average arity is never less than 6. Although Heesch had correctly devised the plan of the proof of the Four Colour Theorem, he was unable to actually carry it out because he missed a crucial element: computing power. The discharge rules he tried gave him a set R containing configurations with a ring of size 18, for which checking reducibility was beyond the reach of computers at the time. However, there was hope, since both the set of discharge rules and the set R could be adjusted arbitrarily in order to make every step succeed. Appel and Haken cracked the problem in 1976 by focusing their efforts on adjusting the discharge rules rather than extending R, using a heuristic due to Heesch for predicting whether a configuration would be reducible (with 90% accuracy), without performing a full check. By trial and error they arrived at a set R, containing only configurations of ring size at most 14, for which they barely had enough computing resources to do the reducibility computation. Indeed the discharging formula had to be complicated in many ways in order to work around computational limits. In particular the formula had to transfer arity between non-adjacent faces, and to accommodate this extension unavoidability had to be checked manually. It was only with the 1994 proof by Robertson et al. that the simple discharging formula that Heesch had sought was found. We took the work of Robertson et al. as our starting point, reusing their optimized catalog of 633 reducible configurations, their cleverly crafted set of 32 discharge rules, and their branch-and-bound enumeration of second neighborhoods[27]. However, we had to devise new algorithms and data structures for performing reducibility checks on configurations and for matching them in second neighborhoods, as the C integer programming coding tricks they used could not be efficiently replicated in the context of a theorem prover, which only allows pure, side effect free data structures (e.g., no arrays). And performance was an issue: the version of the Coq system we used needed three days to check our proof, whereas Robertson et al. only needed a three hours… ten years ago! (Future releases of Coq should cut our time back to a few hours, however.) We compensated in part this performance lag by using more sophisticated algorithms, using multiway decision diagrams (MDDs) [1,12] for the reducibility computation, concrete construction programs for representing configurations, and tree walks over a circular zipper[21] to do configuration matching. This sophistication was in part 11 possible because it was backed by formal verification; we didn’t have to “dumb down” computations or recheck their outcome to facilitate reviewing the algorithms, as Robertson et al. did for their C programs [27]. Even with the added sophistication, the program verification part was the easiest, most straightforward part of this project. It turned out to be much more difficult to find effective ways of stating and proving “obvious” geometrical properties of planar maps. The approach that succeeded was to turn as many mathematical problems as possible into program verification problems. 4 Prover technology The scripts of this proof were developed and checked using the Coq v7.3.1 system [13], under the Proof General front end [7]. This combination of tools provided a very flexible basis for trying out different approaches to the problem. In particular, as the proof advanced, we were able to extend the Coq scripting language to directly support the more effective approach that emerged. These extensions enabled progress and ultimately completion of the project, but unfortunately tie down the scripts to the 7.3.1 version of Coq; the user-level syntax extension facilities of the more recent releases of the system, although easier to use, do not support our extensions (and finishing the proof was more of a priority than porting it). However, the most important feature of the Coq system for this proof was its logic, not its syntax. Specifically, the entire enterprise hinged on the fact that the Coq logical language includes a small but practical programming language, and that the computation of programs embedded in statements is an integral part of the Coq logic. In the Coq logic, the two statements P(2 + 2) and P(4) are not just equivalent, they are identical, meaning that the two statements can be used interchangeably at any point in a formal demonstation. As a consequence, one can use any proof of P(4) as a proof of P(2+2) — e.g., if P(x) is defined as 4 = x, the instance of the reflexivity of = that proves 4 = 4 also proves directly 4 = 2 + 2. This feature is called computational reflection. From the trivial 2+2 example above one may get the impression that computational reflection is neither sound nor useful: A. The correctness of demonstration using computational reflection, depends on the irks and quirks of an arcane programming language. B. If all the feature can do is prove 2 + 2 = 4, there’s not much point to it. Both impressions are utterly wrong. As regards impression A, correctness, the embedded programming language has been carefully trimmed of all the traps and trappings commonly found in general purpose programming languages, to ensure that every expression and statement has a definite, context-independent meaning. The list of features excluded by this principle is long: there are no side effects, since they would make the the outcome of a computation depend on its timing; there are no floating point or primitive integers, since their rounding/overflow behaviour is machine dependent; consequently there are no constant-time access arrays; there are no exceptions, virtual methods, unbounded loops, … In fact the most complex to implement feature of the language is the ability to call functions with arguments. Since definitions with parameters are the main building block for mathematical theories, and thus any pratical formal proof system must include support for replacing 12 parameters with values, admitting computational reflection requires no great leap of faith. On the contrary, the theory of pure functions with arguments, called the λcalculus, has been studied for over 75 years. The logical metatheory of Coq, which was established almost 20 years ago, is in fact directly based on the λ-calculus: in Coq, the consistency of the logic follows from the consistency of computations[13,2,8]. Of course, the fact that the embedded language is so spartan rather strengthens impression B: it seems difficult to write useful code in such a restricted language, despite the well-know fact that in theory, any computation can be described in the pure λ-calculus. We believe our work provides ample evidence that this is definitely not the case, and that on the contrary computational reflection is a very effective proof technique, both in the large and in the small: • In the large, we use computational reflection to implement major parts of the Four Colour Theorem proof, with programs of an unprecedented size and complexity. Most of the previous applications of computational reflection used only relatively modest implementations of formal rewrite systems [11]. • In the small, we use computational reflection to robustly automate many “small steps” of our proof, so we have reasonably short proofs without having to resort to the fickle domain-specific “decision procedures” traditionally used to handle these simple steps. The flagship application of computational reflection “in the large” occurs in the reducibility part of the proof of the Four Colour Theorem. We define a common prologue, in file cfreducible.v, that includes the following: Variable cf : config. Definition check_reducible : bool := … Definition cfreducible : Prop := … Lemma check_reducible_valid : check_reducible -> cfreducible. The variable cf is a parameter of the definitions and lemmas that follow. It stands for the explicit data representation of a configuration map, basically a string of letters and numbers. We define elsewhere a standard interpretation of that string as an explicit program for constructing a mathematical hypermap object (see section 5.3 for an explanation of that construction). The two definitions that follow are of a very different nature; the first is a program, while the second is a logical predicate: • check_reducible is a boolean expression that performs a complex combinatorial reducibility check, using a nonstandard interpretation of the map construction program cf, and the MDD computations mentioned in Section 3. • cfreducible is a mathematical proposition that asserts that the hypermap constructed by the standard interpretation of cf is indeed reducible. Using these definitions, we then prove the check_reducible_valid lemma, which asserts the partial correctness of the check_reducible program with respect to the mathematical specification cfreducible: if the check_reducible returns true, then cfreducible holds. Since cf is a parameter of this proof, the result applies for any value of cf. Using this, we can prove for example Lemma cfred232 : (cfreducible (Config 11 33 37 H 2 H 13 Y 5 H 10 H 1 H 1 Y 3 H 11 Y 4 H 9 H 1 Y 3 H 9 Y 6 Y 1 Y 1 Y 3 Y 1 Y Y 1 Y)). in just two logical steps, by applying check_reducible_is_valid to the concrete configuration above and the trivial proof of true = true, even though the configuration map represented by (Config 11 33 …) has a ring size of 14 and a longhand demonstration would need to go over 20 million cases. Of course the 13 complexity does not disappear altogether — Coq 7.3.1 needs an hour to check the validity of this “trivial” proof. The fact that evaluation is transparent in Coq’s logic has another important consequence for us: it allows us to use freely multiple levels of intermediate definitions to derive specific properties from general ones in a modular way. Because such definitions are automatically expanded and specialized by evaluation, we can directly use the lemmas associated with the generic properties on the derived properties, without having to manually unfold the property definitions or provide new specialized lemmas. For example, when in the scripts above we use the boolean value check_reducible as a logical statement, Coq silently6 interprets this as (is_true check_reducible) where (is_true b) is defined as b=true. — this is why we could directly use reflexivity in the proof of cfred232. We use this “trick” extensively in our proofs, because it allows us to handle a wide range of concepts using a relatively modest set of basic lemmas — we never needed to resort to the lemma search command of Coq. For example, we model paths with a boolean function recursively defined over sequences Variable e : (rel d). Fixpoint path [x : d; p : (seq d)] : bool := if p is (Adds y p') then (andb (e x y) (path y p')) else true. This allows us to directly use the standard lemmas for the boolean “and” function andb to reason about (path x (Adds y p)), (path x (Adds y (Adds z q))), etc. Using boolean functions in lieu of logical predicates has other benefits: we can use rewriting of boolean expressions to apply an equivalence deep inside a complex statement, e.g., to “push” the path predicate over a sequence concatenation, using Lemma path_cat : (x : d; p1, p2 : (seq d)) (path x (cat p1 p2)) = (andb (path x p1) (path (last x p1) p2)). Booleans also make it easy to reason by contradiction even in Coq’s intuitionistic logic. In particular, if a boolean assumption computes or rewrites to false, then the standard tactic Discriminate can be used to conclude the proof. This turned out to be so common that most of our proof tactics attempt to use Discriminate automatically, and this is so effective that almost all our case splits leave at most two nontrivial subgoals. Thus we can make the script layout reflect the structure of the proof just by indenting the script for the first subgoal. Because we employed mostly computable definitions we were able to rely mainly on a “generate-and-test” style of proof. Most of our proofs follow this routine: • Do a case split on relevant variables or expressions, based on their type • Compute out the proof goal, possibly along with some assumptions Quite often this was enough to complete the (sub)proof; most of remaining cases could be resolved just by nudging the computation with selective rewriting (e.g., using the path_cat lemma above). Explicit logical steps came only a distant third (7000 tactic calls, compared to 21,000 stepping/case splitting/rewriting tactic calls); about two thirds of them were backward inference (explicitly forcing a proof schema, spinning off new proof obligations), a third were forward inference (steps that explicitly add new facts). Although there are only 1500 of them, declarative forward 6 is_true is declared as a coercion from bool to sorts. 14 steps explicitly inserting a subgoal were very important because they helped structure the longer proof scripts. We found this approach to machine-assisted theorem proving quite effective, enough so that we made very little use of the (admittedly modest) proof automation facilities of the Coq system. This approach provides a very natural way of harnessing the computational resources of the proof assistant: we write and have Coq execute a set of simple, deterministic programs that compute with concrete (but partial) data. This compares rather favorably with trying to control the assistant with tactics, which are metaprograms manipulating terms, or with a set of nondeterministic rewrite rules. Of course this conclusion is relative to the problem at hand—there is not much symbolic algebra in the proof of the Four Colour Theorem, and even so there were many subproofs that would have benefited from better rewriting support. Despite these caveats, it remains that our simple-minded approach succeeded on a challenging problem. We should point out that the liberal use of intermediate definitions (over 1000) also greatly helped in keeping the development tractable: there are only 2500 lemmas in the entire proof, more than half of which are trivial rewrite equations for integers and list operations. Unfortunately combining definitions also makes much of the Coq automation ineffective: unless definitions are unfolded just at the right level, the primitive Apply, Auto, AutoRewrite and most of the more elaborate tactics will fail to match the goal; user guidance is almost always needed. Moreover it is unclear that more aggressive unfolding in automated tactics would help, since the size of the goal increases geometrically when several layers of definitions are unfolded. The size of the individual lemma proofs varies widely: over half of them are oneliners, 75% are under 5 lines, 90% under 10 lines, but at the other end 40 are longer than a screen page, up to almost 700 lines for the proof of correctness of the compilation of configuration reducibility (lemma cfctr_correct). We have such large proofs because we deliberately avoid the clutter of ad hoc lemmas that often plague formal developments; if an obscure result is only needed as an intermediate step of a higher-level lemma, we use forward reasoning to introduce it and prove at the point at which it is relevant. This often saves a significant amount of bookkeeping, since we do not need to explicitly establish and introduce the context in which the result holds. There is a tradeoff, however: having longer, more significant lemmas greatly helps bookkeeping at the global proof level, but can create bookkeeping problems for individual proofs. Indeed, the proof context (the set of variables, definitions, and assumptions that can be referenced) can easily become uncomfortably large even in a 10-line script if it is not managed. We found it difficult to do systematic context management, using only the standard set of Coq tactics. For example, it takes two separate commands to discharge an assumption H (moving H from the context to the goal G, so that following case, rewrite or reverse chaining commands will operate on H→G). Although this may appear to be a minor annoyance, this makes it very difficult to maintain consistency in a script that explicitly moves assumptions to and from the context—but that is largely what context management consists of. It also didn’t help that several useful variants of this operation required rather different, sometimes arcane sequences of commands. 15 This prompted us to develop our own tactic “shell”, so that we could make better progress with the project. Fortunately, the Coq v6-v7 systems included a rather powerful, if somewhat unwieldly, user syntax mechanism that was sufficient for our purposes (unfortunately, it has been replaced by an easier to use but less expressive notation facility in recent releases of the Coq system). Although we only put in features when they occurred repeatedly in our scripts, we ended up with a complete and rich language. We should point out that such extensions have no impact on the trustworthiness of the proof that was carried out, for two reasons: • There is a small, well-defined kernel of the system that checks the logical validity of proofs carried out with the Coq assistant. The bulk of the Coq code base, and in particular the entire tactic and input syntax subsystems can only prepare candidate proofs and submit them to the kernel for validation. For additional safety, Coq actually stores the proofs that have been approved, so that they can be independently rechecked. • Our extensions only involved modifying the tables of the Coq extensible parser. We did not add any ML code to the system, so our extensions cannot possibly interfere with the kernel code through side effects, languagedependent behaviour, or unsafe code. Our design follows the philosophy of a command shell: we offer a small number of commands, but each of them can perform a variety of related functions, according to its arguments; all commands have identical or at least very similar argument conventions. Nearly 85% of the commands in our scripts are one of the following five7 : • Move: t1 t2 t3 … => p1 p2 p3 …. This is the basic bookkeeping command, for moving assumptions and values from the context to the goal and conversely. It combines the functions of the primitive Coq commands Generalize and Intros, mixed with Clear and Pattern. The arguments t1 t2 t3 are moved to the goal; if ti is a value rather than a logical assumption, “moving ti to the goal” implies first generalizing ti, that is, replacing ti with a new variable xi. After the =>, p1 p2 p3 provide names for the assumptions or values that are moved from the goal to context; p1 p2 p3 can actually be Coq “intro patterns”, allowing tuples, lists conjunctions to be decomposed on the fly. The Move command has several other options for deleting additional assumptions, selecting the exact occurrences to generalize, and performing on-the-fly simplification. • Case: t1 t2 t3 … => [p11 | p12 | …] p2 p3 …. The Case command is a version of Move specialized for the analysis of data structures: the first pattern to the right of => must be a destructuring “intro pattern” [13]. The Case command provides additional facilities for dealing with Coq’s dependent datatypes, but is otherwise interchangeable with the Move command. • Apply: P… => [p11 | p12 | …] …. 7 We reuse the keywords of some of the primitive Coq commands, but ensure backward compatibility by always including a symbol (one of : / =>) in the list of arguments. 16 This is the basic reverse chaining command; it is similar to Coq’s primitive Apply, with two important differences: it is insensitive to intermediate definitions, and the proof method P can contain holes “?” that will be filled by matching the current proof goal. The other command arguments are similar to those of the Case command, except that the patterns p11 , p12 are used on the first, second, etc. subgoal, respectively. • Step p: t By … This is the forward chaining command, similar to Coq’s primitive Assert, except that it can be used to introduce several values (p can be a pattern), and the By introduces a tactic, not a term. The By can be omitted, in which case the command generates a subgoal for the justification of t (only 25% of the Step commands have a By in the four colour proof script). • Rewrite: t1 t2 t3... This command is the main workhorse in our scripts; it is called nearly 10000 times, almost as many times as Move and Case combined. It merges the functionality of most of Coq’s rewriting/computing commands (Rewrite, Unfold, Fold, and Simpl) with a coherent, unified notation, also adding important functionality such as selection of the rewrite occurrences. Most importantly, however, it applies a sequence of rewrite rules rather than a single rule; this feature alone about halves the size of our scripts! These commands and a handful of related ones (Elim, Clear, Injection, Congr, and Def) are defined and extensively documented in the prelude file tacticext.v that is loaded by every file in the development. The Move, Case and Apply commands also have a “view” feature that provides a tie-in for the reflect inductive predicate defined in the boolean prelude file boolprop.v. Briefly, (reflect P b) states that P and b are equivalent (P↔b=true). We prove such properties for all boolean predicates that have a useful logical counterpart; for a predicate foob, the corresponding lemma is fooP. For example in boolprop.v we prove8 Lemma andP : (reflect b1 /\ b2 (andb b1 b2)). Then we can write Move/andP=> [Hb1 Hb2] to decompose an assumption of the form (andb b1 b2) as if it were b1 /\ b2, that is, introducing assumptions Hb1: b1 and Hb2 : b2. Conversely, we can use the sequence Apply/andP; Split to split a proof of (andb b1 b2) in two subproofs of b1 and b2, respectively. Since we adopted a “command shell” design, our scripts do not read as mathematical text. However, given our proof style based on computation and expansion, we don’t think they possibly could, as we tend to get many large, unwieldly subgoals. The important thing for us was that this language allowed us to tackle effectively complex proofs, even when we didn’t have a detailed text proof to guide us. We can present this piece of anecdotic evidence as to the efficiency of our command language: it turned out we needed a variant of Streicher’s axiom K [19], which states that there is only one proof of x = x, for our dataSet structures. Axiom K is derivable in this case, because equality is decidable by definition on dataSet; in fact this derivation is in the Coq standard library (in Logic/Eqdep_dec.v). We tried rolling out our own proof, and came up with a four-line, 50-word proof script; the standard library proof has 8 The actual lemma is called andPx; andP is the corresponding syntactic form with holes “?” for b1 and b2. 17 over 100 non-comment source lines, and 500 words, so we can claim an order-ofmagnitude productivity improvement on this particular example. 5 The computer-checked proof The exercise of formalizing a famous theorem can easily degenerate in the following (somewhat depressing) routine: translate a nice, crisp, mathematical text into computer lingo, at the rate of xx days per page, balk at the long list of trivial mathematical prerequisites that ought to be known to the proof system (but aren’t), write a large ad hoc collection of lemmas to fill the deficiencies. However, our experience with the Four Colour Theorem wasn’t like this. Because we were faced with rather different implementation and performance constraints, we had to come up with our own programs for checking reducibility and unavoidability (although we did reuse as much of the Robertson et al. proof as we could). Because most graph theory proofs rely on the visual analysis faculties of the reader, and a proof assistant like Coq is fully devoid of such faculties, we mostly had to come up with our own proofs (sometimes using visual intuition as guidance, but not always). In addition, as we noted in section 2, we were working with a different, more combinatorial notion of map; while this notion was generally more cumbersome than the more familiar topological one, on several occasions its additional precision provided a clean argument where the “intuitive” situation seemed muddled at best. In particular we found we could do without the notion of “ring completion” and the “omitted straightforward but lengthy proof” of the “folklore theorem (3.3)” in the Robertson et al. paper[26]. Indeed, the biggest payoff was that we could weaken the definition of “appears”, which involves graph isomorphism, into one that involved only morphism, thereby simplifying considerably the configuration occurrence check and its correctness proof. The remainder of this section gives an account of the mathematics and the algorithms involved in the actual computer proof: the theory of planar hypermaps, the reducibility computation, the construction programs for configuration maps and their interpretations, the Birkhoff lemma and the embedding of configurations in a minimal counter example, the enumeration of second neighborhoods, and the discretization of the continuous Four Colour problem to the hypermap one. 5.1 Hypermaps Having ruled out topological definitions of planar maps, we needed a combinatorial replacement for them in order to state (and hopefully prove) the Four Colour Theorem. That structure would have to a) explicitly represent all local geometrical connections b) support a clearly recognizable definition of “planar” c) be easy to manipulate in Coq This combination of requirements ruled out all the structures that had been used in the previous formal developments of graph theory, e.g., simplicial representations, which have separate sorts for edges, faces, and nodes are too complex to meet c), and graphs 18 verifying the Kuratowski minor exclusion [23] fail both a) and c). Rather than combing the literature in search of an appropriate definition, we tried to roll out our own from first principles, and ended up rediscovering hypermaps, which are wellknown combinatorial structures [15,31,32,33,29,16]. We reasoned as follows: 1. Since an essential part of our approach was to define properties by computable programs, we wanted a representation on which graph traversal would be easy to program. A natural starting point was therefore the basic data structure for directed graphs, an array G of lists of integers, where G[i] lists the immediate neighbors of node i (nodes are represented by integer indices). 2. Each cell in these lists corresponds to an edge in the directed graph. Additional edge attributes, such as labels, are often added to the cells; in particular, for symmetric graphs, it is common to add reciprocal pointers between the two cells that correspond to the same undirected edge in the graph. More precisely, when adding an edge e = {i,j} to a symmetric graph, that is, adding j to G[i] and i to G[j], means inserting a cell cj in G[i] and a cell ci in G[j], and putting extra pointers cj → ci and ci → cj in ci and cj, respectively. In this way, the edge e is represented by the linked pair ci ↔ cj: each of the cells ci, cj represents a “halfedge”, called a dart, in the hypermap terminology (some authors use the term flag). 3. The structure can encode geometrical information at no extra cost, simply by taking the convention that G[i] list the darts of node i in geometrical order (say, counterclockwise). Since this is a cyclic order, it is then natural to make the cell lists circular, by having the last cell point to the first. This simplifies considerably geometrical traversals: to go from any dart to the next dart on the same face, clockwise, simply follow, successively, the “edge reciprocal” and “node list” pointers. Moreover, each non-isolated node of the graph is now represented by its circular list, just as the edges are represented by the reciprocal dart pairs. This allows us to further simplify the structure by removing the G array and the integer indices altogether, as the “edge” pointer of a cell cj always points into the same list as G[j]. We assume that we also have some way of iterating over all cells. 4. The resulting structure is easy to represent in Coq: there is only one type of objects, d (for darts), whose only properties is that it is has an equality test and is finite. Pointer attributes like the “edge” and “node” pointers above are just 0 1 3 2 3 3 1 0 0 0 0 1 3 2 3 3 1 0 0 0 1 2 3 19 functions e, n : d → d, and in general any additional attribute (e.g., “colour”) can be represented by a function on d. Here n and e must be, respectively, a permutation and an involution on d. Weakening the requirement for e to just being a permutation, that is, allowing “edges” to have more than two sides, yields the traditional definition of hypermaps: a pair of permutations of a finite set. Hypermaps have been extensively used in the study of graph enumeration [16]. 5. However, even this simplified definition turned out to be impractical in Coq, because “being a permutation” property is moderately difficult to prove and use: both the “there exists an inverse” and the “injective” characterizations require deductive steps to use; in our approach, this is five to ten times more expensive than an equational characterization, because a single Rewrite: command can perform five to ten rewrite steps. For this reason, we extended the structure with a third dart attribute, a pointer to the next counterclockwise dart on the same face, represented by a function f : d → d. The navigation trick pointed out in 3. above turns into an equation e ◦ n ◦ f = Id Because d is finite, this identity characterizes hypermaps; it implies that e, n, and f are all permutations, and that we also have n ◦ f ◦ e = Id and f ◦ e ◦ n = Id. As Tutte pointed out [31], this representation is circularly symmetrical with respect to edges, nodes, and faces. 6. Although their name suggests drawing “darts” as arrows on the original map, this leads to horrendously confused figures when one adds arrows for the e, n, and f functions. In the figures below, we therefore always depict darts as points (we use small hexagonal bullets). When we draw an ordinary map together with the corresponding hypermap, we arrange the darts in circles around the corresponding nodes, halfway between adjacent edges. This way, each node appears at the center of the corresponding n cycle, each f cycle is inset in the corresponding face, and each e cycle is the diagonal of an n-f quadrilateral centered on the corresponding edge, as in the figure below. However, we rarely find it necessary to draw such complex figures; most of the time, we only need to consider the diagram around a single dart, which has the following fixed, regular shape e n f dart node edge map 20 The hypermaps that correspont to plain maps have the property that all e cycles have length 2; this implies the additional identities e = e -1 = n ◦ f. We call such hypermaps plain hypermaps. Although the central part of the proof of the Four Colour Theorem uses only plain hypermaps, most of the basic results on hypermaps are easier to establish for general hypermaps: many of the constructions described in this section create general hypermaps. Although we primarily selected the “one domain, three functions, one equation” definition of hypermaps for its simplicity, its threefold symmetry allows significant reuse of definitions and lemmas. For example, the Euler formula takes a completely symmetrical form for this presentation of hypermaps #e + #n + #f = #d + 2#(e∪n∪f) where #e, #n and #f denote the number of distinct cycles of e, n, and f, respectively, #d the total number of darts, and #(e∪n∪f) denotes the number of connected components of the relation obtained by taking the union of the graphs of all three functions. Because of this symmetry, the graph/map duality is completely trivial in our setting, so there is no point for us in switching from map colouring to graph colouring, as is traditionally done; all the lemmas in our development are therefore phrased in terms of map colouring. We actually chose to define planar hypermaps as those that satisfy this generalized Euler formula, since this property is readily computable. Much of the Four Colour Theorem proof can be carried out using only this formula. In particular Benjamin Werner, who worked with us on the early part of the proof, found out that the proof of correctness of the reducibility part (step I) was most naturally carried out using the Euler formula only. As the other unavoidability part of the proof (steps II) is explicitly based on the Euler formula, one could be misled into thinking that the whole theorem is a direct consequence of the Euler formula. This is not the case, however, because unavoidability also depends on the Birkhoff theorem giving the shape of second neighborhoods. Part of the proof of the latter requires cutting out the submap inside an arbitrary simple ring of 2 to 5 faces in the (hypothesized) counterexample map. Identifying the inside of a ring is exactly what the Jordan Curve Theorem is about, so this calls for a combinatorial analog of that famous theorem. As a side benefit, we will get additional assurance that our definition of planarity is correct by proving that the hypermap Jordan property is actually equivalent to the hypermap Euler formula (about half of that reciprocal proof is a lemma required for the correctness of the reducibility check). Unfortunately, the naïve transposition of the Jordan Curve Theorem from the continuous plane to discrete maps fails. Simply removing a ring from a hypermap, even a connected one, can leave behind any number of components: both the “inside” and the “outside” may turn out to be empty or disconnected. A possible solution, proposed by Stahl [29], is to consider paths (called chords below) that go from one face of the ring to another (loops are allowed). The Jordan Curve Theorem then tells e n f 21 us that such paths cannot start from the “inner half” of a ring face, and end at the “outer half” of a ring face,9 i.e., there are no “Moebius rings” such as the one below. Given careful definitions of the “inner” and “outer” halves of ring faces, this can be made to work (indeed, we have done this in file rjordan.v, as an addendum to the proof), but the resulting definitions are too unwieldly and complex to be practical for machine-checked inductive proofs. Fortunately, a much simpler and slightly stronger version of the Jordan property can be defined directly on the hypermap structure, by considering contours of the hypermap three-function graph, rather than rings in an underlying map. The simplification comes from the fixed local structure of hypermaps, which allows “inner” and “outer” to be defined locally, provided we restrict ourselves to a certain pattern of traversal. Specifically, we exclude one of the three functions (the e permutation), and pick fixed, opposite directions of travel on the other two: from a dart x we can go to either n -1(x) or f(x). We define contour paths as dart paths that only take steps in this n -1 ∪f relation. A contour cycle follows the inside border of a face ring, clockwise, listing explicitly all the darts in this border. Note that n or n -1 steps from a contour cycle always go inside the contour, while f or f -1 steps always go outside. Therefore the Jordan property for hypermap contours is: “no contour path that starts and ends on a duplicate-free contour cycle, without otherwise intersecting it, can start with an n -1 step and end with an f step”, or, more symmetrically, “any such path must start and end with the same type of step”. For example, the figure below is forbidden. This definition is much simpler than the ring one, not only because it avoids the delicate definition of the inner and outer “halves” of a face cycle, but also because 9 More precisely, this follows from the Jordan separation theorem; the nonseparation part of the theorem, which implies that each edge borders at most two regions, is implicitly used to turn the problem of colouring the continuous plane into one of colouring a discrete data structure. contour path 22 “nonintersecting” (resp., “duplicate-free”) have their literal meanings here: not containing the same dart (resp., twice the same dart), whereas for rings we need to also exclude darts in the same f cycle. This also implies that the contour-based property is slightly stronger than the ring property, since it applies to contour cycles that are not derived from simple rings (different darts from the same f cycle can occur in two different sections of a contour cycle). Such contours actually occur in the Four Colour Theorem proof, but only around explicitly matched configuration maps, for which we do not need the Jordan property. Although it is much simpler than the ring property, we found that the definition above was still too complex to carry out inductive proofs in Coq; it still involves three different values (the path, its starting point, and the cycle), along with five to six separate properties. However, we observed that by splicing together the cycle and the path, we could form an equivalent statement that only involves a single value and three properties. We therefore used the following: Theorem (the Jordan Curve Theorem for hypermaps): A hypermap is planar iff it has no duplicate-free “Moebius contours” of the form The x ≠ y condition rules out contour cycles; note however that we do allow y = n(x). As far as we know this is a new combinatorial definition of planarity. Perhaps it has escaped attention because a crucial detail, reversing one of the permutations, is obscured for plain maps (where e -1 = e), or when considering only the cycles of that the permutation. Since, as we show in the Coq development, this Jordan property is equivalent to the Euler identity, it is symmetrical with respect to the choice of the two permutations that define “contours”, despite appearances (we use this in the reducibility proof). Oddly enough, we know no simple direct proof of this fact (the best we can do is to derive it from the equivalence between the “ring” and “contour” versions of the Jordan property, and which is hardly “simple”). We show that our Jordan property is equivalent to the Euler identity by induction on the number of darts. At each induction step we remove some dart z from the hypermap structure. In doing so, we must take care to redefine the permutations so that they avoid z. For two of them we can do this in the obvious way by suppressing z from the cycle in which it occurs: for example, we can define permutations n’ and f’ on the smaller hypermap by n’(n -1(z)) = n(z), f’(f -1(z)) = f(z), and n’(x) = n(x) and f’(x)=f(x) otherwise. For the third permutation, however, the triangular identity of hypermaps leaves us no choice, and we have to either merge two cycles, or split a cycle of that permutation. For example, if that third permutation is e, we do the following transformation on the portion of hypermap surrounding z: e n f z Walkupe x y nx ny n (n-1 ∪ f)* x ≠ y nx ≠ ny 23 Following Stahl [29], we call this operation the Walkup transformation [34]. More precisely, the figure above illustrates the Walkupe transformation; by symmetry, we also have Walkupn and Walkupf transformations. In general, the three transformations yield different hypermaps, and all three prove to be useful. However, in the degenerate case where z is a fixpoint of any one of the three permutations, then all three transformations give the same result (z has at most 3 neighbors to connect), e.g., if e(z) = z, then either Of course we only need to define and study one of the Walkup transformations, as we then get the others (and their properties) for free by symmetry. We chose the Walkupe transformation. Apart from for the removal of z, the Walkupe transformation leaves the cycles of n and f unchanged; however, except in the degenerate cases above, it has a nontrivial effect on the cycles of e: if z and n(z) are on different e cycles, the Walkupe transformation merges the two cycles; if z and n(z) are on the same e cycle, then the Walkupe transformation splits this cycle. The degenerate and merge forms of the Walkup transformation clearly leave the validity of the hypermap Euler equation #e + #n + #f = #d + 2#(e∪n∪f) z Walkupe (split) nz ez nz ez z Walkupe (merge) nz ez nz ez z z Walkup or Walkup nothing Walkupn z Walkupf 24 unchanged (both sides decrease by 1 or 3). The split form only preserves the Euler equation if it disconnects the hypermap; otherwise, it increases the difference between the left and right hand sides by 2. Since repeatedly applying the transformation eventually yields the empty hypermap, for which the Euler equation is trivially valid, we immediately see that all hypermaps satisfy the inequality #e + #n + #f ≥ #d + 2#(e∪n∪f) Planar hypermaps are thus those that minimize the left hand side, so applying any of the Walkup transformations to a planar map always yields a planar map; in the split case, the map is always disconnected (as is obvious from the figure above). Thus, to prove the implication Euler→Jordan we are free to apply any sequence of Walkup transformations to reduce a planar hypermap containing a Moebius contour to the map below, for which the Euler equality obviously fails. We use Walkupe to eliminate darts outside the contour, as this leaves the n -1 and f steps of the contour unchanged, and Walkupf and Walkupn to contract n -1 and f steps on the contour, respectively. In the Jordan → Euler direction, we use only Walkupe transformations, as they leave the contour steps mostly unchanged. We carefully select the removed dart to make sure that we are not in the split case: we use the Jordan property to show that any e cycle C that is closed under n (n(z)∈C for all z∈C) contains a fixpoint of either n or f. We named the latter statement the Euler tree lemma, because it is the hypermap analog of the following: if a planar connected graph has only one face, then it is a tree; this property is used implicitly in the standard “flooding” proof of the Euler formula. We also use all three transformations in the main body of the Four Colour Theorem proof. Since at this point we are restricting ourselves to plain maps (all e cycles have length 2), we always perform two Walkup transformations in succession; the first one always has the merge form, the second one is always degenerate, and always yields a plain map. Each variant of this double Walkup transformation has a different geometric interpretation, and is used in a different part of the proof: • The double Walkupf transformation erases an edge in the map, merging the two adjoining faces. It is used in the main induction of the Four Colour Theorem proof, to replace an identified reducible configuration with a smaller submap. • The double Walkupe transformation concatenates two successive edges in the map; we only apply it at nodes that have only two incident edges, to remove edge subdivisions left over after erasing edges (the node degree condition ensures that the resulting map is plain). • The double Walkupn transformation contracts an edge in the map, merging its two endpoints together. It is used to prove the correctness of the reducibility check, by induction on the size of the remainder of the graph. We use the upper cited Euler tree lemma of the Jordan → Euler proof to find an appropriate dart at which to apply the transformation, so as to avoid the split case of the Walkup transformation. 25 Contours play a central part in our formal development of the Four Colour Theorem proof, because they are the basis for a precise definition of the patch operation, which pastes two maps along a border ring to generate a larger map. This operation is the cornerstone of the proof, since it defines the three-way relation between a configuration submap, whether explicitly identified or computed using the Jordan property, the map in which it occurs, and the remainder of that map. Mathematical articles and books rarely define this operation precisely, probably on the account that it is “intuitively obvious”. We have found, however, that getting the details of this operation exactly right was critical to the successful proof of several key lemmas. Ultimately, this attention to detail paid off handsomely, as it allowed us to remove injectivity checks in the matching configuration, and to omit the analysis of the structure of the second neighborhood altogether. A key observation is that, despite appearances, the patch operation is not symmetrical in its arguments. It does paste two hypermaps along a ring, but in a slightly asymmetrical manner, and requires different assumptions the two maps: • For one of the submaps, which we shall call the disk map, the ring is an e cycle, that is, a hyperedge. This cycle must be simple, i.e., no two darts on it can belong to the same face. • The other submap, the remainder map, the ring is an arbitrary n cycle. The gluing operation itself consists in merging pairwise the darts on the e cycle of the disk map with those on the n cycle of the remainder map, reversed (the two cycles must have the same length). On the merged cycle, the e function is defined as in the remainder map and the n function is defined as in the disk map. The definition of the f function is then adjusted as is to satisfy the triangular identity, as illustrated below. Let us point out that although the darts on the border rings were linked by the e and n permutations in the disk and remainder map, respectively, they are not directly connected in the full map. Indeed, they need not even form a simple ring in the full map. However, because the e cycle is simple in the disk map, it is a subcycle of a contour that delineates the entire disk map (indicated by the mixed dash in the figure above). This contour is preserved by the construction, which, because of this, is reversible: the disk map can be extracted, using the Jordan property, from this contour. This allows us to divide the proof load by two, using equational reasoning to show that geometrical properties in the full map are equivalent to the conjunction of similar properties in the disk and remainder maps: • The full map is planar (resp., connected) iff both submaps are. patch disk remainder contour cycle full map 26 • The full map is plain iff the remainder map is plain, and the disk map is plain except for the border ring. • The full map is cubic (all n cycles of length 3) iff the disk map is cubic, and the remainder map is cubic except for the border ring. • If the full map is bridgeless both submaps are; the converse only holds if the disk map ring is chordless (the only e links between the f cycles of the darts in the e cycle ring are those of the cycle itself). • The full map is four colourable iff both submaps are four colourable with colourings that match on the rings (with a reversal). All the theory exposed so far applies to general hypermaps. However, for the rest of the proof of the Four Colour Theorem we need to specialize to plain (all e cycles of length 2) and cubic (all n cycles of length 3) maps: these properties are required both for the enumeration strategy for the unavoidability part, as well as for the computation of reducibility checks. To do this specialization we show that the task of colouring an arbitrary hypermap can be solved by colouring the hypermap obtained by covering every node and hyperedge with a new face. As depicted in the figure below, the construction of this new map is completely regular and straightforward; the new map has six times as many darts as the original one. 5.2 Reducibility The bulk of the reducibility computation for a given configuration consists in iterating a formalized version of the Kempe chain argument, in order to compute a lower bound for the set of colourings that can be “fitted” to match a colouring of the configuration border, using colour swaps. The actual check consists in verifying that this lower bound contains all the colourings of the map obtained by erasing 1 to 4 specific edges in the complete configuration map; this smaller map is called the e f n f e n original hypermap expansion of central dart expansion of other darts 27 contract map of the configuration. This final check does not require a significant amount of computation (in many cases, the lower bound yields the set of all possible colourings, whence the check is trivial). Although the Kempe chain argument is usually exposed for the dual graph colouring problem, we have found that the form that is actually used for the proof of the Four Colour Theorem is easier to justify for the map colouring problem. The argument uses an alternative formulation of the colouring problem, proposed by Tait[30] in 1880 (one year after Kempe’s original proof, along with a variant of Kempe’s proof that turned out to be equally wrong). Tait suggested replacing the face colouring problem with an edge colouring problem. Suppose we use the integers 0, 1, 2, and 3 to “colour” the map faces; given such a colouring k, we can compute another colouring, this time on the edges of the map: we colour an edge x that separates two faces a and b with the bitwise sum (aka. exclusive or) k(a) ⊕ k(b) of the colours of a and b. This edge colouring has two obvious properties: 1) Since a and b are adjacent we have k(a) ≠ k(b), so k(x) takes only the values 1, 2, or 3 2) The bitwise sum of k(y) for all edges y that are incident to any given node p is 0: expanding the definition of k(y), we see that the colour of every face incident to p occurs twice in the summation. Conversely, any edge colouring that satisfies 1) and 2) can be turned into a face colouring, by choosing an arbitrary colour for one face a0 in each connected component of the map, for each edge x bounding a0, colouring the face across x with k(a0) ⊕ k(x), and so on (the formal proof of correctness involves the Jordan property). Tait further observed that for cubic maps, 2) is equivalent to the much simpler 3) For any node p the colours of the three edges incident to p are a permutation of {1,2,3}. He concluded that four colouring the faces of a cubic planar is equivalent to three colouring its edges. Now, consider the problem of trying to fit the colouring of a submap with one of the remainder of the map, where the submap has been “cut out” across a ring of faces. By Tait’s results, we only need to match the colours of the edges that were cut. 3 0 1 2 1 3 2 1 0 3 1 2 edge colours 28 This simple observation has a significant impact on complexity, since we now have only three colours to consider, and there are only six permutations of them. There is more, however. Consider an arbitrary edge colouring of the remainder part of the map, and erase all the edges with the colour 1. Since this deletes exactly one of the three edges incident to each node, the resulting map is the union of a set of isolated loops with a set of linear paths (“chords”) starting and ending with the 2- and 3- coloured edges dangling at the cutout (see the figure below). Since we are only interested in the colouring of these dangling edges, we shall ignore the loops. The colour of edges on any of the chords must strictly alternate between 2 and 3, so the dangling extremities of a chord must have the same colour if the length of the chord is odd, and different colours if the length of the chord is even. In fact, this is the only constraint on the colouring of chords, since exchanging colours 2 and 3 on any one chord yields a valid edge colouring of the remainder. Since we can do this chord flip simultaneously for any subset of the chords, we can match any edge colouring of the cutout submap that a) assigns colour 1 to the same dangling edges as the remainder colouring b) assigns different colours to the ends of a chord iff the length of the chord is even. We say that such a colouring is consistent with the set of chords. Let us say that an edge colouring of the remainder map is suitable if it can be transformed, through a sequence alternating the chord flips described above with global permutations of the three edge colours, into an edge colouring that matches exactly an edge colouring of the cutout submap on the sequence of dangling edges. We prove the reducibility of the cutout map by showing that every colouring of the remainder map that matches the colouring of a specific, strictly smaller, contract map, is suitable. This implies that it is impossible for a minimal non four colourable map to contain the cutout, since by minimality the map obtained by replacing the cutout with ring starting edge 29 the contract would have a valid edge colouring, whose restriction to the remainder would then be suitable. The bulk of the reducibility check consists therefore in constructing a safe approximation of the set Θ of the sequences of colours assigned to the dangling edges by suitable colourings of the remainder, using fixpoint computatation. We compute an increasing sequence Θ1, Θ2, Θ3 … of safe approximations of Θ until it converges: i. In Θ1 we only put the sequences of colours assigned to the dangling edges by actual colourings of the cutout. ii. If θ∈Θi, we can add ρθ to Θi+1, where ρ is any permutation of {1,2,3}. iii. By the above, we could also add to Θi+1 a sequence θ of colours if we could find some θ’∈Θi and set of chords ξ induced by an edge colouring of the remainder such that both θ andθ’ are consistent with ξ. We can’t really do this, since we don’t know the remainder. However, since any θ induced by an edge colouring of must be consistent with some set ξ of chords, if we can find a θ’ as above for all ξ that θ is consistent with, we can safely add θ to Θi+1. Step iii is practical because the conditions a) and b) above only depend on the endpoints and the parity of chords, and not on their internal edges. This endpoint/parity information can be very simply represented by a formal word in a bracket (or Dyck) language, as follows: For each dangling edge, in counterclockwise order, starting from any fixed edge: • Write down a dash ‘-’ if the edge is not part of a chord. • Write down an open bracket ‘[’ if the edge is the start of a chord. • Write down a close bracket ‘]b’ if the edge is the end of a chord with parity b∈{0,1}. Since the remainder map is planar (more precisely, outerplanar), chords cannot cross; therefore if we read back the codeword while going around the dangling edges, we can unambiguously interpret a closing bracket as marking the end of the last chord for which we have seen the start but not the end (which we’ll refer to as an open chord). We call these four-letter codewords chromograms. For example the chromogram for the last figure, starting with the edge marked by an arrow, is 〈- [ - - [ ]1 ]0 〉. The “consistent with” relation between edge colourings and chromograms can be defined with a simple formal recurrence. Let us write • ε for both the null colour sequence and the null chromogram • θ ≈ γ for “the sequence of edge colours θ is consistent with the chromogram γ” • ν / θ ≈ γ for “the sequence of edge colours θ is consistent with the chromogram γ, given that the sequence of edge colours of the beginning of open chords is ν” We then have • θ ≈ γ iff ε / θ ≈ γ • ν / 1θ ≈ - γ iff ν / θ ≈ γ • ν / cθ ≈ [ γ iff c∈{2,3} and νc / θ ≈ γ • νc / c’θ ≈ ]b γ iff c’ = c ⊕ b and ν / θ ≈ γ • ε / ε ≈ ε Writing [θ]≈ = {γ : θ ≈ γ} for the set of chromograms that a colour sequence is consistent with, we can reformulate step iii of the reducibility computation as 30 iv. Compute Γi = θ∈Θi [θ]≈ v. Add to Θi+1 any θ for which [θ]≈ ⊆ Γi The time and memory required for the fixpoint computation increases geometrically with the ring size of the cutout. For the largest ring size (14) there are about 300,000 different valid edge colour sequences, about 1,000,000 chromograms, and around 20,000,000 colour sequence/chromogram pairs in the ‘≈’ relation. While these numbers are by no means out of reach of modern computers, we need to be somewhat cautious, since the computation needs to run inside the proof checking kernel of Coq, which as a compute engine is several orders of magnitude less efficient than the bare processor. Also, we must select an algorithm that does not use imperative structures such as arrays, since these are not available in the embedded programming language of Coq. These considerations led us to the following design choices: • We compute Γi (step iv) incrementally, like Robertson et al. [26] (Appel and Haken could not do this[4], because computers in the early 70s did not have enough core memory to store a million bits…) • We use 3 and 4-way decision diagrams [1,12] to store Θi and Γi, respectively; each level in the decision tree branches according to a colour or symbol in the sequence or chromogram, respectively. • We store the complements Ξi and Λi of Θi and Γi, respectively, rather than Θi and Γi. This way the computation starts with large sets that are rapidly pruned, rather than with small sets that expand quickly. In addition the large initial sets Ξ0 and Λ0 have a very regular structure, so their decision trees require very little memory, as most of their subtrees can be shared. • Ξ0 is not the set of {1,2,3} sequences whose length is the ring size, but the set of all such sequences whose bitwise sum is 0, and in which the order in which colours appear is a cyclic permutation of 〈1,2,3〉, e.g., if the first colour is 3, the first colour different than 3 will be 1. The first restriction rules out sequences that can’t possibly be induced by edge colourings; the second one is a free way to halve the size of the set, based on the observation that the “consistent” relation ≈ is insensitive to swapping colours 2 and 3. • Likewise, Λ0 is the set of all chromograms that are consistent with some θ∈Ξ0. This corresponds to excluding ill-bracketed words, and words for which the bit sum of the closing bracket parities does not match the parity of the ring size. • To avoid rescanning the large ≈ relation in step v, Ξi actually stores the number of chromograms in Λi consistent with each θ it contains, which allows step v to be carried out in constant time. Thus, unlike the Robertson et al. program [26], our fixpoint computation runs in time linear in the size of the ≈ relation, traversing each θ ≈ γ pair at most twice. • We use dynamic programming to compute a highly compressed representation of Ξ0 and Λ0, but do not try to find ad hoc sharing afterwards; initial experiments with an ML prototype had showed that there was little to gain. We do however omit the last layer of Ξi, as the last colour in a valid sequence is always the sum of the previous ones, and we also use a fixed compression scheme for the last two layers of Λi, using 8 fixed constants (since this is 4- way branching tree, this saves 75% of the memory!). The computation iterates the three following steps: 31 i. Given a set of colour sequences ∆Θ to be added to Θi, remove all the chromograms in Λi that are consistent with some θ∈∆Θ, simultaneously computing Λi+1 and ∆Γ = Λi+1 - Λi, the set of chromograms to be added to Γi. ii. For each θ in the domain of Ξi, decrement the value of Ξi (θ) by the number of chromograms in ∆Γ consistent with θ, simultaneously computing Ξi+1 (θ) and ∆Ξ = {θ : Ξi (θ) > 0 and Ξi+1 (θ) = 0}, the set of colour sequences whose count has just reached 0. iii. Compute ∆Θ = {ρe θ : θ∈∆Ξ, e = ±1}, and feed it back to step i. Here ρ is the cyclic permutation that maps 〈1 2 3〉 to 〈2 3 1〉. We start with ∆Θ = Θ1 and terminate as soon as one of Ξi, ∆Γ, or Γi becomes empty. All the intermediate sets use the same MDD representation, so steps i-iii can all be implemented efficiently with parallel tree walks. Finally, the fixpoint loop is driven by a higher-order tree iterator that can efficiently execute an exponential number of steps. While this is clearly overkill (the number of iterations never goes over 30), it does allow us to do a full correctness proof of the algorithm, sparing us the hassle of having to debug the execution of such a complex algorithm on large data structures in a development environment with absolutely no debugging support (not even print statements!). Furthermore, the full correctness proof is surprisingly straightforward; it is carried out mostly by stepping through all the cases of all the functions. We also needed to prove that the formal, purely combinatorial definition of “suitable” actually implied something about the colouring of planar maps. In principle this implied formalizing all the theory that we exposed in this section: Tait’s edge colourings, the topology of the chords, and the parsing of a chromogram into a chord set. Benjamin Werner, who worked with us on this part of the proof, found out that none of this was needed, except for insight into the solution. It was much easier to ∆Θ Ξ0 Θ1 Λ0 Λi Ξi ∆Λ ∆Ξ ρ∪ρ-1 Λi+1 YES NO Ξi+1 Ξ, Λ restrict Λ decrement Ξ any ∅? NO NO kempe 32 prove directly that if a remainder map has a suitable colouring, then it has a colouring that matches a border colouring of the configuration; the main lemma, which states the existence of a consistent chromogram, is proved by induction on the size of the remainder map. Indeed this result can be proved directly from the Euler formula, whereas we needed to use the Jordan property twice in our informal discussion! When we finally decided to port his outline to our development on hypermaps (he had had some trouble with the definition of the node contraction needed to carry out the induction), it only took us a day to complete the proof10. 5.3 Configuration maps Since the reducibility check is based solely on the edge-colouring of the configurations and their contracts, it would appear that it is not necessary to describe the exact geometry of the maps to prove the Four Colour Theorem: the colourings can be computed from the adjacency graphs of the configuration maps, and in the unavoidability part of the proof these adjacency graphs are matched against the graphs of the explicitly enumerated second neighborhoods submaps of an arbitrary counter example; the latter are, by construction, geometrically correct. This is how Robertson et al. outline their proof [26], and indeed they use adjacency lists to describe their 633 configurations. We naively followed this line of reasoning in our initial investigations, which focused on the feasibility of the reducibility proof, since it was straightforward to compute the set of colourings of a graph. However when we started working on the graph theory linking the two computational parts of the proof, we realized that this approach was wrong: • Several small geometric side conditions are imposed on configurations and their contracts; while most of them are simply sanity checks, some are used in the actual proof, and therefore must be checked. • Enumerating just the arity of faces in neighborhoods is already fairly complex; enumerating their full geometry would have been one step up. • Most of all, the enumeration strategy proposed by Robertson et al. does not explicitly identify complete submaps, because it is limited to the second neighborhood, and the configurations needed to prove the Four Colour Theorem do not lie entirely within the second neighborhood: only their kernel does. This is sufficient, if one can apply the “folklore theorem” that any kernel has a unique free ring completion; however this theorem only applies if the adjacency graph of the completion is geometrically consistent. Of course the Robertson et al. data is geometrically consistent: it gives a planar embedding for each configuration, for which adjacency lists are oriented clockwise. The programs supplied by Robertson et al. also perform a combinatorial sanity check. Nevertheless, we had to revise our approach, and explicitly define a hypermap for each configuration. We first did this directly from the adjacency data, using pairs of integers for the darts; this allowed us to keep the colouring and contract colouring functions, along with their correctness proofs. However, this approach also involved encoding geometric reasoning as obscure arithmetic identities, and didn’t support very well doing induction on the geometry of the map, which was required for establishing the correctness of the geometric side condition checks. 10 We cheated by reusing the euler_tree lemma, which we had used to prove Jordan→ Euler. 33 A little analysis revealed that all 633 configuration submaps could be built by following very simple, inside-out construction program, so we switched to that description. This change cut down the size of the configuration data by a factor of 5, but, more importantly, it provided a clean framework for the various checks we needed to perform on the maps. We could rephrase all the operations we needed to perform on a configuration map (computing the set of colourings, the contract, checking the radius, even compiling an occurrence check filter) as nonstandard interpretations of its program (the standard interpretation being the map construction). This approach both made the implementation efficient, and the correctness proofs straightforward, although sometimes lengthy: these proofs always consisted in establishing some simulation relation between the standard and nonstandard interpretations. Although the main use of configuration maps is the reducibility argument, which calls for disk maps, we chose to construct remainder maps, for various technical reasons: • The proof of the Birkhoff lemma [10] requires a handful of explicitly constructed remainder maps (see §5.4 below). • It’s easy to show that the construction yields a remainder map with a simple border n cycle; stripping this cycle trivially yields a disk map with the same colourings. • The remainder map construction is more uniform because it starts with a nonempty map. • Remainder maps are plain and quasicubic (cubic except on the n cycle) whereas disk maps are cubic and quasiplain (plain except on the e cycle), and the plain identities, e(e(x) ) = x and n(f(x) ) = e(x), are more useful than the cubic ones. The construction actually yields a pointed remainder map, in which one dart on the border n cycle has been selected. The construction starts from a single edge and applies a sequence of steps to progressively build the map. For configuration maps, it turns out we only need three types of steps: • Rj (rotate) steps that only change the selected ring dart x to n-j(x) for some j. • Y steps that insert a node between x and the border ring. • H steps that insert a linked pair of nodes between x and n -1(x) and the border ring. The first construction step is always a Y step. Construction sequences are applied right to left, so the construction sequences for configurations always end with a Y (they also start with an H). For example, the program 〈H R3 Y Y〉 constructs a map consisting of a single square surrounded by a ring of four faces: base map ring ring Rj step ring Y step ring H step before step added by step ring boundary (before/after) selected dart (before/after) j … 34 The data for a configuration also includes a contract, that is, a set of edges that should be erased to get, by induction, a suitable colouring of the remainder. As these edges must not contain border ring darts, they must have been disconnected from the border ring and connected to an internal node by a Y or H step. We therefore denote contract edges by the left-to-right index of this connexion, counting that each Y makes one such connexion, and each H makes three (because the initial step is a Y, the feet of the H belong to different edges). These indices appear before the H that starts the construction sequences (the other indices in the sequence are R steps). This representation allows us to easily find contract edges while scanning the program left to right. Since the correctness proof of the function that computes configuration border colourings is rather long, we decided not to try to combine it with the definition of contracts. Instead, we added three new construction steps that allowed us to construct a map that had the same set of colourings as the contract map (or, more precisely, the same “contract colouring” as the configuration map). This allows us to produce the set of contract colourings by compiling the configuration program into a contract program, then running that program with the “generate colourings” interpreter. The new steps are: • the U step, which inserts a single loop between x and n(x) • the K step, which adds an inverted Y above x and n -1(x) • the A step, which connects x and n -1(x) with a degree 2 node (the resulting map is not quasicubic; in fact, it might not even be plain – see below) We also use the U to construct some of the remainder maps used in the proof of the Birkhoff lemma [10]. These new steps are more primitive than the H and Y steps; indeed, we can define the H, Y, and K steps in terms of U and a rotation variant N = R ◦ K ◦ R-1 of K, as we then have Y = N ◦ U, H = N ◦ Y, and K = R-1 ◦ N ◦ R. Thus we only need to give precise hypermap constructions for the base map and the U, N, and A steps: U step ring ring A step ring K step R3 H Y Y base map 35 The A and N steps have degenerate cases in which these definitions are incorrect. For the N step, this occurs if the ring size is less than three: if x is the selected dart, we have n -2(x) = x, but the definition above gives different values for n’(x) and n’(n -2(x)). The degenerate case for the A step occurs when x and n -2(x) are already on the same f cycle: in that case the definition above splits the cycle and disconnects the hypermap (this is similar to the split case of the Walkup transformation). Although these degenerate cases do not occur with our data, this is not easy to check for the A step, and at any rate is is easier to provide consistent definitions for these cases than to deal with partial constructions throughout the proof, so we use the following: Note that both these definitions are still valid in the completely degenerate case where the ring size is 1. The contract compilation is defined in the figure below, which shows the effect of erasing one or two edges from a Y or H step. The correctness proof of the contract compilation would seem straightforward from these diagrams, but it is the longest proof (close to 700 lines of script) in the entire development, with the second largest compilation time (excluding the reducibility and unavoidability computation lemmas). degenerate N step ring ring degenerate A step ring base map U step N step A step ring ring ring ring added by the step deleted by the step unchanged by the step selected ring dart (old/new) 36 The correctness proof of the edge colouring procedure defined in the figure below (where ρ is the permutation defined in §5.2) is comparatively easier, because we can use the correctness lemmas for the U and K steps to show the correctness of the Y and H steps, since the Y and H steps can be defined in terms of U, K, and R steps.. Observe that while some of the colouring rules have side conditions, only the A and K step can fail; hence the colouring of a non-contracted configuration never backtracks. The unavoidability part of the four colour theorem proof requires yet another interpretation of the map construction programs: we need a filter that will efficiently test for the occurrence of a configuration in the data structure used to represent a second neighborhood pattern, which, following Robertson et al., we shall call a part [26]. Now Robertson et al. computed this test by somewhat arbitrarily assigning integers to faces of the configuration and part, precomputing the geometrical relations (in their case, the clockwise triangles) for both maps, and then using yet another array to store the partial morphism in a traversal of a list of triangles of the configuration [27]. As we have pointed out already, the Coq language does not have arrays, and the idea of emulating them in order to implement such an indirect algorithm, and proving the correctness of the combination, was not appealing. It seemed that a more geometrical approach should exist, and we did indeed find one, by combining the insights provided by our analysis of hypermaps and of the incremental construction of the configuration maps. As we shall see in the next section, we strengthened one of the auxiliary results of Robertson et al [27]. and showed that if it was possible to perform a geometrically b b c c b b c c b b c c’ b ⊕ c b c b ρ1 ρ b 1 b U step K step (b≠c) A step (c= c’) Y step H step (b≠c) H step (b=c) b c ρ1 ρ b 1 b b c ring colouring ring U step ring no step ring no step ring no step ring K step ring K step ring Y step ring Y step ring A step ring no step ring no step ring U step edge erased by the contract node erased by the contrac 37 connected traversal of the faces of the kernel of the configuration in the part, while generating the same sequence of arities, then the geometrical properties of configurations and counter example maps directly implied the existence of a one-toone embedding of the disk map of the configuration into any counter example map fitting the part. It follows that there is no need to construct the isomorphism, hence, no need for arrays. All we needed was a data structure to represent a map traversal, and we found that the most obvious one worked. We ensure that our traversal is connected by strictly following kernel edges of the map. Each node in the traversal must be cubic, since it is reached by a kernel edge, so there are two other edges leaving that node, and a face across those two edges, whose arity might need to be checked. The traversal can end with this check, or continue with either or both of the two edges. It is also possible that the face across is a ring face; in that case the arity check must be skipped by the traversal (there may not be a corresponding face in a matching part). This will certainly happen once for the many configurations whose kernel is not two-connected, and in fact this is the only case in which this happens. In those instances, the traversal should continue either left or right, check the face there (it’s always a kernel face) and then continue in the same direction (that is, the traversal does either two lefts or two rights). Hence, the structure we need is the 7-case variant of the binary tree, shown below. In turns out that it is always possible to cover exactly the configuration map kernel, using two such trees, and starting from the two faces incident to the initial edge of the construction. Thus, we can generate an optimal test (called a quiz in the Coq scripts) for all 633 configurations. We compile this pair of trees outside-in, that is, by interpreting the construction program in reverse, left-to-right, using the rules in the figure below. Conveniently, this is also the right direction for computing the face arities, which are stored in the tree nodes, and for keeping track of which ring darts belong to the kernel in the full configuration map, and which ones belong to the border ring. Qask0 Qask1 QaskL QaskR QaskLR QaskLL QaskRR non-tree edge tree node arity check parent node 38 Note that the rules above are not quite complete: we do not handle the case where there are non-empty trees for both arms of an H, which would have required a QaskRR tree node (the other apparently missing cases can be ruled out using the geometric properties of configurations). It turned out that only 4 of the 633 configurations required that case, so it was easier to do a trivial modification of those 4 construction programs than to prove the correctness of an additional complex case. 5.4 Embedding configurations The main result of the graph theory part of the Four Colour Theorem proof is that no dart of a minimal counter example can fit the quiz of a (combinatorialy) reducible configuration. The few weeks it took us to prove this represented only a fraction of the effort involved; much more time was spent developing and experimenting with the basic concepts – sequences, paths, hypermaps – to see which formalization would allow us to complete this part of the proof. This attention to detail paid off in the end: it turned out that half of the final part of the proof could be skipped. Specifically, we were able to eliminate cartwheels, an intermediate combinatorial structure used to represent second neighborhoods. Roberston et al. use cartwheels as follows [26, 27]: • They recall a theorem of Birkhoff [10], stating that second neighborhoods in a minimal counter example are isomorphic to cartwheels. • They do the unavoidability enumeration on cartwheels, defining both part matching and discharging on cartwheels. • Although they use an array-based version of our quizzes to check for configurations occurrence in parts, they only do so as a heuristic. A positive check is always backed by the explicit construction of an injective simplicial map from the configuration kernel to the “skeleton” of the part, as well as an ad hoc check that the configuration is “well-positioned”. They state without proof that these checks imply the existence of a morphism from the configuration to the map. outer ring outer ring outer ring outer ring outer ring kernel kernel kernel kernel kernel kernel kernel kernel kernel kernel traversed not traversed tree node 39 Since we were bent on doing a fully formal proof, using cartwheels would not give us the benefits of reusing well-known results. On the contrary, they would just introduce another layer of definitions and of correspondence lemmas, which were not really needed. Since it is easy to navigate precisely in hypermaps (just use the right sequence of e, n, and f), we could define discharging and part fitting directly on the full counter example map just as easily as on a cartwheel (in fact we derive discharging from part fitting). More importantly, we could do completely without the Birkhoff theorem, by showing directly that the mapping derived from a successful quiz check was automatically an embedding. This was a boon, because it relieved us from having to prove in every case the injectivity of all the intermediate mappings – from configuration to part, and part to map – that composed this mapping: we only needed to show that these were morphisms. To summarize, we used the following proof outline: 1. We prove the Birkhoff lemma [10] (every minimal counter example map is internally five connected). 2. We deduce from this that any face of such a map is surrounded by a simple ring (called a spoke ring) of at least five faces (we say the map is pentagonal). 3. Given a successful quiz, we construct a preembedding φ from the kernel darts of the configuration map to the counter example map, that is, a mapping that preserves all f arrows and arities, and preserves enough e arrows to connect its domain (i.e., the kernel is connected by the relation whose graph is the union of the f arrows and the set of e arrows x → e(x) such that φ(e(x)) = e(φ(x)). 4. We show that, provided the configuration map satisfies two simple geometric conditions, φ can be extended into an embedding φof the entire configuration map, injective except on the border n ring. 5. Following Robertson et al. [26], we don’t try to replace directly the image under φ of the contract map, as we would have no simple way of proving that the resulting map is bridgeless – the contract map need not be chordless, and its border ring may not even be simple. Rather, we apply directly the image under φ of the configuration contract to the counter example map, and show that colouring this map induces via φ -1 a colouring of the contract map. We show that the geometric properties of the contract are carried over by φ, and that they imply that applying the contract to the counter example map yields a bridgeless map. 6. We conclude as indicated in subsection 5.2. For each step except the last, we ended up inventing our own proofs. For Step 1, we use a variant of the reflection method used for reducibility, reusing much of the code from that part of the proof. The formal statement for Step 1 is “there are no simple m-nontrivial rings of size k ≤ 5”, where m=1 if k=5, and m=0 if k < 5, and a ring is m-nontrivial if both its inside and outside have more than m faces. Let Ωk be a list of configuration maps with ring size k and at most m kernel faces. Suppose r be such an m-nontrivial ring of size k in a minimal plain counter example map g, and consider the set Ξ (resp. Ξ’) of border colourings of the remainder map h (resp. h’ ) of the inside (resp. outside) of r. Since g is plain and not four colourable, Ξ is disjoint from the set Θ of colourings suitable for h, hence disjoint from the safe approximation of Θ computed from Ξ’ as in §5.2, i.e., by symmetry, we must both have Ξ ⊆ kempe(Ξ0,Ξ’,Λ0) and Ξ’ ⊆ kempe(Ξ0,Ξ,Λ0). We check by enumeration that 40 for any such pair Ξ, Ξ’ there is some h”∈ Ωk whose set of border colourings Ξ” is disjoint from either Ξ or Ξ’. This contradicts the minimality of g, since if, for example, Ξ and Ξ” are disjoint, then the map obtained by replacing h’ with h” in g is strictly smaller but not four colourable. Using the monotony of kempe, the enumeration of the Ξ, Ξ’ pairs and the Ωk checks are carried out simultaneously by the two mutually recursive procedures below. While the lemmas proved in Step 2 are not original (they are a subset of the consequences of the Birkhoff lemma), their proofs are somewhat different from traditional mathematical argument, because they rely on the (abundant) computing power of Coq rather than its (nonexistent) geometrical intuition. For example, we establish exact formulas for navigating about the first neighborhood of a face: e.g., x is a dart on the spoke ring iff n(x) is on the central face cycle, and the next dart on the ring is f(f(e(x))). We then use these formulas to replace visual reasoning on figures with equational reasoning. To prove Step 2 we exhibit an explicit colouring function and use brute force enumeration to prove that it’s a valid colouring; this way we don’t need to prove that a chordless 4-face ring must be 0-nontrivial, as in steps d) and e) in Section 3. Ξ0 Λ0 pick {θ}⊆ Ξ∩ Ξ’ Ξ Λ Ξ’ Λ’ & return check call subcheck call subcheck Ξ {θ} Λ return true Ξ ∩ Ωk? call check ∃∅ ∀≠∅ subcheck Ξ’ Λ’ call kempe 41 For step 3, we define φ as follows: let x1…xn be the sequence of darts traversed by running the quiz on the configuration map. Since this sequence contains exactly one dart in each face cycle in the kernel of the configuration map, for any dart x in this kernel we can find a unique i such that x = f k (xi) for some k. We then set φ(x) = f k (ui), where u1…un is the sequence of darts traversed by running the quiz on the counter example map. Since both runs succeed, xi and ui have the same arities, so φ trivially preserves arities and f arrows. Since the two question trees composing the quiz span the configuration map kernel, we can then establish that φ is a preembedding by showing that it preserves one e arrow for each branch in these trees. This follows from the equational characterization of the “left” and “right” steps that were presented graphically in 5.3: • a “left” step takes us from x to n(e(n(x))) = e(f -2(x)) • a “right” step takes us from x to n(e(x))) = f(e(f(x))) However, we can’t use these equations directly, because the traversal may leave the kernel during double left and right step. We get around this issue by slightly adjusting the definition of double left steps, so that the following property holds • φ(e(xi)) = e(φ(xi)) for every xi in x1…xn. This follows from a property of the quiz tree traversal • φ(e(n -1(x)) = e(φ(n -1(x)) for every x at which the recursive quiz tree traversal is called (because of the double steps, x is not necessarily one of the xi). This holds at the tree roots because the traversals start from an edge cycle in both the configuration and counter example maps, and inductively because of the geometric identities. Since φ preserves f arrows, the first of these equations implies that φ preserves at least one e arrow connecting the face cycle of x to the face reached by a simple left or right step. The second equation is used for the double-left and double-right steps; it implies the existence of another e arrow connecting the face reached by the double step to the lower part of the tree. x left right x double left double right x face ring spoke ring 42 Step 4 is new to our proof, although it was inspired by the justification for the configuration search heuristic of Robertson et al [27]. The two additional geometrical constraints on configuration maps are: a) The kernel must have radius 2, that is, there must be a face in the kernel whose second neighborhood contains the entire kernel. b) The arity of the ring faces must be between 3 and 6, inclusive. Condition a) is fairly natural, since we are trying to embed the kernel of the configuration map inside a second neighborhood of a counter example map; condition a) was mentioned by Robertson et al. Condition b) is analogous to Robertson et al.’s “well-positioned” condition, but is simpler in that it does not depend on how the configuration embeds in the second neighborhood. We have an informal argument that condition b) should hold for all configurations useful for the Robertson et al. proof technique, but our proof does not depend on this argument, since we explicitly check for condition b) during the quiz compilation; we check condition a) separately. The conclusion of step 4 rests on three geometrical lemmas: i. φ preserves all e arrows in the kernel. ii. φ maps kernel e arrows faithfully . iii. The border ring of the configuration is chordless. The proof of each of these uses induction on the size of the ring interior disk. The proof of lemma i depends only on the connexity of the domain φ, and the preservation of f arrows; it doesn’t use conditions a) or b), or arities. It proceeds as follows: • Consider a kernel dart x such that e(x) is also in the kernel, but φ(e(x)) ≠ e(φ(x)). • By the connectivity property of φ, there must be some simple path p of faces in the kernel that goes from e(x) to x, and is preserved by φ; i.e., p is a ring in the configuration map kernel, but φ(p) is only a path in the counter example map. • We use induction on the size of the disk map delimited by p; by exchanging x and e(x), we may assume that this disk map lies inside the kernel. • Consider the arrows n(x) ↔ e(n(x)) = f -1(x) and f(e(x)) ↔ e(f(e(x)) = f -1(n(x)). If they are both preserved by φ, then we get φ(e(x)) = e(φ(x)) by equational reasoning, using the fact that the counter example map is cubic. • Hence, one of these e arrows is not preserved; note that these correspond to the edges between the face containing n(x) and those containing x and e(x), respectively. • The face containing n(x) is on or inside p, so n(x) is in the kernel, and there must be a simple path q, not meeting p, from some face y∈ p to n(x) (y may be x or e(x)). Dividing p at y and combining one of the pieces with q gives us a ring with a smaller disk, so we can apply the induction hypothesis. e(x) n(x) y p q 43 The proof of lemma ii follows a similar argument, where the roles of the configuration and counter example maps have been swapped. • We want to prove that if φ(x) = e(φ(y)) for some kernel darts x and y, then x = e(y), so assume some x and y for which this does not hold. • Condition a) implies that there a simple path p in the kernel of the configuration map going from y to x, of length at most 5 (at worst, we go from x to the central face and then to y). Because of i and the assumptions on x and y, φ(p) is a ring in the counterexample map, while p is only a path in the configuration map. • Exchanging x and y and reversing p if necessary, we can assume that the interior of p is no larger than its exterior. • If φ(p) has an empty interior then we can use a simplified variant of the induction used for lemma i to derive a contradiction (n(φ(x)) must be in φ(p)). • By the above, φ(p) is 0-nontrivial, so by the Birkhoff lemma it must have length 5 and its interior must consist of a single face. • Thus φ maps p to a spoke ring; since φ preserves the formula for the next dart in a spoke ring, p must be a spoke ring arc in the configuration map, wrapping around a single face of the configuration map. That face cannot be in the kernel, for then it would have to be a pentagon, since it is mapped to the central pentagon and φ preserves arity on the kernel, and this would imply x = e(y). Hence it is a ring face, adjacent to the five different faces in p. Since it must also be adjacent to the faces respectively preceding and following it on the border ring, it must be at least a heptagon; but this violates condition b). The proof of lemma iii uses yet another variant of the same argument: • Any chord of the border ring of the configuration map splits it in two rings with disconnected interiors. Since the union of these interiors is the kernel, which is connected, one of the subrings must have a disjoint interior. • By induction on the length of the subring with an empty interior, there must be a subring of length 3. • The middle face of that subring is only adjacent to two other faces in the subring – this violates condition b). The combination of lemmas i–iii implies that φ faithfully injects the kernel of the configuration map into the counter example map. Since the configuration map is plain, and cubic except on the border n cycle, we can define φe i (nj (x)) = e i (nj (φ(x))) to get the required embedding: it follows from iii that each border ring face is adjacent to at least one kernel face, hence the equation above defines φtotally. φ(x) φ(y) φ(p) ? φ(x) φ(y) n(φ(x)) φ(p) 44 Finally, for step 5 we follow closely Robertson et al., using the following conditions to ensure that the application of a contract yields a smaller bridgeless map: a) The contract is sparse: no two darts of its e-closure belong to the same n cycle, or to the border n cycle. b) The contract has between 1 and 4 darts. c) If the contract has 4 darts, then it has a triad: there is a kernel dart of the configuration map that is adjacent to at least 3 darts, belonging to different faces, in the e-closure of the contract, but not to all such darts. Our condition c) is slightly more restrictive but easier to check than the corresponding condition in [26]. We show that conditions a), b), and c) are preserved by φ, with the non-border and kernel requirement dropped in a) and c), and that the combination of a), b), and c) implies that no simple ring r of a minimal counter example can consist almost entirely of contract edges, i.e., be such that all but one dart of r belong to the eclosure of the contract: • By induction on the size of r, condition a) implies that r cannot have an empty interior; the proof is similar to that of lemma iii in step 4. • By symmetry, r is thus 0-nontrivial; hence by the Birkhoff lemma and condition b), r must be the spoke ring of a pentagon, and contain all the darts of the e-closure of the contract, which must be of length 4. • Condition c) states that there is a dart adjacent to three different darts in r, yet is not in the inner pentagon cycle (it would then be adjacent to all of r). This violates a corollary of the Birkhoff lemma. We conclude by showing, by induction on the contract size, that the absence of such almost-contract rings implies that erasing all the contract edges yields a smaller bridgeless map, whose colouring induces a contract-colouring of the counter example map, and thus of the contract map. 5.5 Enumerating parts Recall that the unavoidability proof consists in proving that in a minimal counter example the second neighborhood of any face whose “discharged” arity is less than 6 contains a copy of the kernel of one of the 633 reducible configurations. This is done by going through a symbolic enumeration of all possible shapes of second neighborhoods, checking for each shape that either a) The central face has arity at least 6, after “averaging” with the discharge rules. b) The quiz for one of the 633 configurations runs successfully at one of the darts of the neighborhood. Both of these checks depend only on the arity of the neighborhood faces, so a “shape” is just a mapping from these faces to integer intervals. We call these mappings parts, following Robertson et al. Using their terminology [26], we distinguish four kinds of faces in the domain of a part: • The central face is called the hub; it always has a fixed arity. • Faces in the first neighborhood of the hub are called spokes. • Faces in the second neighborhood that are adjacent to two (consecutive) spokes are called hats. • Other second neighborhood faces, which are adjacent to only one spoke, are called fans. We represent a part as a counterclockwise sequence of subparts, where each subpart is a record containing the arity intervals for a spoke, the hat clockwise from it, and the 45 counterclockwise sequence of fans that follows that hat. The length of the subpart sequence is the arity of the hub. The intervals in a part take only a very limited range of values: • The lower bound is at least 5, since a counter example map is pentagonal. • The upper bound is either less than 9 or infinite, because the discharge rules don’t discriminate between faces with more than 8 sides, and configuration maps have at most one face with more than 8 sides, which it is only matched with the hub. For the same reason, the lower bound is less than or equal to 9. Hence there are only 15 possible intervals, and the interval [5,+∞) means “no constraint”. We always assign the [5,+∞) interval to fan faces that are not known to be distinct, for example when the exact arity of the corresponding spoke is not known. Indeed we have four kinds of subpart records, according to whether there are 0, 1, 2, or 3 known distinct fans; in the latter cases the arity of the spoke must be 6, 7, and 8, respectively. We check that a part “fits” at a particular dart x of a counter example map by using the hypermap functions to navigate around the part and its subparts, verifying each nontrivial arity constraint. Crucially, this does not involve checking for a one-to-one correspondence between the faces listed in the part and the face cycles in the second neighborhood of x (by subsection 5.4, this is not needed). Because the definition of “fits” is purely equational, it can be manipulated efficiently by our Coq scripts. Since “fits” relates parts to the counter example map, we can directly use parts to specify how arities are discharged. We can turn the 32 rules of Robertson et al. into a list p1,.., pn of parts (n=71), and then define the discharged arity δ(r) of an f cycle r as δ(r) = #r + 1 /10∑x∈r #{i : pi fits f -2(x)} − 1 /10∑x∈r #{i : pi fits f -2(e(x))} Since e is an involution, we have, from the Euler formula, ∑r δ(r) = ∑r #r = #d = 6#f − 12 < 6#f = 6#{r : r is an f cycle} Hence δ(r) < 6 for some r, and this holds for any sequence pi. In comparison, the definition used by Robertson et al. [26] depended on the Birkhoff theorem and on additional assumptions on the discharge rules. Given a part p that fits a dart x, we can efficiently check whether some part pi fits some dart f k (x) in the f cycle r of x by doing a memberwise comparison of the intervals in pi and p rotated left k times. Thus we can get a lower bound on the first sum in the definition of δ(r). To compute a lower bound on δ(r) we also need to find an upper bound on the second sum. As illustrated in the figure below, we use using pattern matching to compute a part pi such that pi fits f -2(e(x)) only if pi fits f -2(x). hub spoke spoke spoke hat fan hat fan subpart 46 (This is the reason for the f -2 in the definition of δ(r).) We can therefore get the desired upper bound by comparing pi with all rotations of pi. The above gives us an efficient way of checking for condition a). To efficiently run a quiz, i.e., to check condition b), we need to be able to navigate efficiently around a part. This is an issue because the Coq programming model does not include randomaccess arrays. However, running a quiz only requires local displacements (left and right steps), so we can use Huet’s zipper data structure [21] to solve our problem. The data structure we use to search and check for reducible configurations is a zpart (for zipped part). A zpart is a representation of a pair of a part and of a dart position in that part, optimized for fast access to the arity intervals of nearby darts. It consists of • the (counterclockwise) sequence of subparts that starts with the subpart s containing the dart position • the reversed sequence of subparts before s • a subpart location pinpointing the dart position in s This representation makes it easy to move one step up or down the sequence of subparts in constant time, keeping the initial part p and its reverse in global constants to handle steps that wrap around the beginning or end of p (to minimize the occurrence of such cases, the total size of the forward and reverse sequences of a zpart is always twice that of p). Because the construction of all our configuration maps always starts with a Y step, all quizzes start with a three-way branch from the (central) node created by the Y step: Thus quizzes can be decomposed into the triple of arities of the faces incident to the central node, and a triple of questions rooted at the edges incident to the central node. We use a decision diagram structure to sort all the question triples according their respective arity triple, for all rotations and reflections of the central node (omitting reflections for configurations that have an axial symmetry). This speeds up considerably the configuration search: • While traversing the part, we only need to consider complete nodes, such that all the three faces incident to the node are mapped to an interval of arities by the part. There are at most 6, and usually less than 3 complete nodes in each subpart. s0 s1 s2 s4 s3 s5 h01 h12 h23 f1 1 f2 1 f3 1 hub hub s1 s0 s2 s3 s4 s5 h23 h01 h34 h12 f1 1 f3 1 f2 1 h34 subpart of pi h34 f2 1 face index in pi face index in pi h34 f2 1 hub spoke i hat over i and j j th fan over i and j hub si hij fi j 47 • We only need to look up the triple of the top values of these intervals in the decision diagram to get the list of question triples that should be tested. This list is usually quite short (it can contain up to 239 triples, though). In addition, a little analysis reveals that these questions only traverse a very limited subset of the dart positions of the part: only 14 positions are reachable in each subpart, so we can use a 15-element enumeration to represent subpart locations in a zpart (reserving one value for out-of-bounds locations). Coding and verifying the correctness of the configuration occurrence check was therefore a straightforward11 exercise in brute-force enumeration that only took a few days. We use the clever branch-and-bound strategy devised by Robertson et al. [26,27] to drive the enumeration of parts. They use explicit scripts to guide the initial decomposition of the part search space, then turn to either i. blind enumeration to check that if we fix the arity of every face that is mapped to a finite interval by the part, then a reducible configurations occurs, ii. automated branch and bound to check that 10δ(r) − 60 ≥ 0. iii. explicit matching to check that this branch is a subcase of the rotation/reflection of a previously explored branch. Check ii uses an explicit hint to break the bound check into checks that a well-chosen multiset of the T[ j] = #{i : pi fits f j-3(x)} − #{i : pi fits f j-3(x)} and T[j1,j2] = T[j1] + T[j2] obey specific bounds that sum up to less than 20(m – 6) + 2, where m is the arity of the hub x. The branching strategy for these searches consists in enumerating all the sublists of pi that have a non-empty intersection with the appropriate rotation of the part p, and check that this intersection is included in enough pi to respect the bound. If 11 Except for a devious trick that allows double steps to wrap around a spoke whose arity interval is [5,6] – see file redpart.v. u u l u r u l u r u r u l u u h h h l h l h r h r f0 r f1 r f2 r f1 l f1 l f2 l f0 l f0 r hub spoke spoke spoke hat hat fan fan left step right step subpart unreachable dart spoke 48 this fails, then check i is performed on the intersection (in fact, most of the configurations searches occur this way). Robertson et al. provide a single C program[27] that both interprets the explicitly supplied scripts and performs the automated checks i-iii. We used a hybrid strategy to implement this in Coq • We used reflection, as we had done for the reducibility and Birkhoff lemmas, to perform the automated checks: we wrote (rather short) recursive Coq programs that performed the checks, and proved their partial correctness. • We used a so-called deep embedding for the explicit scripts: we added five Coq tactics that allowed us to directly interpret the explicit scripts12 as Coq proof scripts. We chose to use a deep embedding mainly because it seemed silly to duplicate part of the proof search engine of Coq in a reflection program. However, this decision had another much more important, advantage: we could use Coq’s interactive facilities to step through the scripts. This turned out to be crucial, because, although we had formally proved their correctness, most of our reflected functions needed to be debugged! The reason for this apparent contradiction was that we only formally verified partial correctness: that if a reflected check returned true, then the condition it checked for actually held. By that measure, a check that returned false, would be correct, albeit useless. As it turned out, a handful of trivial programming errors (e.g., off-by-one arithmetic, misspelled constants, and even clerical errors in entering the discharge rules) had crept in the development, and needed to be weeded out. As we pointed out earlier, debugging with a completely pure language is awkward, so the ability to step through the script was vital. Finally, the deep embedding allowed us to use the proof assistant to develop our own scripts for parts of length 5 and 6. In the Robertson et al. proof these cases are dispatched with manual proofs using graphical arguments that would have been difficult to carry out in a fully formal system, so we decided instead to reuse the scripting infrastructure to come up with our own scripts. Although the script for size 6 is 640 lines long, this was completely straightforward. 5.6 From maps to hypermaps Like the Alexander proof of the Jordan Curve Theorem [25], our proof that an arbitrary map colouring problem can be reduced to a set of of hypermap colouring problems uses discrete grids to approximate the continuous plane. Specifically, we use grids based on points with coordinates (x,y) that are binary fixed point numbers, i.e., such that 2s x and 2s y are both integers, for some fixed non-negative integer s (the scale of the grid). We approximate the open connected regions of the map with simple polygons that are the union of grid squares, which we call mattes (i.e., the vertices of a matte are grid points, and the sides of a matte are parallel to one of the axes). We really only need to construct discrete approximations of finite maps, since it is well-known that the compactness theorem for propositional logic implies that the Four Colour Theorem for infinite maps follows from the Four Colour Theorem for finite maps. However, the proof of the compactness theorem relies on the axiom of choice, which we wished to avoid. Therefore we specialized the compactness argument to the Four Colour Theorem, making use of the fact the every open region 12 Actually, a pretty-printed and somewhat more readable version of these scripts. 49 contains a grid square for a suitable scale (this gives an effective enumeration of the regions of a map). The discrete colouring problem for a finite map is constructed in several steps: 1) Pick, for each pair of adjacent regions, a common border point that is not a corner; also pick a point in each region. 2) Pick, for each of the border points chosen in step 1), a small rectangle that contains the point but that is disjoint from other regions and the other rectangles chosen in this step. 3) Pick grid rectangles (rectangles whose vertices are grid points) that approximate from inside the rectangles chosen in step 2); the approximation of each rectangle must contain the corresponding border point from step 1). 4) Pick mattes that appoximate the map regions from inside; the matte approximation of a region R must contain the point chosen in step 1) and meet the inside of every grid rectangle from step 3) that corresponds to a border point of R. 5) Now the interior of each border rectangle B meets two region mattes; call them Q and R. Extend Q with grid squares included in B until Q contains a square adjacent to a square contained in R; do this for each B. 6) Choose a grid rectangle that contains all the extended mattes from step 5) – a bounding box. 7) Contruct the hypermap that corresponds to the map whose regions are the individual grid squares contained in the bounding box from step 6), and the exterior of that box. 8) Each matte from step 5) now corresponds to a simple ring cycle in the hypermap from step 7); use the converse of the “patch” operation defined in §5.1 to construct the remainder map of that ring (in essence, erasing the darts corresponding to grid segments inside the matte). Repeat for each matte; the result is the desired hypermap. To summarize, we take the hypermap for a regular rectangular grid and punch out approximations of the map regions. It is interesting to note that in this construction, the planarity of the final hypermap is a direct consequence of the planarity of the Step 1) Steps 2)-3) Step 4) Step 5) Step 8) Steps 6)-7) 50 rectangular grid hypermap. The latter is established simply by counting: the map for an m×n rectangle has (m+1)(n+1) nodes, mn+1 faces, m(n+1) + (m+1)n edges, hence N + F – E = (m+1)(n+1) + (mn+1) - m(n+1) - (m+1)n = 2 so the Euler formula holds. The Jordan Curve Theorem plays no part in this. In step 8) the mattes and bounding box should all use the same scale. This can obviously always be achieved by subdivision. In the Coq proof, we reduce everything to a common scale just after step 4), and perform steps 5) to 8) on the integer grid; even in steps 3) and 4), we use only rectangles and mattes on the integer grid, introducing the scale factor separately: for example, we use the following definition: an integer matte Q contains a point P = (x,y) at scale s iff (2 s x, 2 s y) ∈ Q. Steps 1), 2) and 3) follow from the topological definitions, classical reasoning, and the fact that grid points form a dense subset of the plane. Step 4) is carried out by choosing, for each region R of the map, an increasing sequence of mattes approximating R from inside, Q0 ⊆ Q1 … ⊆ Qn ⊆ R. We take for Q0 a single grid square containing the point chosen in step 1). For any i, if there is some border grid rectangle B from step 3) that meets R but not Qi, we choose some point P in BR, and choose a matte Qi+1 such that Qi {P} ⊆ Qi+1 ⊆ R; if there is no such B then we are done (we set n=i). The proof that Qi+1 exists is the only point in the construction where we need to use topological reasoning. We show that the union Qi of all mattes Q such that Qi ⊆ Q ⊆ R is a nonempty clopen of R; since R is connected, this implies Qi = R, hence P∈ Qi, and the existence of Qi+1 then follows from the definition of Qi. That Qi is both closed and open in R follows from two combinatorial properties of mattes; for any matte Q and grid recatngle N, 1. For any point P∈N, there is a matte Q’ such that Q {P} ⊆ Q’ ⊆ Q N . 2. For any point P∈NQ there is a matte Q’, such that Q ⊆ Q’ ⊆ Q N and Q’ contains all grid squares adjacent to P. Property 1 implies that Qi is closed, and 2 implies that is Qi open. Both properties are immediate consequences of a more specific property of integer mattes 3. For any matte Q whose vertices have even integer coordinates, any rectangle N that meets Q and whose vertices have integer coordinates, and any unit square S included in N and not touching the border of N, there is a sequence S0... Sn=S of unit squares of N such that S0 ⊆ Q, and Q S1 ... Si is a matte for all i ≤ n. In fact, property 3 is the only significant combinatorial property we need to proove on mattes, because it also provides the means of carrying out step 5) of the construction. Property 3 is nontrivial because of the restriction that mattes must be simple polygons; that restriction is needed in step 8). This restriction prevents us from blindly adding all the unit squares in N to Q: we have to carefully select the sequence S0... Sn so that it stays away from Q after S0 and S1. This is possible because of the assumption that the vertices of Q have even coordinates. Consider the 4×4 square with even coordinates that covers S and all the unit squares adjacent to S. The assumption on Q implies that each of the four 2×2 quadrants of the 4×4 square are either contained in or disjoint from Q. 51 If S is contained in Q we can take n=0; if not, the 2×2 quadrant that contains S is disjoint from Q. If one or two of the quadrants adjacent to S meet Q then they are contained in Q so we can take n=1, letting S0 be the square of that quadrant adjacent to S. If Q meets only the quadrant that touches S diagonally, then we can take n=2, letting S0 be the square of that quadrant that touches S (diagonally), and S1 be either of the two squares adjacent to both S0 and S. Finally, if the entire 4×4 square is disjoint from Q then by induction13 on the size of N there is a sequence of unit squares that extends Q, ends with a unit square S’ adjacent to S, and lies entirely on one side of the line separating S from S’, i.e., if S’ is just below S, then the sequence contains only squares below S. In that last case we can simply append S to the sequence. Finally, step 6) is a straightforward computation, and step 7) is just a matter of coining arithmetic formulas for the e, n, and f permutations, given the following system of coordinates : • grid points have integer coordinates • grid unit squares have the same coordinates as their lower lefthand corner • darts of the hypermap are unit vectors between two grid points. The coordinates of a dart are obtained by adding the coordinates of its origin to the coordinates of the unit square that lies counterclockwise of the dart. The dart coordinates enjoy several useful identities. A dart wth coordinates d belongs to a square with coordinates d/2has origin d-d/2, and its orientation is given by the remainder of that division (e.g., (0,0) for left-to-right). The f (resp. n) permutation simply cycles through the four remainder values while keeping the square (resp. origin) coordinates unchanged. In the Coq script these coordinates and their properties are used throughout this part of the proof; for example, they are used to carry out the complex matte extension proof outlined above in an orientation-independent way. 6 Looking back This was a project that started out with a modest objective – trying out the internal execution engine of Coq on a “real” benchmark – but became more ambitious over time, as each successfully reached objective revealed that a new one might just be within reach. We had come across the Robertson et al. proof[26] while looking for programming project for students in a basic computer science course. We wondered how a “modern” approach to the reducibility computation, based on decision diagrams, would fare against the idiomatic array-based C program supplied by Robertson et al., so we tried rolling out our own, in OCaml. The C program turned out to be faster, but not by much; however we noted that the usually crucial caching and hash-consing 13 In the formal proof, the induction comes before the case analysis on S and its neighbors; furthermore in the induction hypothesis we need to make a distinction between the rectangle N in which the Si sequence is taken, and a larger rectangle N0 that contains all the neighbors of S, but is such that Q and N0-N are disjoint. S 52 optimizations had little impact for this problem. This meant that the reducibility computation could be carried out efficiently by a purely functional program – such as an internal Coq program. This looked like a nice benchmark for Coq or other provers, so we decided to give it a try. We decided to start modestly by just enumerating the colourings of a configuration, using explicit adjacency lists. Our program14 was reasonably fast, even though it made liberal use of higher-order iterators, so we tried generating the decision tree, primarily to check memory requirements. The experiment showed that: • The full computation would barely fit in memory, using Coq’s internal data representation. The lower layers of the chromogram tree would have to be compressed, and this would complicate the algorithm enough to make a correctness proof worthwhile. • Proving the correctness of such purely combinatorial programs was surprisingly easy in Coq, using the generate-and-test approach outlined in section 4. For example, a somewhat tricky function that computed in a single pass the normalized edge-colouring of a ring from one of its face colourings was harder to program than to prove correct! These observations prompted us to switch to a more ambitious goal: a fully checked formal proof of the (computation-intensive) reducibility part of the Four Colour Theorem. We enlisted the help of Benjamin Werner, and divided the work: we would port and verify the decision diagram based check, and he would formalize the Kempechain proof of the validity of the “consistent” relation presented in section 5.2. Our “program check” part turned out to be easier than expected, once we had hammered out a crude first prototype of the tactic shell described in section 4 to organize the bookkeeping that cluttered our scripts. Having finished early, we took a look at the bigger picture – which definition of planar graphs could plausibly be used to actually do a full proof of the full Four Colour Theorem in Coq? It appeared that an explicit construction program definition would work best for the reducibility part. However, using this definition for the general statement seemed contrived. In particular, the proof of correctness of the contract validity check clearly depended on the Birkhoff theorem and hence some form of the Jordan Curve Theorem, which seemed difficult to derive by induction on the construction program15. The line of reasoning exposed in 5.1 lead us to hypermaps, and we decided to try to validate the concept by proving the equivalence between the Euler formula and the Jordan property. This was much harder than anticipated, because of the gap between the intuitive, picture-rich proof outline, and the very precise logical statement that had to be fed to the Coq proof assistant. We went through several styles of definition for the basic notions, such as paths. At the outset, the “boolean predicate with reflection” approach, outlined in section 4, emerged as a clear winner. Our tactic shell proved to be surprisingly effective in this setting; in particular, we could usually derive an induction hypothesis directly from a goal with a single command, so we rarely needed 14 Sadly, this program does not appear in the final version of the development. It was scrapped when we changed the representation of configurations to the explicit construction programs presented in 5.3. 15 Other efforts at formalizing graph theory had stumbled upon this difficulty. 53 spell out such hypotheses. With the addition of occurrence selection to our tactic shell, this even turned out to be nearly always true. With the formal setting in place, we decided to tackle the contract validity check, a corner of the reducibility proof that we had originally planned to cut. The correctness of the check follows easily from the Birkhoff lemma (this is one of the cleverest innovations of the Robertson et al. proof). Proving the Birkhoff lemma was enticing because it would be an example of using the reflected reducibility computation to prove a tangible graph theoretic result. However, the proof of the Birkhoff lemma required the graphically obvious operations of cutting up a map along a cycle and conversely of gluing back two map pieces. Defining these operations formally, and proving such intuitively obvious properties as “the whole map is planar if and only if the two pieces are” turned out to be much harder than expected, again because of the gap between pictorial intuition and computerized formalism. Indeed, the asymmetric “patch” relation described in section 5.1 emerged directly from the formal statement of the Jordan property – it seems completely counterintuitive when one only looks at plain maps or graphs. This “patch” relation is one of the enabling concepts of the formal proof; as the proofs of most of the key embedding lemmas depend on its precise formulation and properties (see section 5.4, steps 1 and 4). The “obvious” symmetry properties of plain map cutouts in a plain map are actually not so trivial to prove formally (see file revsnip.v). At this point we needed to integrate in our setting Benjamin Werner’s work on the correctness of the consistency relation between edge colourings and chromograms, in order to be able to use the reflected reducibility computation in the proof of the Birkhoff lemma. The work wasn’t quite complete; some lengthy case analyses were unfinished. Our own experience suggested that these manual analyses could be avoided by replacing the relational characterisation of the consistency predicate with an effective one, and replacing lemmas that asserted the existence of chromograms with explicit functions for computing them. Indeed, it took us less than a day to write these functions and prove their correctness (using our tactic shell), and only another day to complete the entire proof, reusing lemmas from the Jordan↔Euler proof to build on-the-fly the construction program he needed to postulate. The entire development fits in a single file (kempe.v). We also needed to relate the adjacency list data used by the reducibility check functions to our hypermap model. Extracting geometrical relations directly from the adjacency data turned out to be complex and impractical, so we decided to exploit the construction program structure that was built into the Robertson et al. data (it was used in their colouring function), and to generate both the adjacency data and a dependently-typed hypermap from it. The correspondence proofs were feasible, but not very elegant; they used a large number of ad hoc arithmetic manipulations, as the maps dart were defined as pairs of indices in the construction. We were now faced with a software engineering problem: we needed to integrate formal theories whose development had spanned several years. This meant that while their main results were still usable (after all, this was a formal development), their proofs tended to rely on slightly different collections of facts about basic data 54 structures such as integers, lists, and paths. We might have been able to complete the Birkhoff lemma proof, but a major refactoring was clearly needed for any further progress. This actually took several (part-time) months to complete. With the cleaned-up basis in place, the Birkhoff lemma and its immediate consequences (e.g., that any face should have at least five sides) followed easily (the “immediate consequences” actually required more work). The last bit of graph theory that remained in the Four Colour Theorem proof was the construction of the embedding. It was only at this point, when we examined how we could formalize the unpublished proof of correctness [27] of the configuration search heuristic used by Robertson et al., that we realized how much the unavoidability proof could be simplified, and that a complete formal proof of the Four Colour Theorem was close at hand. After devising the quiz compilation algorithm, we realized that we could also compile the contract map colourings, using extended construction programs as an intermediate representation. That was the death knell for the adjacency list representation and the mess of index calculations it required. We completely removed it from the proof, along with the graph colouring procedure that had been the first step of the proof. After this, completing the combinatorial proof was mostly a (purely functional) programming exercise, described in section 5.5, and refactoring (packaging sets of geometrical assumptions in records, which we had deferred, since we didn’t know which exactly sets would be useful). Finally, the first part of the proof – the discretization of the continuous problem – was the last one to to be completed. We wanted to avoid getting heavily involved in real analysis and the proof of the Jordan Curve Theorem, so we had planned to settle for an informal proof of this reduction. While writing the first draft of this paper we realized both that we would have a much stronger result if we completed this last part, and that it was possible to do this while relying mostly in the combinatorial methods we had developed for the rest of the proof – so we went ahead and did it. 7 Looking ahead As with most formal developments of classical mathematical results, the most interesting aspect of our work is not the result we achieved, but how we achieved it. We believe that our success was largely due to the fact that we approached the Four Colour Theorem mainly as a programming problem, rather than a formalization problem. We were not trying to replicate a precise, near-formal, mathematical text. Even though we did use as much of the work of Robertson et al. as we could, especially their combinatorial analysis, most of the proofs are largely our own. Most of these arguments follow the generate-and-test pattern exposed in section 4. We formalized most properties as computable predicates, and consequently most of our proof scripts consisted in verifying some particular combination of outcomes by a controlled stepping of the execution of these predicates. In many respects, these proof scripts are closer to debugger or testing scripts than to mathematical texts. Of course this approach was heavily influenced by our starting point, the proof of correctness of the graph colouring function. We found that this programs-as-proof style was 55 effective on this first problem, so we devised a modest set of tools (our tactic shell) to support it, and carried on with it, generalizing its use to the rest of the proof. Perhaps surprisingly, this worked, and allowed us to single-handedly make progress, even solving subproblems that had stumped our colleagues using a more orthodox approach. We believe it is quite significant that such a simple-minded strategy succeeded on a “higher mathematics” problem of the scale of the Four Colour Theorem. Clearly, this is the most important conclusion one should draw from this work. The tool we used to support this strategy, namely our tactic shell, does not rely on sophisticated technology of any kind, so it should be relatively easy to port to other proof assistants (including the newer Coq). However, while the tactic shell design might be the most obvious byproduct of our work, we believe that it should have wider implications on the interface design of proof assistants. If, as this works seems to indicate, the “programming” approach to theorem proving is more effective than a traditional “mathematical” approach, and given that most of the motivated users of poof assistants have a computer science background and try to solve computer-related problems, would it not make interface of a proof assistant more similar to an program development environment, rather than strive to imitate the appearance of mathematical texts? Acknowledgements The work of Benjamin Werner on reducibility correctness was an integral part of this development, even though it does not appear in the final proof script; we benefited from extended discussions with him throughout this work, several of which also involved Vincent Danos. We thank Kenneth Appel for a historical perspective on the original Four Colour Theorem proof, Robert Cori for pointing out the related work on hypermaps, and Hugo Herbelin and Bruno Barras for always patiently helping us (mis)use the Coq assistant. We are also grateful to Cédric Fournet for his careful proofreading of the initial draft of this report. Rerefences 1. S. B. Akers, ‘Binary Decision Diagrams’, IEEE Transactions on Computers c27(6) (1978), 509–16. 2. T. Altenkirch, ‘Constructions, Inductive types and Strong Normalization’, PhD Thesis, University of Edinburgh, 1993. 3. K. Appel and W. Haken, ‘Every map is four colourable’, Bulletin of the American Mathematical Society 82 (1976), 711–12. 4. K. Appel and W. Haken, ‘Every map is four colourable, Part I: Discharging’, Illinois Journal of Mathematics 21 (1977), 429–90. 5. K. Appel and W. Haken, ‘Every map is four colourable, Part II: Reducibility’, Illinois Journal of Mathematics 21 (1977), 491–567. 6. K. Appel and W. Haken, Every map is four colourable, American Mathematical Society, 1989. 7. D. Aspinall, ‘Proof General: A Generic Tool for Proof Development’, Proceedings of Tools and Algorithms for the Construction and Analysis of Systems, (TACAS) 2000, Springer-Verlag LNCS 1785, 2000. 56 8. G. Barthe, J. Hatcliff and M.H. Sørensen, ‘CPS translations and applications: the Cube and beyond’, Higher Order and Symbolic Computation 12(2) (1999), 125– 70. 9. Y. Bertot and P. Castéran, Interactive Theorem Proving and Program Development, Coq'Art: The Calculus of Inductive Construction, Texts in Theoretical Computer Science, an EATCS series. Springer Verlag, 2004. 10. G. D. Birkhoff, ‘The reducibility of maps’, American Journal of Mathematics 35 (1913), 115–28. 11. S. Boutin, ‘Using reflection to build efficient and certified decision procedures’, Proceedings of Theorethical Aspects of Computer Science (TACS) 1997, SpringerVerlag LNCS 1281, 1997. 12. R. E. Bryant, ‘Graph-Based Algorithms for Boolean Function Manipulation’, IEEE Transactions on Computers c-35(8) (1986), 677–91. 13. The Coq Development Team, ‘The Coq reference manual, version 7.3.1’, system and documentation available from ftp://ftp.inria.fr/INRIA/coq/V.7.3.1. 14. T. Coquand and G. Huet, ‘The Calculus of Constructions’, Information and Computation 76(2/3) (1988), 95–120. 15. R. Cori, ‘Un code pour les graphes planaires et ses applications’ Astérisque 27 (1975), 169. 16. R. Cori and A. Machí, ‘Maps, hypermaps and their automorphisms: a survey I, II, III’, Expositiones Mathematicae 10(5) (1992), 403–67. 17. H. Geuvers, R. Pollack, F. Wiedijk and J. Zwanenburg, ‘A Constructive Algebraic Hierarchy in Coq’, Journal of Symbolic Computation 34(4) (2002) 271–86. 18. H. Heesch, Untersuchungen zum Vierfarbenproblem, 810/a/b, Bibliographishes Institut, Mannheim-Wien-Zürich, 1969. 19. M. Hofmann and T. Streicher, ‘The groupoid interpretation of type theory’, Proceedings of Twenty-five years of constructive type theory, Oxford University Press, 1998. 20. J. E. Hopcroft and R. E. Tarjan, ‘Efficient planarity testing’, Journal of the Association for Computer Machinery 21 (1974), 145–54. 21. G. Huet, ‘The Zipper’, Journal of Functional Programming 7(5) (1997), 549–54. 22. A. B. Kempe, ‘On the geographical problem of the four colours’, American Journal of Mathematics 2 (part 3) (1879), 193–200. 23. K. Kuratowski, ‘Sur le problème des courbes gauches en topologie’, Fundamenta Mathematicae 15 (1930), 271–83. 24. A. Miquel, ‘A Model for Impredicative Type Systems, Universes, Intersection Types and Subtyping’, Proceedings of the fifteenth annual IEEE symposium on Logic in Computer Science (2000). 25. M. H. A. Newman, Elements of the topology of plane sets of points, Dover Publications (1992). 26. N. Robertson, D. Sanders, P. Seymour, and R. Thomas. ‘The Four-Colour Theorem’, Journal Combinatorial Theory, Series B 70 (1997), 2–44. 27. N. Robertson, D. Sanders, P. Seymour, and R. Thomas. ‘Discharging Cartwheels’, unpublished manuscript, available at ftp://ftp.math.gatech.edu/pub/users/thomas/fcdir/discharge.ps. 28. T. L. Saaty and P. C. Kainen, The Four-Colour Problem: Assaults and Conquest, McGraw-Hill, 1977. 29. S. Stahl, ‘A Combinatorial Analog of the Jordan Curve Theorem’, Journal of Combinatorial Theory, Series B 35 (1983), 28–38. 57 30. P. G. Tait, ‘Note on a theorem in the geometry of position’, Transactions of the Royal Society of Edinburgh 29 (1880), 657–60. 31. W. Tutte, ‘Duality and trinity’, Colloquium Mathematical Society Janos Bolyai 10 (1975) 1459–72. 32. W. T. Tutte, ‘Combinatorial oriented maps’, Canadian Journal of Mathematics 31 (1979), 986–1004. 33. T. R. S. Walsh, ‘Hypermaps versus bipartite maps’, Journal of Combinatorial Theory, Series B 18 (1975) 155–63. 34. D. W. Walkup, ‘How many ways can a permutation be factored into two ncycles?’, Discrete Mathematics 28 (1979), 315–19. 35. B. Werner, ‘Une Théorie des Constructions Inductives’, PhD Thesis, Université Paris VII, 1994. 36. R. Wilson, Four Colours Suffice, Penguin books, 2002.Voir également :
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actu.fr Bretagne Grand Est Hauts-de-France Île-de-France Normandie Nouvelle Aquitaine Occitanie Pays de la Loire Polynésie Française 20.4° Menu 78actu Yvelines. La manufacture de masques en tissu de Vernouillet ouvre un chantier d’insertion Île-de-France 78actu Société Économie Faits divers Politique Coronavirus Loisirs-Culture Sports Insolite Monde Lifestyle Annonces légales Yvelines. La manufacture de masques en tissu de Vernouillet ouvre un chantier d’insertion Lancée jeudi 23 avril au matin, la manufacture de masques en tissu à Vernouillet (Yvelines) s'apprête à ouvrir un chantier d'insertion pour une durée de quatre mois. Publié le 24 Avr 20 à 17:32 Pierre Bédier le vendredi 24 avril 2020 à la manufacture de masques en tissu à Vernouillet. Pierre Bédier le vendredi 24 avril 2020 à la manufacture de masques en tissu à Vernouillet. (©Le Courrier des Yvelines / Florian Dacheux) C’est désormais officiel. Lancée hier matin au sein du Gymnase de l’Amandier à Vernouillet (Yvelines), la manufacture de masques en tissu ouvrira dès la semaine prochaine un chantier d’insertion. Lire aussi : Yvelines. Une manufacture de masques en tissu à Vernouillet Composé pour l’heure d’une vingtaine de bénévoles, cet atelier de confection a pour objectif de produire 12 000 masques, de manière à fournir les populations vernolitaine (10 000) et médanaise (2000) avant le déconfinement. Accueilli par Karine Kauffmann et Pascal Collado, respectivement maires de Médan et Vernouillet, Pierre Bédier, le président du Département des Yvelines qui était également accompagné du sous-préfet Stéphane Gauvrogel, a confirmé le financement avec le soutien de la DIRECCTE (Direction régionale des entreprises, de la concurrence, de la consommation, du travail et de l’emploi) d’un chantier d’insertion à hauteur de 118 000 euros. Une manufacture du même genre à Mantes Spécialisée dans ce domaine, l’association ACR Equalis de Conflans chapeautera le recrutement de 16 contrats d’insertion pour une durée de 4 mois. Une dizaine de jeunes du centre social Les Résédas de Vernouillet étaient par ailleurs sur place ce vendredi après-midi pour prêter main forte aux bénévoles et pourquoi pas se découvrir une nouvelle passion pour la couture. Certains d’entre eux pourraient intégrer le chantier d’insertion. Cliquez sur une miniature pour agrandir et parcourir les photos. Stéphane Gauvrogel, sous préfet, et Pierre Bédier, président du Département des Yvelines, étaient sur place ce vendredi 24 avril pour confirmer la mise en place de la manufacture et annoncer officiellement le lancement d’un chantier d’insertion. (© Le Courrier des Yvelines / Florian Dacheux) Karine Kauffmann et Pascal Collado, respectivement maires de Médan et Vernouillet, ont fait part au sous-préfet de l’entente des deux villes sur ce projet. (© Le Courrier des Yvelines / Florian Dacheux) Un important stock de tissu a été offert par l’association franco-turque de Carrières-sous-Poissy. (© Le Courrier des Yvelines / Florian Dacheux) Les hommes sont aussi de la partie question couture. (© Le Courrier des Yvelines / Florian Dacheux) La table de découpage. (© Le Courrier des Yvelines / Florian Dacheux) Des jeunes du centre social Les Résédas à Vernouillet sont venus prendre quelques cours de couture. (© Le Courrier des Yvelines / Florian Dacheux) Présent au côté de Pierre Bédier, Raphaël Cognet, le maire de Mantes-la-Jolie, était là en repérage. Une manufacture du même genre doit ouvrir dans les prochains jours afin de fournir en masques la population mantaise. Florian Dacheux Par : Rédaction Saint-Germain| +78actu +Coronavirus +Société Ailleurs sur le web Sophie Marceau: cette photo confirme sa prise de poids hallucinante ! Life Indigo | Sponsorisé Le visage de Sophie Marceau aujourd'hui confirme les rumeurs, la preuve en image ! Game Of Glam | Sponsorisé Ces entreprises travaillent sur une cure pour le coronavirus - et vous pouvez investir dedans eToro Blog Post | Sponsorisé Ailleurs sur le web Là où Nagui vit aujourd’hui à l’âge de 58 ans est difficile à croire Cash Roadster | Sponsorisé Le fils de Jamel Debbouze choque par sa beauté aujourd'hui, la preuve en image ! PsychicMonday | Sponsorisé A lire aussi sur Actu.fr Le juge François-Marie Giacomoni est mort Île-De-France VIDÉO. Danse déconfinée à Paris : "J'ai fait une connerie, j'ai été hypnotisée par un moment de joie" Île-De-France Toulouse. Ces VIP de la prison de Muret (3/3) : Marc Cécillon, un capitaine des Bleus aux idées noires Occitanie Ailleurs sur le web 14 215 876 y jouent ! Le nouveau jeu de construction qui rend tout le monde accro. 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Deux promeneuses violées, un appel à la vigilance est lancé Île-De-FranceActu Seine-Saint-Denis Newsletter Recevez chaque jour 10 infos proches de vous Email * Nom * Prénom * En renseignant vos informations, vous avez pris connaissance de notre Politique de confidentialité, vous acceptez de recevoir chaque jour, par email, une lettre d'information personnalisée des derniers articles publiés sur actu.fr J'accepte de recevoir des communications de la part de notre groupe Publihebdos Je m'inscris Dernières actus 13:22 Yvelines. Les Fées des blouses ont fabriqué plus de 2000 blouses pour l'hôpital de Rambouillet 13:22 Yvelines. Le transport à la demande élargi au Sud de Rambouillet 12:05 Yvelines. La Poste va rouvrir 69 bureaux cette semaine 11:10 Yvelines. Décès du Dr Jacques Fribourg, urgentiste à l’hôpital de Trappes, victime du Covid-19 9:02 Yvelines. Plaisir : une permanence pour les femmes victimes de violences conjugales 8:26 Yvelines. 60 couturières à l'œuvre pour fabriquer des masques à Vélizy-Villacoublay 7:52 Yvelines. Les déchetteries de Versailles Grand Parc rouvertes Hier Yvelines. Mantes-la-Jolie : distribution de masques pour le personnel médical libéral Hier Yvelines. Voisins-le-Bretonneux : des denrées collectées pour les jeunes actifs en difficulté Hier Yvelines. Coignières : les premiers masques en tissu distribués aux plus fragiles Hier Yvelines. Des pommes d'amour pour remercier les soignants du centre hospitalier de Poissy / St Germain-en-Laye Hier Yvelines. Affaire Christian Maréchal à Chambourcy : un homme placé en détention ! Hier Yvelines. Aubergenville : le marché couvert va rouvrir le 2 mai Hier Yvelines. Une journée de la Déportation particulière Hier Yvelines. La course cycliste Paris-Mantes annulée : une première depuis 1956 Avant-hier Yvelines. Soli’Malte, une maraude alimentaire, sanitaire et sociale à Versailles Avant-hier Yvelines. Découvrez le fitennis à 18h avec le Tennis Club de Poissy en live sur Facebook Avant-hier Yvelines. Issou : mordue par un serpent, Alexandra est hospitalisée depuis 48 heures Avant-hier Yvelines. « Soyons solidaires », une plateforme d’entraide à Guyancourt Avant-hier Yvelines. Flins-sur-Seine : comment la réouverture de l'usine Renault a-t-elle été préparée Avant-hier Yvelines. À Versailles, l'institut Vedecom a un nouveau président Avant-hier Yvelines. Une vidéo pour rompre l'isolement aux Eaux-Vives à Saint-Rémy-lès-Chevreuse Avant-hier Yvelines. Ecquevilly : la Protection Civile prête main-forte à la ferme de la Muette Avant-hier Yvelines. La solidarité s'organise à Louveciennes pour fabriquer des masques Avant-hier Yvelines. Maule : durant le confinement, Distrikt livre ses bières à domicile Avant-hier Yvelines. A 10 heures, on se bouge sur le Facebook de Versailles sport 24/04/2020 Yvelines. Course pédestre : le Paris-Versailles annulé 24/04/2020 Confinement. A Bois-d'Arcy dans les Yvelines, la famille Le Pann donne des balconcertos tous les soirs 24/04/2020 Yvelines. France médical industrie, à Sartrouville : « Au plus fort de la crise, je recevais 100 appels par jour » 24/04/2020 Yvelines. La manufacture de masques en tissu de Vernouillet ouvre un chantier d'insertion Météo à Versailles Averses Vent Sud 20.4° 8 km/h 54.1% Voir les prévisions 78actu Contact Mentions légales Données personnelles Cookies et consentement Gérer mon consentement Sites du groupe Publihebdos Annonces légales Examens Élections Municipales 2020 Emag Publihebdos Agence APEI Hebdoscom Presse Régionale Sites du groupe Sipa-OF Ouest-France Autres sites © 2020 actu.fr, détenu et coexploité par Publihebdos et ses filiales. Hébergement dédié : Group DIS Digital Ad Trust ACPM Switch site Exclusives Google Android Auto TV Wear OS Chrome YouTube Alphabet G Suite Videos More social networks Submit a Tip / Contact Us Trade In Toggle dark mode Search SearchToggle search 9TO5MAC 9TO5TOYS ELECTREK DRONEDJ YESTERDAY Google wants to help cure your boredom with its most popular Doodle games Kyle Bradshaw - Apr. 26th 2020 7:58 pm PT @SkylledDev 1 Comment FacebookTwitterPinterestLinkedInReddit Right now, so many of us around the world are stuck at home, eager for a way to fill the time. For the next two weeks, Google is going to do its part to help cure your boredom by showcasing a popular game from a past Doodle every day. Over the years, Google has featured an impressive number of games and minigames on its homepage, all of which have been carefully preserved and archived on the Google Doodle Blog. Starting April 27 and running for two weeks, Google is launching a new series of ten Doodles, each one a callback to one of the company’s popular games. As COVID-19 continues to impact communities around the world, people and families everywhere are spending more time at home. In light of this, we’re launching a throwback Doodle series looking back at some of our popular interactive Google Doodle games! The first day’s featured popular Google Doodle game, “Coding for Carrots,” dates back to 2017, and originally celebrated the 50th anniversary of Logo, the first ever programming language designed for use by children. In the game, you’ll create simple combinations of commands that will tell your rabbit how to collect the carrots on each level. If you or your children are particularly interested in today’s throwback Doodle, consider introducing them to Scratch, a kid-oriented programming language which the game is loosely based on. popular-google-doodle-games-1-codingcoding-for-carrots As for why Google is staggering the games out over the two week period, you may remember that some of the best and most popular Doodles in the last few years have actually had multiplayer support. For example, the Google Doodle in celebration of the Mexican bingo-like game Lotería allowed you to compete either with just your friends or with random players online. By staggering them out, Google can ensure each day’s focused game has a variety of players. Combined with the recent decisions to make Stadia free for two months and feature Stadia on the Google.com homepage, it’s clear that Google believes that promoting gaming is one of the most effective ways to encourage people to “Stay Home. Save Lives.” What past popular Google Doodle game would you like to see featured on the homepage? Let us know down in the comments. More Google Doodles: Google Doodle celebrates Earth Day with relaxing game about honeybees Google Doodle offers coronavirus tips, ‘Stay Home. Save Lives.’ [Updated] Two-week Google Doodle series thanks coronavirus helpers [Updated] FTC: We use income earning auto affiliate links. More. You’re reading 9to5Google — experts who break news about Google and its surrounding ecosystem, day after day. Be sure to check out our homepage for all the latest news, and follow 9to5Google on Twitter, Facebook, and LinkedIn to stay in the loop. Don’t know where to start? Check out our exclusive stories, reviews, how-tos, and subscribe to our YouTube channel Check out 9to5Google on YouTube for more news: Guides Google Doodle Google Doodle Google's homepage is one of the most-viewed web pages around the globe, and often, the company uses that page to draw attention to historic events, celebrations, or current events such as "coronavirus helpers" and more using Doodles. The colorful drawings are changed on a regular basis. About the Author Kyle Bradshaw @SkylledDev 9to5’s resident Fuchsia fanatic. Got a tip or want to chat? Twitter or Email. Kyle@9to5mac.com Kyle Bradshaw's favorite gear Best Chromebooks Best Chromebooks Nest Wifi review Nest Wifi review Powered by WordPress.com VIP MODE BEAUTÉ CUISINE DÉCO JARDIN COUPLE MARIAGE MAMAN SANTÉ SOCIÉTÉ STARS CULTURE HOROSCOPE Le Journal des femmes SANTÉ CORONAVIRUS ACTUS SANTÉ MALADIES EXAMENS MÉDICAMENTS QUOTIDIEN SEXO-GYNÉCO FORUM FichesFiches maladiesCoronavirusGestion de l'épidémie Masque coronavirus : la vente en pharmacie autorisée dès ce lundi Article mis à jour le 27/04/20 11:13 Partager sur MASQUE EN TISSU - En pleine épidémie de coronavirus, mettre un masque de protection est recommandé. Selon un arrêté publié ce weekend au Journal Officiel, la vente de masques grand public, en tissu lavable homologué et réutilisable est autorisée en pharmacie. Leur prix : moins de 5 euros l'unité. Le Conseil Scientifique estime que le port du masque doit être obligatoire dans les lieux publics, les collèges et les lycées après le déconfinement. Masque coronavirus : la vente en pharmacie autorisée dès ce lundi© phoenix021 - Adobe Stock SOMMAIRE Définition Masque grand public Où s'en procurer ? Masque pour enfant Masque lavable Comment le mettre ? Fabriquer son masque maison Où est-il obligatoire ? Quelle efficacité ? Lequel choisir ? • Chirurgical, canard • Respiratoire FFP2, FFP3 Jetable ou réutilisable ? Où l'acheter ? Prix NOS INFOS ACTU & CORONAVIRUS voir un exemple Entrez votre email [Mis à jour le lundi 27 avril à 11h13] Chirurgical, en tissu homologué, à "bec de canard", 3 plis, bleu ou blanc... Mettre un masque de protection permet de limiter la propagation de l'épidémie de Covid-19. D'abord réservé en priorité aux soignants, il est désormais recommandé pour le grand public. Son port pourrait même devenir obligatoire dans les transports en commun d'ici le 11 mai. Dans ses recommandations du samedi 25 avril, le Conseil Scientifique préconise que le port du masque doit être "systématique" dans les lieux publics pendant les mois suivant la levée du confinement. Cela nécessite donc que des "masques alternatifs de production industrielle ou artisanale anti-projection" soient "disponibles pour l'ensemble de la population et distribués en priorité aux personnes en contact régulier avec le public". Par ailleurs, un arrêté publié ce weekend au Journal Officiel autorise la vente de masques grand public dans les pharmacies dès ce lundi 27 avril. Leur prix ne doit pas dépasser 5 euros l'unité et il ne faudra désormais plus d'ordonnance pour s'en procurer. La distribution auprès des Français devrait avoir lieu à partir du 4 mai "par les canaux de distribution les mieux adaptés", a annoncé Agnès Pannier-Runacher, la secrétaire d'Etat à l'économie dans un article des Echos ce vendredi 24 avril. Mais pour cela, encore faut-il avoir des stocks suffisants... A date, selon la Direction générale de Santé publique France : 2.25 milliards de masques ont été commandés en France (25 contrats ont été signés) : "ce sont huit millions de masques qui ont été produits la semaine écoulée dans notre pays et on prévoit une production d'environ 17 millions de masques par semaine en France", a tenu à préciser le ministre de la Santé, Olivier Véran. Anne Hidalgo, maire de Paris, promet la distribution de 500 000 masques fin avril. 5 millions de masques seront débloqués dans la semaine (du 20 avril), qui seront destinés à "un certain nombre de professionnels de santé, les ambulanciers, les préparateurs en pharmacie, les sages-femmes, les techniciens de laboratoire, les aides à domicile, les manipulateurs radio...", ajoute le ministre de la Santé. Et de préciser qu'il sera ensuite possible pour "les malades" et "les personnes fragiles de solliciter des masques", toutefois "ça va prendre du temps". 1.5 milliard de masques doivent être livrés pour le 31 mai. Dès leur réception, ces masques permettront de combler la pénurie et les tensions d'approvisionnement auxquelles fait la France fait face depuis plusieurs mois. Chirurgical, médical, FFP2, 3 plis, en tissu, avec des fils de cuivre... Quel modèle porter ? Comment bien le mettre et quand le jeter ? Peut-on le fabriquer soi-même ? Avec quel tissu ? Quel tuto pour rapidement et facilement le confectionner ? Conseils. Définition : qu'est-ce qu'un masque de protection ? Le masque de protection, aussi appelé "anti-virus", à usage médical, "chirurgical", anatomique, type "bec de canard"... est un dispositif médical destiné à filtrer les bactéries et à éviter de contracter un virus, comme celui de la grippe ou toute autre maladie virale. "Ces masques sont généralement portés au bloc opératoire pour éviter d'abord que les bactéries de la bouche du chirurgien ne soient projetées sur plaie chirurgicale du patient, mais peuvent aussi être portés par le grand public pour se protéger des micro-organismes dans un contexte d'épidémie, comme la grippe ou le Covid-19 par exemple", indique le Dr Pierre Parneix, médecin hygiéniste et praticien hospitalier en Santé publique au CHU de Bordeaux et responsable du Centre d'appui à la prévention des infections associées au soins (CPIAS) de Nouvelle Aquitaine. Masque grand public : c'est quoi, normes, tissu homologué, efficacité... Promis par Emmanuel Macron lors de son allocution du 13 avril, le masque "grand public" ou "alternatif" sera un complément aux gestes barrières et chaque Français pourra s'en procurer un d'ici le 11 mai. Ce masque grand public ou alternatif est en tissu lavable et donc réutilisable plusieurs fois. Il n'a pas le même degré de protection que le masque chirurgical ou médical, que l'on trouve en pharmacie. Et encore moins efficace que le masque respiratoire de type FFP2 ou FFP3 utilisé par le personnel médical. Il répond toutefois à des normes. Il peut être fabriqué de manière artisanale (fait-main à partir de tutoriels ou modèles de patron disponibles sur Internet, diffusé par l'Afnor ou le CHU de Grenoble par exemple), à trois plis ou à "bec de canard", mais aussi de manière industrielle. Les autorités sanitaires françaises ont d'ailleurs validé plus de 80 prototypes à près de 50 entreprises issues de la mode et du textile. L'objectif étant d'en confectionner 17 millions par semaine d'ici le 11 mai. "Il est probable, s'agissant des transports publics, qu'il soit obligatoire pour les utiliser de porter un masque grand public", ont ré-insisté Edouard Philippe et Olivier Véran lors de leur conférence sur les grands principes du plan de sortie du déconfinement. Où s'en procurer dans chaque ville ? Même si les modalités de distribution n'ont pas clairement été évoquées (sera-t-il distribué gratuitement ou sera-t-il payant ?), le ministre de la Santé envisagerait "un amorçage de l'Etat" et un travail "avec les maires et pourquoi pas avec la grande distribution". Autrement dit, ces masques seraient "probablement disponibles dans les mairies des villes ou les collectivités locales", a évoqué Olivier Véran, toujours sur RTL. Mais ce n'est pas tout "plusieurs modes de distribution ont été identifiés pour permettre à un maximum de Français d'y avoir accès. (...) Nous regardons toutes les hypothèses : pharmaciens, mairies, grande distribution, buralistes, plateforme Afnor, e-commerce...", indique aux Echos Agnès Pannier-Runacher la secrétaire d'État à l'Economie, le 24 avril. Plusieurs villes commencent à distribuer des masques "grand public" : A Nice, les premiers masques sont distribués aux Niçois à partir de vendredi 24 avril, a annoncé le maire de la ville, Christian Estrosi. A Toulouse, les 37 communes de la métropole se sont associées pour que chaque habitant puisse bénéficier d'un masque en tissu d'ici le déconfinement. La distribution sera assurée par chaque mairie. 800 000 masques ont d'ores et déjà été commandés par Toulouse Métropole. A Lyon, 2 millions de masques vont être distribués en mai aux Lyonnais. Ces masques seront en tissu, lavables et donc réutilisables. A Deauville, les premiers 5 000 masques ont été distribués depuis le samedi 18 avril à la salle des fêtes de la ville. La distribution se fait les mardis et les vendredi de 9h à 16h ainsi que les jours de marché. A Paris, deux millions de masques seront prochainement distribués aux Parisiens, a annoncé Anne Hidalgo, maire de Paris le 7 avril. La Présidente d'Île-de-France Valérie Pécresse a quant à elle promis, sur BFMTV, de distribuer gratuitement des masques aux voyageurs das les transports en commun "pendant les premières semaines du déconfinement" A Reims, 300 000 masques grands publics sont attendus afin d'être distribués à chaque habitant de Reims et du Grand Reims à la sortie du confinement. A Nantes, 600 000 masques "certifiés selon les recommandations de l'AFNOR" vont être distribués à chaque habitant d'ici 3 semaines. Chaque masque coûtera entre 3.5 et 4 euros à la commune, mais sera totalement gratuit pour les Nantais, assure Johanna Rolland, la maire de Nantes. A Valence, un masque en tissu sera distribué gratuitement à chaque habitant de la ville à partir du 26 avril. A Joinville, 11 000 masques seront distribués dans toutes les boîtes aux lettres de la ville à partir du lundi 27 avril. Dans la communauté de communes de Lacq-Orthez (Pyrénées-Atlantiques), 55 000 masques ont été commandés et vont être distribués à chacun des habitants des 61 communes concernés à partir du 11 mai. Les enfants devront-ils porter des masques à l'école ? Un retour progressif à l'école est prévu à partir du 11 mai et sur la base du volontariat des parents des élèves. A l'heure actuelle, il n'y aucune recommandation scientifique pour faire porter des masques aux enfants, a assuré Olivier Véran, le ministre de la Santé le vendredi 24 avril sur France Inter. "Je considère que c'est très compliqué de demander à un enfant de 10 ou 11 ans, qui est à l'école primaire, d'aller porter un masque toute la journée, à l'intérieur, en extérieur, de jouer avec dans la cour de récré quand on va commencer à dépasser les 30 °C" a-t-il précisé. Il a rappelé toutefois l'importance pour les enfants de respecter les gestes barrières : le lavage des mains, le fait de tousser dans son coude, de jeter ses mouchoirs après usage, s'éloigner des autres élèves quand cela est possible... Un stock de masques chirurgicaux est néanmoins prévu pour les enfants malades, fragiles qui en auraient besoin et "quand cela se justifie". Selon le Conseil Scientifique qui a livré ses recommandations samedi 25 avril, le port systématique d'un masque dans les semaines suivant la levée du confinement serait "impossible" en maternelle, "adaptable en fonction de l'âge" en primaire et "obligatoire" pour les adolescents, des collèges et des lycées. Des masques lavables aux normes AFNOR : quel tissu utiliser ? L'Association française de normalisation (AFNOR) met à disposition des particuliers et des industriels un tutoriel de fabrication de masques barrières (non médicaux) téléchargeable. Ce dispositif, conçu pour faciliter et accélérer la fabrication en série ou artisanale d'un nouveau modèle de masque, n'est pas destiné à être utilisé par les personnels soignants au contact des patients. "Ce masque n'exonère à aucun moment des gestes barrières. C'est une protection supplémentaire pour les personnes saines lorsqu'elles se déplacent ou travaillent", résume Rim Chaouy, responsable de pôle santé et sécurité au travail d'AFNOR et pilote du projet.Ces masques à fabriquer soi-même doivent être lavés à 60 degrés, en machine sur un cycle "coton", après chaque utilisation. Ces masques, validés par la Société Française d'hygiène hospitalière, peuvent être fabriqués avec : deux couches de toile de coton (ou un tissu non-tissé en polypropylène), enserrant une couche de viscose (soie artificielle), de polyester ou de polyamide pour les masques classiques deux couches de popeline de coton pour les masques 3 plis. La liste des matériaux testés, validés et utilisables va continuer à être complétée au gré des résultats des essais menés en laboratoire. L'AFNOR doit réaliser cette semaine des tests pour savoir quels tissus utilisés et définir s'il est possible de confectionner un masque à partir d'un tissu non homologué (vieux draps, tissu de récupération...) "Nous avons beaucoup de contacts tous azimuts avec les pharmacies, La Poste, la grande distribution, pour des mises à disposition. On cherche des points de relais et on travaille également sur les modalités de vente à prix coûtant ou de gratuité. [...] Mais il faut s'armer de patience, la distribution n'est pas encore réalisée. On en est toujours à la phase de confection", explique un porte-parole de l'Association à Libération. Comment bien mettre un masque : côte bleu ou blanc ? Le masque de protection n'est efficace que s'il est bien porté. Les conseils du Dr Parneix pour le positionner correctement : Porter le masque dans le bon sens. En général, la face colorée doit se trouver vers l'extérieur et non contre la bouche. Le côté blanc est la face absorbante : elle doit être appliquée contre la bouche. Le côté rembourré de la barrette nasale doit être placé sur la bosse du nez pour bien protéger le nez. Préférer les masques avec des attaches élastiques plutôt que des lanières : le masque sera plus simple à enfiler. Le masque doit être bien enveloppant et passer sous le menton pour bien protéger la bouche et le nez. comment mettre un masque de protectionEtapes pour bien mettre un masque de protection © AFNOR Masque en cuivre, une bonne idée ? Face à la pénurie de masques de protection dans le monde, le Chili estime avoir trouvé une alternative. Deux entreprises chiliennes ont mis au point un modèle de masque en polymère dans lequel ont été ajoutées des nanoparticules de cuivre, un métal antiseptique et bactéricide capable de détruire très efficacement les acides nucléiques de l'ADN d'un virus, comme le coronavirus. Ces masques en fils de cuivre certifié sont lavables et donc réutilisables. Premier producteur de cuivre au monde, le Chili a effectivement rapidement fait certifier les propriétés antiseptiques du cuivre. Plusieurs études scientifiques avaient déjà montré que le coronavirus responsable de l'épidémie Covid-19 survivait très peu de temps sur le cuivre, contrairement sur les autres surfaces comme le plastique, le carton ou l'inox. Le cuivre est d'ailleurs un matériau déjà très utilisé dans les unités intensifs des hôpitaux (les surfaces en cuivre permettraient de réduire de 40% le risque d'infection). Fabriquer son masque de protection en tissu : efficace ? Certains experts estiment que porter un masque artisanal est "mieux que rien". Les masques normés étant désormais réservés aux soignants et aux malades (suspectés ou testés), de nombreux tutoriels pour fabriquer son masque à la maison fleurissent sur Internet. "Pour être efficace, le masque doit être conforme à la norme française et européenne "NF EN 14683" rappelle le Dr Parneix. Alors masque fait-maison ou pas ? Les experts sont partagés. Certains affirment que les masques fait-maison ne sont absolument pas efficaces, "faussement rassurants" et qu'ils ne protègent ni celui qui le porte ni son entourage. La Société française d'hygiène hospitalière confirme, dans un avis du 14 mars, qu'il faut "éviter d'utiliser d'autres types d'écrans à la place des masques chirurgicaux (masques en tissu, masques en papier, chiffons noués derrière la tête type bandana) du fait de données scientifiques concernant leur efficacité (étanchéité) très rares". D'autres estiment que porter un masque artisanal est "mieux que rien".Pour trancher, Olivier Véran, ministre de la Santé a "demandé aux autorités compétentes de nous rendre des avis sur la question". Et de mettre en garde sur l'utilisation de ces protections homemade : elles ne sont, selon lui "pas l'alpha et l'omega de la protection". Quid des écharpes et des bandanas qu'on enroule autour de son visage ? On se croit protégé alors que non : ils filtrent certes les grosses poussières et le pollen par exemple, mais pas les virus qui sont des micro-organismes microscopiques (leur taille varie de 0.02 µm à 0.3 µm). Ils ne protègent donc pas du Covid-19. Coudre son masque en tissu : patron AFNOR et tutos faciles Si la vente de masques se profile, l'heure est encore au do it yourself. Pour fabriquer son masque de protection efficace contre le coronavirus à la maison, en tissu ou en papier, nous avons réunis les meilleurs tutos. Masque obligatoire pour sortir ? Pour le moment, le port du masque n'est pas obligatoire dans l'espace public en France. En France, pour le moment, le port du masque n'est pas obligatoire dans l'espace public. A partir du 11 mai, le port du masque sera "très probablement obligatoire dans certains lieux publics et dans les transports en commun", a indiqué Edouard Philippe lors d'une conférence de presse du dimanche 19 avril. En attendant cette éventuelle obligation, porter un masque dans la rue, dans les transports en commun ou dans n'importe quel lieu public est recommandé par l'Académie de Médecine. Des règles concernant une potentielle obligation de porter un masque et les modalités du réseau de distribution des masques "grand public" devront être établies et dévoilées d'ici deux semaines, a informé Olivier Véran sur RTL le 14 avril. Par ailleurs, le ministère de la Transition écologique et solidaire, en charge des Transports, est en train de discuter sur d'éventuelles règles mises en place dans les transports en commun pour le personnel et les passagers. Au Luxembourg, porter un masque, un foulard ou une écharpe autour du visage est obligatoire dans l'espace public depuis le lundi 20 avril, a indiqué son Premier ministre Xavier Bettel. Près de 7 millions de masques vont être distribués aux citoyens du pays et ceux qui y travaillent comme les travailleurs frontaliers. En Italie, le port du masque, ou de "tout instrument (écharpes, foulards) qui permettent de couvrir le nez et la bouche" est obligatoire dans l'espace public en Lombardie et en Toscane, depuis le 5 avril. En Espagne, pays d'Europe le plus touché par l'épidémie, distribue des masques à l'entrée des stations de métro. Le port du masque n'est toutefois pas obligatoire pour sortir dans la rue. En République Tchèque, en Slovaque, en Solvénie : le port du masque est obligatoire dans tous les espaces publics. En Autriche, le port du masque est obligatoire dans tous les supermarchés à partir du 1er avril. Cette obligation sera certainement étendue dans tous les espaces publics. En Bulgarie, porter un masque est "très fortement" recommandé et pourrait devenir obligatoire pour tous les passants. Aux Etats-Unis, qui ont dépassé la barre des 200 000 contaminations, ils envisagent également de faire porter des masques à tous les Américains une fois que les stocks seront à nouveau disponibles. Au Sénégal, le port du masque est obligatoire depuis dimanche 19 avril dans "les services de l'Administration, du secteur privé, de commerce et de transport". Une mesure prévue pour durer pendant tout l'état d'urgence, soit au moins jusqu'au début du mois de mai. "Tout manquement aux présentes dispositions sera puni par les peines prévues par les lois et règlements", apprend-on dans un arrêté du ministère de l'Intérieur. A quoi sert vraiment le masque ? Le port de masques médicaux n'est pas une pratique culturelle en France à l'inverse d'autres pays comme en Asie. "Au lieu de remercier les gens qui portent des masques car elles nous protègent de leurs virus, on a plutôt tendance à les juger comme "dangereuses". C'est un regard qu'il faut vraiment changer ! Porter un masque, est un très bon réflexe", indique le Dr Pierre Parneix. Une personne suspectée de présenter des symptômes d'infection respiratoire ou avérée malade peut porter un masque chirurgical pour protéger les autres de ses symptômes (à la maison, dans les lieux publics...). Le masque peut également servir dans un contexte de prévention de l'exposition au virus. Il a un "effet barrière" qui va empêcher le passage des particules bactériennes et virales. "Le mode de transmission du coronavirus est sensiblement le même que celui de la grippe, c'est-à-dire qu'il se transmet d'homme à homme lors de contacts rapprochés (se toucher ou se serrer la main par exemple) et par voie aérienne en toussant ou en éternuant (gouttelettes de salive, postillons)" précise le médecin hygiéniste." Les masques ont donc un intérêt pour s'en protéger mais "n'ont aucun sens si on ne se lave pas les mains puisqu'en pratique on touche des surfaces contaminées" a rappelé le Pr Jérôme Salomon le 26 février 2020 au Sénat. Pour tous les autres : inutile de porter un masque de protection. Le meilleur moyen de se prémunir contre le coronavirus est de respecter les gestes barrières : se laver les mains régulièrement et particulièrement après chaque sortie, utiliser du gel antibactérien s'il n'y a pas de point d'eau à proximité, éternuer et tousser dans son coude, se moucher avec un mouchoir à usage unique que l'on jette immédiatement, maintenir une distance de sécurité avec ses interlocuteurs d'au moins un mètre... masque de protection pour quiMasques : à qui sont-ils réservés ? © Ministère de la Santé Masque chirurgical, FFP2, FFP3 : quelles différences et lequel choisir ? Il existe plusieurs types de masque qui ont des niveaux de filtration variables. "Pour définir le niveau de filtration, on réalise un test avec un aérosol contenant des souches de staphylocoque doré", explique le Dr Parneix. On distingue : Les masques chirurgicaux (bleu) de type I qui filtre 95% des bactéries et de type II qui filtre plus de 98% des bactéries. On distingue les masques de type 2 normaux et ceux de type R qui sont plus étanches et résistants aux projections. Dans le cas de l'épidémie de coronavirus, ce sont des masques chirurgicaux qui sont réservés aux malades ou aux personnes en contact avéré avec un malade. Le masque de protection respiratoire (masque coque, canard, masque FFP2 ou FFP3) est un masque de sécurité avec un très haut niveau de filtration qui est utilisé en milieu de soins pour des agents infectieux comme celui de la tuberculose par exemple. Ce masque, qui pour être efficace doit avoir une forme adaptée au visage de la personne qui le porte, ne filtre plus uniquement des bactéries, mais également des aérosols. Il est réservé aux professionnels de la santé de toute la chaîne (transport sanitaire, pompiers, médecins libéraux, infirmiers...) et n'est pas à destination du grand public. "Cela n'a pas de sens d'en porter quand on est un citoyen, pour aller faire ses courses par exemple, souligne le Pr Salomon. Ce sont les gestes à risque qui contaminent du coronavirus : quand on intube un patient, quand on l'examine…" différences masques de protection coronavirusLes différents masques de protection © Olena - Adobe Stock Le masque est-il jetable ou réutilisable ? S'il n'est pas en tissu, le masque est jetable. Le masque chirurgical anti-projection (vendu en pharmacie ou sur internet) a une durée de vie de 3 à 4 heures. "Au-delà, il faut le jeter à la poubelle. Le masque est à usage unique et en aucun cas lavable ou réutilisable. Après chaque sortie, dès le retour à la maison, il ne faut pas le manipuler et le jeter car il est potentiellement contaminé", prévient le Dr Parneix. Il ne faut donc pas réutiliser un masque dès lors qu'il a été manipulé et ôté du visage. Le masque de type FFP a quant à lui une durée de 8 heures. Masque périmé : encore efficace ? Soutenu par la direction générale de la Santé, le Ministère du Travail a autorisé, jeudi 26 mars 2020, l'usage de masque de protection (de type FFP2) périmé, à condition que la date de péremption ne dépasse pas 6 mois, et que "des consignes strictes sont respectées avant leur utilisation". Ces masques périmés doivent : "Avoir été stockés dans les conditions de conservation conformes à celles prévues par le fabricant ou le distributeur". "Avoir fait l'objet de quatre tests successifs" avant leur utilisation. Ces tests ont pour but de vérifier l'intégrité des conditionnements, l'apparence (la couleur d'origine), la solidité des élastiques et de la barrette nasale de maintien du masque, et l'ajustement du masque sur le visage. Le masque chirurgical ou anatomique a une durée de vie de 3 heures. On le jette après. Achat : pharmacie, internet, sur ordonnance ? Il vaut mieux acheter un masque chirurgical en parapharmacie ou en pharmacie que sur Internet pour deux raisons : avoir la garantie que le masque soit bien conforme à la norme NF EN 14683 et pouvoir bénéficier des conseils du pharmacien, notamment sur le port du masque. "Évitez de vous procurer des masques chirurgicaux sur Internet sans la mention NF EN 14683 car on n'est jamais sûr de leur efficacité", conclut l'expert. De plus, en cas d'épidémie, certains vendeurs en profitent pour gonfler les prix des masques sans en garantir la qualité. Il faut donc rester vigilant et privilégier le circuit pharmaceutique. Selon des informations divulguées par le Parisien le 23 avril, des masques pourront être vendus dans les bureaux de tabac à partir du 30 avril au prix indicatif de 5 euros. Chaque buraliste devra passer sa propre commande de masques sur une plateforme conçue par la Confédération des buralistes. A partir du 27 avril, les officines sont autorisées à vendre des masques grand public à leurs clients - sans ordonnance médicale - à un prix ne dépassant pas 5 euros l'unité, précise un arrêté publié au Journal Officiel du 25 avril. Prix d'un masque de protection Un masque chirurgical coûte en moyenne entre 20 et 50 centimes l'unité (hors pandémie de coronavirus) Un masque respiratoire de type FFP2 peut coûter jusqu'à 20 euros. Un masque respiratoire de type FFP3 peut coûter jusqu'à plusieurs de dizaines d'euros. Mise en garde : les prix n'étant pas réglementés se sont envolés sur Internet et dans certaines pharmacies. NEWSLETTER Entrez votre email Voir un exemple Merci au Dr Pierre Parneix, médecin hygiéniste et praticien hospitalier en Santé publique au CHU de Bordeaux et responsable du Centre d'appui à la prévention des infections associées au soins (CPIAS) de Nouvelle Aquitaine. Anaïs Thiébaux Mis à jour le 27/04/20 11:13 Partager sur GESTION DE L'ÉPIDÉMIE Masque coronavirus > Guide Gestes barrières anti coronavirus : "pour longtemps en France" > Guide Plaisir, hygiène, taille, profondeur : tout savoir sur le vagin > Guide Test coronavirus : sérologique, PCR, salivaire, rapide... Stop Covid : c'est quoi cette application, quelles sont ses limites ? Quels seront les effets psychologiques du déconfinement ? Quels sont les effets psychologiques du confinement et de l'isolement ? Prime aux soignants du coronavirus : quelles conditions, quand ? Coronavirus à l'hôpital : bilan, courbe des cas, droit de visites Covid-19 et isolement à domicile : les consignes jusqu'à la guérison R0 et coronavirus : moins de nouvelles personnes infectées en France Epidémie de coronavirus au stade 3 : conséquences, combien de temps ? Confinement et sommeil agité : une technique d'hypnose à essayer Coronavirus et pilule : comment faire pendant le confinement ? 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Pandémie : définition, différence avec une épidemie Immunité collective et coronavirus : définition, efficacité, létalité Les 6 conséquences du confinement sur la santé Plateau de l'épidémie de coronavirus : est-il atteint en France ? Réanimation : c'est quoi, quels soins, quelle chance de survie ? Curare : origine, mode d'action, quelle utilisation en réanimation ? Anxiété, stress : les impacts psychologiques de l'épidémie de coronavirus Enterrement des défunts du coronavirus : les conditions funéraires Plan blanc : c'est quoi, dans quel hôpital ? Urgence de santé mondiale : définition, conséquences, dans quels cas ? 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Son prototype est réutilisable à l'infini et jugé cinq fois plus rentable que les fameux FFP2. 25 avr. 20:15 - Pour l'instant, seuls les masques en tissu sont disponibles pour le grand public dans l'Hexagone. Mais des entreprises travaillent également sur des prototypes beaucoup plus performants. C'est notamment le cas du groupe Michelin, plutôt spécialisé dans les pneus. Son modèle 100% Français est composé d'une partie masque qui va servir à l'infini et une partie filtre, utilisable et lavable 20 fois. Ce prototype qui coûtera 28 euros s'est déjà prévendu à 350 000 exemplaires. TOUTE L'INFO SUR LE WE 20H LIRE AUSSI CORONAVIRUS - Suivez les dernières actus en direct sur l'épidémie Déconfinement : un plan, six chapitres Coronavirus : un respirateur "made in France" produit à un coût défiant toute concurrence Ce sujet a été diffusé dans le journal télévisé de 20H du 25/04/2020 présenté par Anne-Claire Coudray sur TF1. Vous retrouverez au programme du JT de 20H du 25 avril 2020 des reportages sur l'actualité politique économique, internationale et culturelle, des analyses et rebonds sur les principaux thèmes du jour, des sujets en régions ainsi que des enquêtes sur les sujets qui concernent le quotidien des Français. Mis à jour : Hier à 18:02Créé : samedi dernier à 20:15 CONTENUS SPONSORISÉS Jusqu'où peut-on faire des courses sans risquer une amende ? Jusqu'où peut-on faire des courses sans risquer une amende ? Sponsorisé par Les top vidéos du moment Douleurs cervicales ? Les habitants de Taverny ne peuvent plus se passer de cet appareil Douleurs cervicales ? Les habitants de Taverny ne peuvent plus se passer de cet appareil Sponsorisé par NeckRelax Avec l’arrivée de cette version de Windows 10, Microsoft mise tout sur les PC portables Avec l’arrivée de cette version de Windows 10, Microsoft mise tout sur les PC portables Sponsorisé par 01net A VOIR AUSSI Droguée, séquestrée et violée : la chanteuse Duffy raconte sa disparition médiatique Droguée, séquestrée et violée : la chanteuse Duffy raconte sa disparition médiatique Droguée, séquestrée et violée : la chanteuse Duffy raconte sa disparition médiatique "Beaucoup de gens vont sortir fracassés de cette crise", alerte le Secours catholique "Beaucoup de gens vont sortir fracassés de cette crise", alerte le Secours catholique "Beaucoup de gens vont sortir fracassés de cette crise", alerte le Secours catholique Patrick Bruel et CharlElie Couture se remettent du coronavirus Patrick Bruel et CharlElie Couture se remettent du coronavirus Patrick Bruel et CharlElie Couture se remettent du coronavirus À CONSULTER AUSSI Wyritol lingettes Wyritol lingettes Sponsorisé par JPG Le mystère de l'orgasme féminin enfin résolu...grâce aux lapins Le mystère de l'orgasme féminin enfin résolu...grâce aux lapins Le mystère de l'orgasme féminin enfin résolu...grâce aux lapins Nettoyeur haute pression 19,99€ - Nettoyez voiture, maison, trottoirs, mur, toiture,... Nettoyeur haute pression 19,99€ - Nettoyez voiture, maison, trottoirs, mur, toiture,... nettoyeur-haute-pression SUR LE MÊME SUJET LE WE 20H PARLONS… HIGH TECH 03:08 Fermeture des marchés : les producteurs s'organisent 01:20 Les titres du JT de 20H 01:15 Attaque au couteau à Romans-sur-Isère : que sait-on de l'assaillant ? 03:20 Coronavirus : une très lente réadaptation du patient après la réanimation 02:21 Les passagers du Zaandam enfin à quai 02:18 Pénurie de masques : les masques "alternatifs" et les écharpes sont-ils efficaces pour se protéger ? ET AUSSI Les titres du JT de 13H Les titres du JT de 13H Contrôles de police : le point sur la situation à la porte d'Orléans Les prévisions météo du JT de 13 heures du 31 mars 2020 L'édito Aphatie : "Cette mission d'information ressemble beaucoup à une commission d'enquête" Fraude au chômage partiel : le gouvernement renforce les sanctions LIRE ET COMMENTER ACCUEIL>HIGH-TECH FIL INFO EN DIRECT LA PANDÉMIE DE COVID-19 12:26 A la veille des annonces d'Edouard Philippe, La France Insoumise propose son propre plan de déconfinement 12:22 VIDÉO - Première apparition en public de Boris Johnson depuis sa guérison du Covid-19 11:40 Présentation du plan de déconfinement : l'opposition critique la méthode du gouvernement 10:29 EN DIRECT - Coronavirus : +7,1% de chômeurs au mois de mars, du jamais-vu 10:17 Le PS pleure Henri Weber, figure de Mai-68, mort du coronavirus TOUT LE FIL INFO A VOIR, A LIRE Secteur automobile : des voitures à prix cassés pour éviter l'effondrement des ventes ?Secteur automobile : des voitures à prix cassés pour éviter l'effondrement des ventes ? Secteur automobile : des voitures à prix cassés pour éviter l'effondrement des ventes ? Ce rituel simple aide à brûler la graisse du ventre et des cuisses (faites ceci tous les jours) Ce rituel simple aide à brûler la graisse du ventre et des cuisses (faites ceci tous les jours) Brûleur Oméga Comment protéger son épargne en cas de crise économique ou sanitaire ? Comment protéger son épargne en cas de crise économique ou sanitaire ? capital-placement.info Coronavirus : 8500 respirateurs produits pour rien ? Le gouvernement se défend02:53Coronavirus : 8500 respirateurs produits pour rien ? Le gouvernement se défend Coronavirus : 8500 respirateurs produits pour rien ? Le gouvernement se défend Coronavirus : dans la peau du Dr Anthony Fauci, Brad Pitt s'en prend avec humour à Donald Trump02:07Coronavirus : dans la peau du Dr Anthony Fauci, Brad Pitt s'en prend avec humour à Donald Trump Coronavirus : dans la peau du Dr Anthony Fauci, Brad Pitt s'en prend avec humour à Donald Trump VIDÉO - Inès, l’infirmière de "Koh-Lanta" : "Travailler sans matériel au début de la pandémie, ça m'a mise hors de moi"03:12VIDÉO - Inès, l’infirmière de "Koh-Lanta" : "Travailler sans matériel au début de la pandémie, ça m'a mise hors de moi" VIDÉO - Inès, l’infirmière de "Koh-Lanta" : "Travailler sans matériel au début de la pandémie, ça m'a mise hors de moi" LES ACTUS DU MOMENT DÉCONFINEMENT : LE CASSE-TÊTE DE L'EXÉCUTIF NOS CONSEILS FACE AU CONFINEMENT CORONAVIRUS : L'IMPACT ÉCONOMIQUE DE LA PANDÉMIE CORONAVIRUS : LA PANDÉMIE QUI BOULEVERSE LA PLANÈTE LES SOLDES D'ÉTÉ 2020 TOUS LES SUJETS NEWSROOM International Planète Météo Politique Police Justice Terrorisme Football Autres sports Insolite Résultats Élections Municipales 2020 SOCIÉTÉ Social Population Education Santé DAILY Famille Vie de couple Psycho Loisirs MA SANTÉ Bien-être Nutrition Beauté CULTURE People Médias Sorties ÉVASION Régions Voyages SOS Villages CONSO Mon argent Immobilier Automobile Shopping WORK Open-space Emploi INNO Sciences Green High-tech PODCAST Happy Work Futur & Co Impact Positif Les gens qui lisent sont plus heureux MYTF1 TF1 TMC TFX TF1 SÉRIES-FILMS Actualité en continu SOS VILLAGES Direct SUIVEZ-NOUS PARTENAIRE DE LCI Copyright © 2020 Mentions légalesConditions générales d'utilisationPolitique de protection des données du service LCICookiesEspace PresseAccessibilitéLa médiatrice vous répondNewsletter et alertes Acces direct au contenuAcces direct au menu Toms Guide : actualités high-tech et logiciels ACTU GUIDE D’ACHAT TEST DOSSIER TUTO FORUM TÉLÉCHARGER Accéder à mon compte Rechercher THÈMES : Smartphones PC portables Apps gratuites Jeux vidéo Windows Accès Internet Cinema Bons plans Voir plus Accueil » Actualité » Tesla : l’Autopilot se met à jour pour reconnaître et réagir aux feux tricolores et aux panneaux Stop Tesla : l’Autopilot se met à jour pour reconnaître et réagir aux feux tricolores et aux panneaux Stop par BRUNO CLAIRET, le 26 avril 2020 15:27 Facebook Twitter Partager par emailImprimer Tesla met à jour son système de conduite automatisée. L’Autopilot détecte désormais les feux tricolores et les panneaux Stop pour adapter la vitesse du véhicule. La fonctionnalité apporte un niveau de sécurité supplémentaire, mais reste en bêta. Elle commet encore trop d’erreurs. Tesla vient de déployer une mise à jour de son assistant de conduite. Avec le patch 2020.12.6, l’Autopilot est maintenant capable de reconnaître les feux de signalisation et les panneaux Stop. Une fonctionnalité qui marque une étape supplémentaire vers la conduite entièrement autonome qu’Elon Musk avait déjà annoncée pour l’année dernière. En accès anticipé depuis deux mois, le patch est désormais disponible pour tous les clients américains. Une mise à jour est prévue pour les autres pays, mais on ignore encore le planning exact de Tesla. La firme doit en effet l’adapter à leurs signalétiques spécifiques avant d’envisager une distribution plus large cet été. Image 1 : Tesla : l’Autopilot se met à jour pour reconnaître et réagir aux feux tricolores et aux panneaux Stop Crédit : Dirty Tesla La note de version précise que la mise à jour permet d’identifier les feux tricolores et les panneaux Stop pour induire de nouveaux comportements aux intersections. Lorsqu’un feu tricolore est vert, orange, clignotant ou éteint, le véhicule ralentit automatiquement. S’il repère un feu rouge ou un panneau Stop, il ralentit et s’arrête. Une pression sur le boîtier de vitesse ou l’accélérateur permet d’indiquer que le véhicule peut poursuivre sa route sans danger. Tesla rappelle que la fonctionnalité est encore en bêta, et qu’il arrive parfois qu’elle n’interprète pas correctement une situation. Il convient donc de rester vigilant, les premiers retours des possesseurs de Tesla le confirment. L’Autopilot progresse, mais commet encore trop d’erreurs pour être autonome Dans sa première version, la fonctionnalité provoquait l’arrêt des Tesla même au feu vert. La mise à jour est beaucoup plus fiable, et les premiers essais montrent qu’elle apporte un niveau de sécurité supplémentaire. Elle reste néanmoins en version bêta. La chaîne YouTube Dirty Tesla a mis en ligne une série de vidéos qui illustre le comportement de l’Autopilot dans différentes situations. Le système réagit plutôt bien, mais il commet encore des erreurs. Tesla va bien sûr l’améliorer, mais la route semble encore longue avant d’obtenir un système de conduite entièrement autonome. Comme à son habitude, Elon Musk fait preuve d’un optimisme à toute épreuve, il vient en effet de confirmer que ses taxis autonomes seront prêts pour être déployés cette année. Votre Tesla se garera automatiquement avant fin 2020, c’est le souhait d’Elon Musk Source : Teslarati Thèmes : tesla, Voiture Electrique La maison de Christian Clavier choque le monde entier, la preuve en image ! PsychicMonday | Sponsorisé La femme de Marc Lavoine choque le monde entier, la preuve en image ! LawyersFavorite | Sponsorisé La gélule brûle-graisse dont tout le monde parle Brulafine.com | Sponsorisé Michel Drucker: sa maison va vous piquer les yeux Boite a Scoop | Sponsorisé Le fils de Sophie Marceau nous fait halluciner aujourd'hui ! Life Indigo | Sponsorisé 15 utilisations du dentrifice que vous ne connaissez pas ! Que Des Astuces | Sponsorisé Conservée dans une bulle depuis 1997, une BMW 740i neuve est mise aux enchères Tom's guide Tesla : l’Autopilot détecte un piéton sur l’autoroute et écrase le frein Tom's guide Réagissez avatar Démarrer la discussion... Suivre la discussion Article Liés 25/04 Kia accélère l’électrification de sa gamme avec un crossover inédit prévu pour fin 2021 24/04 Un moteur Tesla se retrouve dans un Land Rover Defender 24/04 Ferrari doit présenter deux modèles cette année, et pourquoi pas une électrique ? 23/04 Tesla : qu’arrive-t-il si l’on sort d’une Model 3 en Autopilot ? 22/04 Taycan : Porsche prépare une version d’entrée de gamme pour sa sportive électrique 22/04 Tesla Model Y : ses places dans le coffre sont-elles utilisables ? Nos derniers dossiers Articles les plus lus 1 PS4 : les 15 meilleurs jeux gratuits à télécharger sur le PlayStation… 2 Quels smartphones Huawei auront les mises à jour d’Android pour le… 3 Tesla : qu’arrive-t-il si l’on sort d’une Model 3 en Autopilot ? 4 The Walking Dead : comment Jadis aurait-elle affronté Alpha ? 5 Kia accélère l’électrification de sa gamme avec un crossover inédit… Abonnez-vous à notre newsletter E-mail * SUIVEZ-NOUS Facebook Twitter Youtube InstagramRSS À PROPOS Mentions légales et CGU Confidentialité et Cookies Préférences Cookies ENTREPRISE Qui sommes nous Contact © 2020 Galaxie Media Tous droits réservés PHONANDROID 04/26 Windows 10 : Microsoft fait de la pub pour Edge dans la barre de recherche et le site d’Outlook 04/26 Non, OnePlus ne va pas licencier 80% de son personnel en Europe pendant la crise 04/25 Netflix : les séries et films à regarder ce week-end 04/25 Android : l’enregistrement des appels audio arrive bientôt, mais pas pour tout le monde Tous les bons plans Vie du net EN CE MOMENT : Meilleures tablettes Néobanques Confinement PC portables Casques-micro FORUM CONNEXION Les Numériques TÉLÉPHONIE TV & SON MAISON INFORMATIQUE PHOTO LOISIRS VOITURES & CO TESTS NEWS TÉLÉCHARGEMENT COMPARATIFS Publicité AccueilVie du net Facebook investit plus de 5 milliards dans le principal opérateur 4G indien et affiche ses ambitions sur le commerce en ligne Par Johann Breton Publié le 26/04/20 à 11h24 Partager : L'opérateur Jio, qui a démocratisé l'accès à la 4G en Inde, vient de laisser entrer Facebook à son capital. Quatre ans après l'échec de l'initiative Internet.org, le réseau social consolide ainsi sa position centrale dans la vie numérique indienne. © iStock © iStock Dans une relative discrétion, Facebook a annoncé cette semaine le deuxième plus gros investissement de son histoire après l'acquisition de WhatsApp. Certes loin des plus de 20 milliards de dollars finalement dépensés pour croquer le service de messagerie, le réseau social vient de signer un chèque de 5,7 milliards de dollars (soit environ 5,3 milliards d'euros) pour s'offrir 9,9 % de Jio Platforms, groupe indien qui détient notamment l'opérateur éponyme. Dans le pays, Jio est tout sauf un poids léger : il compte près de 400 millions de clients sur ses offres mobiles, lancées à prix très bas et exclusivement sur le réseau 4G en 2015. Cela représente approximativement deux tiers du marché, et l'opérateur passe maintenant à l'offensive sur la fibre. Ni Facebook ni l'opérateur ne se cachent concernant les raisons de ce rapprochement. Le réseau social, de son côté, voit dans la transaction une voie royale pour façonner un peu plus les habitudes des mobinautes indiens, qui sont déjà largement acquis à la cause de WhatsApp (400 millions d'utilisateurs) et d'Instagram. Dans son communiqué, la firme parle notamment de développer l'activité des petites entreprises, avec en toile de fond la démocratisation de WhatsApp en tant que plateforme de commerce en ligne, en particulier derrière la solution de Jio, JioMart. Jusqu'ici, Amazon et Flipkart se taillaient la part du lion sur le marché indien (chacun plus de 30 % du marché). Jio et son actionnaire principal, Reliance Industries, saluent pour leur part l'arrivée d'un partenaire expert dans l'analyse et la monétisation des données, n'hésitant pas à souligner la masse que représente l'audience cumulée de tous les services des deux groupes en Inde. Aucun plan concret n'aurait encore été défini, mais Reliance Industries serait très ouvert aux suggestions et nouvelles idées envisageables avec le soutien de Facebook, les opportunités couvrant un très vaste spectre du secteur de l'éducation à celui de la santé. Facebook s'achète un retour en bonnes grâces Pour Facebook, cette entrée au capital de Jio Platforms est par ailleurs doublement stratégique. En effet, le réseau social a des relations très difficiles avec le gouvernement indien depuis 2015 et l'épisode Internet.org (devenu Free Basics). Pour rappel, Facebook offrait gratuitement un accès à Internet via réseau mobile, mais restreint à une poignée de services – dont, bien évidemment, son réseau social. Les autorités n'avaient pas apprécié cette vision tronquée d'Internet et, début 2016, avaient de fait banni le service en s'appuyant sur les arguments de la neutralité du Net et d'une pratique discriminatoire pour les opérateurs. En injectant plusieurs milliards en Inde, Facebook se rachète en quelque sorte une image auprès des mêmes autorités, mais la firme gagne surtout un allié colossal. Reliance Industries est un titan, la première capitalisation d'Inde, dont les activités ne s'arrètent pas aux télécommunications, mais incluent aussi l'énergie (pétrole) ou encore le commerce en général. Son patron est, en outre, l'homme le plus riche d'Asie, et il ne se prive pas de conseiller le gouvernement indien. Autant dire que le poids politique de ce nouveau partenaire devrait garantir à Facebook une certaine tranquilité dans ses activités locales et fournir un levier de choix si, d'aventure, des négociations devaient être nécessaires avec les autorités. Tag associé : Partager : JOHANN BRETON Rédacteur noté 3 étoiles lors de son test. Très grande autonomie, mauvais en photo. Ergonomie correcte. Néobanques : le top 10 Les Numériques-Challenges Play Video Publicité ACTUALITÉ Faut-il encore commander sur Amazon, Cdiscount, Fnac-Darty et autres boutiques en ligne ? 29 VIE DU NET Faut-il encore commander sur Amazon, Fnac Darty, Cdiscount et les autres boutiques en ligne ? Hier à 14:12 VOIR TOUTES LES ACTUALITÉS DOSSIER CES 2020 : retour sur les grandes tendances et les innovations les plus marquantes VIE DU NET il y a 3 mois CES 2020 – Retour sur les grandes tendances et les innovations les plus marquantes Dossier par Mathieu Chartier VOIR TOUS LES DOSSIERS Suivez-nous Contact FAQ La rédaction Comparaison de prix Préférences cookies Abonnez-vous à la Numsletter Mon adresse mail Vie privée Cookies Conditions Générales d’Utilisation Tout droits réservés © 2020 Sunday, April 26 2020 Trending Automatic Self-Piercing Rivets Market Global Market Share, Regional Growth Analysis, Upcoming Trends, Technology Updates and Development Analysis Research Report 2026 Contact Bandera County Courier Business Health Science Technology World Search for Home/Health/Latest Report on Healthcare CRM Market with a Growing CAGR During 2020-2027 | Top Companies: IBM, salesforce.com, Siemens Healthineers Health Latest Report on Healthcare CRM Market with a Growing CAGR During 2020-2027 | Top Companies: IBM, salesforce.com, Siemens Healthineers him April 25, 2020 Healthcare CRM Healthcare CRM The report titled, “Healthcare CRM Market” boons an in-depth synopsis of the competitive landscape of the market globally, thus helping establishments understand the primary threats and prospects that vendors in the market are dealt with. It also incorporates thorough business profiles of some of the prime vendors in the market. The report includes vast data relating to the recent discovery and technological expansions perceived in the market, wide-ranging with an examination of the impact of these intrusions on the market’s future development. Healthcare CRM Market research reports growth rates and the market value based on market dynamics, growth factors. The complete knowledge is based on the latest innovations in the industry, opportunities, and trends. In addition to SWOT analysis by key suppliers, the report contains a comprehensive market analysis and major player’s landscape. Ask for Sample Copy of This Report: https://www.healthcareintelligencemarkets.com/request_sample.php?id=57690 Top Key Players Included in This Report: IBM, salesforce.com, Siemens Healthineers, SAP, Accenture, Oracle, Microsoft, Amdocs Ltd., Anthelio Healthcare Solutions, AllScripts Healthcare Solutions Inc., NetSuite, Cerner. Major highlights of this research report: In-depth analysis of the degree of competition across the globe. Estimation of Healthcare CRM Market values and volumes. Healthcare CRM Market analysis through industry analysis tools such as SWOT and Porter’s five analysis. Detailed elaboration on global market value, volume, and penetration. Healthcare CRM Market growth projections. Detailed description on development policies and plans. Analytical study on drivers, restraints, opportunities, hindrances, inadequacies, challenges and strengths. The report on the Healthcare CRM Market has newly added by IT Intelligence Markets to its huge repository. The global market is expected to increase from 2020 to 2027. Primary and secondary research methodologies have been used for curating this research report. Get Discount on This Report: https://www.healthcareintelligencemarkets.com/ask_for_discount.php?id=57690 The competitive landscape of the Healthcare CRM Market is described in terms of the players and their statistics. For each key player, the report reveals production rates, costing, overall pricing, revenue generation, and market share within the Healthcare CRM Market. The research on the Healthcare CRM Market will be applicable to investors, business owners, industry experts, and various c level peoples. Profiling of the several top-level industries has been included in this informative report. *If you have any special requirements, please let us know and we will offer you the report as per your requirements. For Any Customization, Ask Our Experts: https://www.healthcareintelligencemarkets.com/enquiry_before_buying.php?id=57690 Table of Contents: Healthcare CRM Market Overview Economic Impact on Industry Market Competition by Manufacturers Production, Revenue (Value) by Region Supply (Production), Consumption, Export, Import by Regions Production, Revenue (Value), Price Trend by Type Market Analysis by Application Manufacturing Cost Analysis Industrial Chain, Sourcing Strategy and Downstream Buyers Marketing Strategy Analysis, Distributors/Traders Market Effect Factors Analysis Healthcare CRM Market Forecast About Us: HealthCare Intelligence Markets Reports provides market intelligence & consulting services to a global clientele spread over 145 countries. Being a B2B firm, we help businesses to meet the challenges of an ever evolving market with unbridled confidence. We craft customized and syndicated market research reports that help market players to build game changing strategies. Besides, we also provide upcoming trends & future market prospects in our reports pertaining to Drug development, Clinical & healthcare industries. Our intelligence enables our clients to take decisions with which in turn proves a game-changer for them. We constantly strive to serve our clients better by directly allowing them sessions with our research analysts so the report is at par with their expectations. Contact Us: Marvella Lit Address: 90, State Office Center, 90, State Street Suite 700, Albany, NY 12207 Email: sales@healthcareintelligencemarkets.com Web: www.healthcareintelligencemarkets.com Phone: +44-753-712-1342 FacebookTwitterLinkedInStumbleUponRedditShare via EmailPrint him Related Articles April 22, 2020 5 Massive Growth of Natural Killer (NK) Cell Therapeutics Market to Witness a Pronounce Growth at CAGR during 2020-2027| Chipscreen Biosciences (China), Affimed NV (Netherlands), Altor BioScience Corporation (USA), Innate Pharma SA Medical 3D Printing Market February 27, 2020 28 Trending 2020 Medical 3D Printing Market Worldwide Survey with Top growing Companies: Arcam AB, Autodesk, Stratasys, ExOne Company, H Intressenter AB, Optomec, Organovo Holdings, Voxeljet AG Anti-Aging Products and Therapies Market February 25, 2020 5 Anti-Aging Products and Therapies Market Competitive Dynamics & Global Outlook 2027| Key Players Allergan, Ipsen, Lanzhou Institute, Corneal(Allergan), Galdermal Medical Oxygen Concentrators February 24, 2020 6 Growth Hacking Strategies for Global Medical Oxygen Concentrators MARKET with Report during year 2020-2027 with Major Key Players: Chart Industries, Inogen, Invacare Corporation Aller au contenu principal Rechercher Politique International Société Vox Économie Sport Culture Voyage Style Madame Figaro Live Se connecter Lire Tech & web en PDF en ligne Lire le journal Abonnez-vous 1€ le premier mois ActualitésStart-upTestsPratiqueJeux vidéos AccueilTech & Web Essais cliniques : l’acquisition de Medidata, le pari gagnant de Dassault Systèmes La crise actuelle souligne la pertinence de la plus grosse acquisition réalisée par le groupe français. Par Elsa Bembaron Publié hier à 18:58, mis à jour hier à 18:58 Dassault Systèmes Nul triomphalisme chez Bernard Charlès, le PDG de Dassault Systèmes, à l’occasion de la présentation des résultats du premier trimestre 2020. Mais une pointe de satisfaction de disposer des bons outils au bon moment. Le groupe a déboursé près de 6 milliards de dollars pour mettre la main sur Medidata. «Il y a à peine six mois nous avons fait cet investissement pour faire évoluer les processus collaboratifs dans la santé, développer les jumeaux numériques... Aujourd’hui, nous sommes au centre de la situation pour les essais cliniques. Près de 60% des essais cliniques en cours, dont ceux sur le Covid-19, sont faits sur Medidata», constate Bernard Charlès. La pandémie place sur le devant de la scène de nouveaux sujets et apporte un nouvel éclairage sur la stratégie de Dassault Systèmes. Medidata est notamment reconnu pour les «bras de contrôle synthétiques». Cette méthode est utilisée dans le cadre d’essais cliniques. Traditionnellement, deux groupes de patients sont constitués. La moitié des patients se voient administrer une thérapie, l’autre un placebo. «Le bras de contrôle synthétique permet de réduire la taille des cohortes de patients. Une population virtuelle est établie, en lien avec le réel. Cela permet aussi de mieux cibler les profils qui ont le maximum de chances de bénéficier de l’efficacité d’un traitement», explique Bernard Charlès. L’objectif final étant de gagner en efficacité, en pertinence et en rapidité. Les défis posés par le Covid-19 mettent en avant ces impératifs. Sponsorisé par FRANFINANCE Faites signer électroniquement vos contrats entreprises depuis Flashlease Faites signer électroniquement vos contrats Entreprises depuis Flashlease ! Découvrez notre solution de signature électronique ! #digitalbanking #digital #finance #digitalisation #efficacité FRANFINANCE VOIR PLUS Des équipes à 150% La branche santé de Dassault Systèmes n’est pas la seule à avoir été très sollicitée avec la crise. «Nos clients industriels nous ont demandé de l’aide pour mettre en place des solutions de télétravail, ajoute Bernard Charlès. Nos équipes travaillent à 150%». Non seulement Dassault Systèmes n’a pas eu recours au chômage partiel, mais en plus, le groupe s’engage à conserver tous ses salariés, même s’il peut «y avoir une redirection de certaines activités». Pour son PDG, le Dassault Systèmes qui sortira de la crise ne sera plus le même que celui qui y est entré. Il n’a pas remisé pour autant ses objectifs de croissance, même s’il les a revus à la baisse. «Notre modèle est résiliant, avec 85% de nos revenus récurrents», mentionne Bernard Charlès, même s’il s’attend à une nouvelle vague de consolidation dans la sous-traitance aéronautique et automobile. Des opérations qui pourraient profiter à son groupe, puisque dans ce cas, les outils sont harmonisés. Au final, Dassault Système table sur une croissance de 14 à 15% contre un objectif de 24% auparavant. Les prévisions initiales de bénéfice net par action tablaient sur une croissance de 12%. Désormais, le groupe anticipe une stabilité, voire une progression de 2%. Au premier trimestre, le chiffre d’affaires du groupe a progressé de 17% , reflétant l’intégration de Medidata, à 1,13 milliard d'euro. La marge opérationnelle du groupe est passée de 22,5% à 13,3% un an plus tôt. Le groupe maintien son dividende à 0,70 euro par action. «Sans abaisser notre dividende, nous avons un taux de distribution de 30% quand de nombreux groupes sont encore au delà de 50%», mentionne Bernard Charlès. La rédaction vous conseille Dassault Systèmes met le cap sur la santé Dassault Systèmes change de dimension Bernard Charlès: «Nous vivons une nouvelle révolution industrielle» Avec Medidata, Dassault Systèmes réalise la plus grosse acquisition de son histoire Sujet DASSAULT SYSTEMES par Taboola Contenus Sponsorisés 5 entreprises qui développent un vaccin contre le coronavirus : laquelle sera la première à le commercialiser ? - eToro eToro Blog Post Ce nouveau placement à 7% qui séduit de plus en plus de Français Top Economie 14 formations à domicile financées à 100% L'École Française Nouveau 2020 : plus de conditions de revenus pour bénéficier des aides à l’isolation Mes Combles Gratuits + 55ans : Classement des meilleures mutuelles 2019 Meilleurtaux.com Le nouveau jeu de construction qui rend tout le monde accro. Pas d'installation Forge Of Empires - Jeu en ligne gratuit C’est sur ce site que les célibataires de plus de 50 ans trouvent l’amour à Versailles Nos Belles Années Découvrez la nouvelle Opel Corsa-e, élue Best Buy Car of Europe Opel France par Taboola Recommandé pour vous Confiné en Maine-et-Loire, Gérard Depardieu est «remonté contre Macron» Le Figaro Trouvez votre Volkswagen Tiguan d'occasion sur ParuVendu.fr ParuVendu La mystérieuse disparition de Kim Jong-un Le Figaro Premium 4 commentaires fafia le 24/04/2020 à 09:10 la maladie doit payer après un tel investissement ... nous n'en n'avons pas fini des virus à répétition !!! DORSAIS123 le 23/04/2020 à 20:14 Dassault un des derniers groupe debout ,issus de l'épopée des Bâtisseurs des"trente glorieuse") L’OTAN impose à l’Allemagne l’achat d’avions américains POUR renouveler une partie de sa flotte d’avions de chasse, quelque 45, F-18 de l’américain Boeing.Car l’OTAN l’exige, pour être en droit de transporter les bombes nucléaires mises à disposition par Washington,??? "Ça fait Flipper " Berlin est forcée de s’équiper au moins en partie en matériel américain au lieu de pouvoir choisir des avions européens.et devient dépendante des technologies,US. Déja Madame Lagarde militait aux États-Unis au Center for Strategic & International Studies (CSIS).Qu'elle co-présidait avec Zbigniew Brzezinski,la commission Action USA/UE/Pologne et suivait plus particulièrement le groupe de travail Industries de défense USA-Pologne (1995-2002)Bruce P. Jackson, fondateur(US Committee for Expand NATO).représentait les intérêts de l’avionneur Lockheed Martin,en avait été vice-président.c'est au travers des commissions présidées par Christine Lagarde que M. Jackson monta le contrat du siècle:la vente,04/2003, de 48 chasseurs F-16 Lockheed-Martin à la Pologne 3,5 milliards $.la logique eut voulu que Varsovie s’équipe de Mirage 2000-5 MK2 Dassault ou de Jas-39 Saab JPR92200 le 23/04/2020 à 19:34 Brillante société française de haute technologie qui mérite son succès : " le hasard ne favorise que les esprits préparés " (Pasteur) LIRE LES 4 COMMENTAIRES Chargement de l'article suivant... 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Skip to content COLE REPORTS COLE MARKET RESEARCH MARKET REPORTS INDUSTRY ANALYSIS MILITARY NEWS MORE Search for: SEARCH … GLOBAL OPTICAL NETWORKING AND COMMUNICATION MARKET RESEARCH ANALYSIS REPORT 2020 | HUAWEI TECHNOLOGIES CO (CHINA), NOKIA (FINLAND), CISCO APRIL 23, 2020SANTOSH Global Optical Networking and Communication Market Research Analysis Report 2020 | Huawei Technologies Co (China), Nokia (Finland), Cisco Optical Networking and Communication Market: Industry Growth ...An exclusive research report Global Optical Networking and Communication Market 2020 by Manufacturers, Countries, Type and Application, Forecast to 2026 added to the dataset of MarketsandResearch.biz especially covers global market top players, types, applications, pattern, and size. The report analyzes various market dynamics, the quickest and slowest market segments. The report gives autonomous data about the global Optical Networking and Communication market which includes analysis on size and patterns, inhibitors, elements, drivers, openings and difficulties, condition and arrangement, cost outline, porters five power investigation, and key organizations’ profiles including business review and improvement. The report assists the specialists to make choices dependent on the market investigation. The report presents segmentation of the global Optical Networking and Communication market by end-users, regional countries, product types and key manufacturers based on differentiable validation and industrial analysis. The document demonstrates an essential diagram of the business including definitions, groupings, applications and industry chain structure. On the basis of segments by manufacturers, this report focuses on the sales, price of each type, the average price, revenue, and market share, for key manufacturers. DOWNLOAD FREE SAMPLE REPORT: https://www.marketsandresearch.biz/sample-request/25435 Geographical data will help the reader understand the best performing regions. This report has added an examination and increment pace of the market in these districts covering: North America (United States, Canada and Mexico), Europe (Germany, France, UK, Russia and Italy), Asia-Pacific (China, Japan, Korea, India and Southeast Asia), South America (Brazil, Argentina, Colombia etc.), Middle East and Africa (Saudi Arabia, UAE, Egypt, Nigeria and South Africa) Our best analysts have surveyed the market report with the reference of inventories and data given by the key players: Huawei Technologies Co (China), Nokia (Finland), Cisco (US), Ciena (US), ADTRAN (US), Broadcom (US), Finisar (US), Fujitsu Optical Components (Japan), Infinera (US), ADVA Optical Networking (Germany), NEC (Japan), Juniper Networks (US), Ericsson (Sweden), Lumentum Operations (US), On the basis of product, this report displays the production, revenue, price, market share, and growth rate of market types split into: Fiber, Transceiver, Amplifier, Switch, Splitter, Circulator On the basis on the end users/applications, this report focuses on the status and outlook for major applications/end users, sales volume, market share and growth rate of market applications, includes: Telecom, Data Center, Enterprise, Others, Further, the report delivers a comprehensive analysis of industry trends along with the restraints, drivers, and opportunities in the market. The prominent vendors are also studied with SWOT analysis, fiscal overview and major developments. Different graphical presentation techniques such as charts, graphs, tables, and pictures have been used while making the report. The overall report includes research on historical and forecasts global Optical Networking and Communication market data, demand, application details, price trends, and company shares of the leading company by geography. ACCESS FULL REPORT: https://www.marketsandresearch.biz/report/25435/global-optical-networking-and-communication-market-2020-by-manufacturers-countries-type-and-application-forecast-to-2026 Why Purchase The Report? The researchers have offered projections about the global Optical Networking and Communication industry size based on value and volume in this part of the report. The report throws light on the recent industry developments and upcoming prospects that are likely to foster the overall market growth. The study gives reliable estimations regarding the growth potential of varied industry segments including product type, vertical, and application. Customization of the Report: This report can be customized to meet the client’s requirements. Please connect with our sales team (sales@marketsandresearch.biz), who will ensure that you get a report that suits your needs. You can also get in touch with our executives on +1-201-465-4211 to share your research requirements. About Us Marketsandresearch.biz is a leading global Market Research agency providing expert research solutions, trusted by the best. We understand the importance of knowing what global consumers watch and buy, further using the same to document our distinguished research reports. Marketsandresearch.biz has worldwide presence to facilitate real market intelligence using latest methodology, best-in-class research techniques and cost-effective measures for world’s leading research professionals and agencies. 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Contact Us Mark Stone Head of Business Development Phone: +1-201-465-4211 Email: sales@marketsandresearch.biz Web: www.marketsandresearch.biz Post navigation Global Tea Packaging Market Research Analysis Report 2020 | Starship Technologies, Robby Technologies, Nuro, IncGlobal Sports & Energy Drinks Market Research Analysis Report 2020 | Red Bull, Monster, Rockstar, Pepsico, Big Red, Arizona, National santosh VIEW ALL POSTS BY SANTOSH PAGES About Us Contact Us Our Staff Privacy Policy Search for: Search … RECENT POSTS Global PEX Tubing Tools Market Size, Analytical Overview, Growth Factors, Demand, Trends And Forecast To 2026 Global Medical Sector Contract Electronic Manufacturing Market 2025 Top Manufacturers : Benchmark Electronics, FLEX, Celestica, Jabil Circuit, Sanmina, Nortech Systems Comprehensive Report on Load Bank Rental Market, Size, Share, Status and Forecast 2020-2026 | VIBSpECTRUM International, Aggreko, Vardha Technical Services LLC, Crestchic Limited, Power House Solar Panel Module Market (Impact of COVID-19) Growth, Revenue Status & Forecast Report 2025 | Trina Solar, Canadian Solar, JinkoSolar, JA Solar, Hanwha Q CELLS, First Solar, Yingli Green, SFCE, ReneSola Global Mechanical Energy Storage Market 2025 Top Manufacturers : Voith, Sulzer, DeWalt, Ingersoll Rand, Hitachi, Porter-Cable Theme of Rigorous Themes ABOUT US PRIVACY POLICY OUR STAFF CONTACT US Technology and gaming content trusted in North America and globally since 1999 8,575 Reviews & Articles | 60,314 News Posts CORONAVIRUS NEWS REVIEWS ARTICLES GUIDES GAMING ASK THE EXPERTS NEWSLETTER FORUMS Cases, Cooling & PSU CPUs, Chipsets & SoCs Motherboards Peripherals RAM Storage Video Cards Home News Video Cards NVIDIA is making something on 5nm, and we don't know what it is NVIDIA's mysterious 5nm product probably isn't a next-gen GPU, but TSMC is making something for Team Green on 5nm Anthony Garreffa | Apr 23, 2020 at 10:47 pm CDT (0 mins, 45 secs time to read) So, NVIDIA is making something on a fresh new 5nm node -- ordering TSMC to make it for the GeForce giant. Digitimes is reporting from its "industry sources" that NVIDIA is tapping TSMC's fresh new 5nm process node -- and it won't be for NVIDIA's next-gen Ampere GPU architecture or the GeForce RTX 3080 and GeForce RTX 3070 graphics cards. There is absolutely nothing else to this news other than Digitimes reporting that its sourcesh ave told them, that NVIDIA is making something new and mysterious on TSMC's new 5nm process node. It could be something for the Tegra side of things, or here's a wild rumor -- so prepare your salt shakers peeps -- the Nintendo Switch 2 SoC. Just throwing that out there. ASUS ROG STRIX GeForce RTX 2080TI Overclocked (ROG-STRIX-RTX-2080TI-O11G) Today Yesterday 7 days ago 30 days ago $1279.99 $1279.99 $1279.99 $1279.99 Buy * Prices last scanned on 4/23/2020 at 10:47 pm CDT - prices may not be accurate, click links above for the latest price. We may earn an affiliate commission. NEWS SOURCES:tomshardware.com, digitimes.com, s.thestreet.com ABOUT THE AUTHOR - Anthony Garreffa Anthony is a long time PC enthusiast with a passion of hate for games built around consoles. FPS gaming since the pre-Quake days, where you were insulted if you used a mouse to aim, he has been addicted to gaming and hardware ever since. Working in IT retail for 10 years gave him great experience with custom-built PCs. His addiction to GPU tech is unwavering. Similar News TSMC making 5nm ASICs for Bitmain and Canaan in 2020 AMD's next-gen Zen 4 could arrive on new 5nm or 6nm in 2021 TSMC makes progress on 5nm, should have 5nm chips in 2020 Intel promises process node leadership with 5nm... but in 2022+++ Apple's new iPhone will be powered by A14 chip made on 5nm from TSMC < PREVIOUS STORY: Intel's upcoming Ponte Vecchio Xe GPU turns up in add-in card form Related Tags NVIDIA TSMC 5nm NVIDIA 5nm node NVIDIA 5nm tech TSMC 5nm Nintendo Newsletter Subscription Latest News NVIDIA is making something on 5nm, and we don't know what it is Intel's upcoming Ponte Vecchio Xe GPU turns up in add-in card form NVIDIA, AMD purchase ALL excess capacity at TSMC for next-gen GPU/CPUs Intel revenue up 23% to $19 billion in Q1, operating income spikes 69% PlayStation 5 may use vapor chamber cooling, new Sony patent suggests View More News Latest Reviews Synology DS220j Two-Bay Consumer NAS Review ARCTIC Liquid Freezer II 280 CPU Cooler Review NZXT C850 Gold ATX Power Supply Review Seagate IronWolf 510 m.2 NAS SSD Review Star Wars: The Rise of Skywalker 4K Blu-ray Review View More Reviews Latest Articles GeForce Ti Showdown: GTX 980 Ti vs GTX 1080 Ti vs RTX 2080 Ti Best Gaming Monitors in 2020 (4K, UltraWide, 1440p, and 1080p) Best GPUs for PC Gaming in 2020 DOOM Eternal Benchmarked at 8K - Not Even The Devil Can Run This Half-Life: Alyx Buyers Guide: PC Hardware & VR Headset Recommendations View More Articles SITEMAP ADVERTISING ABOUT CONTACT RSS FEEDS PRIVACY POLICY © 1999-2020 Tweak Town Pty Ltd. All Rights Reserved. Read more: https://www.tweaktown.com/news/72034/nvidia-is-making-something-on-5nm-and-we-dont-know-what-it/index.html Search Reviews Smart Home & Gadgets Mobile Car Tech Online Science & Health Cameras Computing Gaming Entertainment Apple iPhone SE: Australian First Look Review Apple's iPhone SE: Every Phone Plan In Australia Every Telstra Plan For The New iPhone SE Apple iPhone SE: Australian First Look Review Tegan Jones | Apr 24, 2020, 2:25pm⋅ Filed to: apple Share Image: Tegan Jones/Gizmodo Australia Last year, Google turned the mid-range phone market on its head by introducing the ludicrously-priced and well-specced Pixel 3a. A few other brands have followed suit since then, but none have been quite as exciting as the new iPhone SE. Now it truly seems like flagship inclusions at lower price points are here to stay - and it's about damn time. The trend of $1,500 - $2,000 becoming the norm for new phones over the last few years has been bad for buyers. A new middle ground has been long overdue and we welcome it. But is the resurrected iPhone SE actually a good phone to buy in 2020? It's worth noting that I have only been playing with the iPhone SE for a day and a half, so these are my first impressions. Keep your eyes peeled for the full review next week. iPhone SE Specs But before we dive into things, here's a quick reminder of the specs. Display: 4.7 inches (1334 x 750) CPU: A13 Bionic Chip Storage: 64GB, 128GB, 256GB Battery: Up to 13 hours video playback, Qi wireless charging, fast charge Rear camera: 12MP (f/1.8) Front camera: 7MP (f/2.2) Video recording: 4K video up to 60 fps Water resistance: IP67 Size: 5.45 x 2.65 x 0.29 inches Weight: 148g Colour: Black, White, PROJECT(RED) iPhone SE Price In Australia In Australia the iPhone SE starts at $749 for the 64GB. 12GB will bring the price up to $829 and 256GB comes in at $999. You can grab one from the Apple Store now. If you'd prefer a plan, we have all of them compiled in one handy place for you. What's good about the iPhone SE? The iPhone SE is such a good size for smaller hands. Image: Tegan Jones/Gizmodo Australia iPhone SE Size In a world where 6.5-inch phones have become par for the course, it's refreshing for a premium phone manufacturer to offer something a bit smaller. While large phones screens are good for some things – streaming shows, reading, general browsing – they can also be incredibly uncomfortable. While I have gotten used to the iPhone 11 Pro Max, I have found the SE far more comfortable to use as a daily driver. The screen size isn't too small, the device nestles perfectly into my hand and it can actually fit into my lady pockets. A true miracle in 2020. And from a tactile perspective, it's the same as flagship phone – you won't find cheap plastic here. The material is lovely to touch and feels exactly the same as my 11 Pro Max, just smaller. While it isn't as small as the previous SE or some other older iPhones, I think that anyone upgrading from one of those previous generations will still be pleased – especially if you compare it to something like the Pro or Pro Max. iPhone SE Camera I'll have a far more in-depth dive into the iPhone SE camera in my full review, but I have a few initial thoughts. It's important to set your expectations if you're coming to the SE from a flagship device. It only has a single lens on the front and back, so it simply can't have the same functionality as a triple or quad camera array. That being said, it still pretty darn good so far, and this is primarily due to the processor. The iPhone SE has the same A13 bionic chip as iPhone 11 devices, which is a crazy move that I'm here for. The signal processor and N=neural engine in the A13 boosts the computational photography of the iPhone SE, which helps make up for what it lacks in hardware. But what does that actually mean when it comes to real world functionality? To put it simply, the single camera lenses are more capable thanks to the processor and software. The SE has Smart HDR, optical image stabilisation, subject recognition and the ability to capture clearer faces. It's also able to offer the same six lighting affects in portrait mode as the iPhone 11, as well as depth control adjustment you usually need a second lens for. When it comes to video, the rear camera is still able to capture 4K and 30 fps with cinematic stabilisation. As for the selfie cam, you still get the lovely bokeh effect – the soft, out-of-focus background in photos – of pricier iPhones thanks to the inclusion of monocular depth estimation. So far I have mostly messed around with the portrait and regular photo modes - and they really do a great job for single lenses. Sure, they're not as good as the iPhone 11 Pro Max, but at a price point of $750 - $1000 that was never going to happen. Here are a few comparisons between the iPhone SE and iPhone 11 Pro Max: iPhone SE Regular Photo Mode There's a noticeable different between the iPhone SE (left) and iPhone 11 Pro Max (right) in terms of brightness, sharpness and warmth. However, some of this could be brought back into the final product with some editing. iPhone SE Portrait Mode Left: iPhone SE, Right: iPhone 11 Pro Max iPhone SE Low Light Camera The iPhone SE (left) doesn't have a dedicated night mode like the iPhone 11 Pro Max (right) but it's low-light capabilities seem quite decent so far. iPhone SE Selfies Left: iPhone SE, Right: iPhone 11 Pro Max If you want to best camera phone on the market, the iPhone SE is not what you should opt for. But so far, I'm quite impressed and I'm looking forward to putting it through its paces more. What's not so good about the iPhone SE? Image: Tegan Jones/Gizmodo Australia iPhone SE Battery Life Unlike Android manufacturers, Apple doesn't reveal its battery sizes - but we do know that it's smaller than any device in the iPhone 11 lineup. For example, Apple quotes the SE as having up to 13 hours of video playback, where as the baseline iPhone 11 has up to 17. It didn't take long for me to notice the battery drain, particularly when I went out for a walk. In 30 minutes, the battery dropped by 13 per cent while I had my AirPods Pro connected via bluetooth, location enabled and Spotify playing. During that time I also sent some messages via Facebook Messenger and Slack. I went on a similar walk the next day and lost 10 per cent battery in 20 minutes - this time around I had added some light browsing into the mix. That being said, the battery still go me through the whole day. I started using the device at 7:30am and by 11:30pm it was down to 3 per cent. As a moderate to heavy user, this was okay. However, if Australia wasn't in lockdown right now I would probably have run out of battery earlier due to heavier music listening and reading on my morning commute, as well as connecting my Apple watch for gym workouts and taking the phone out after work. If the world was normal right now, I would definitely need to be put the SE on a charger during the day. Apple's iPhone SE: Every Phone Plan In Australia Apple mic-dropped the mobile world last week by unveiling a brand new iPhone SE for just $749. Not only that, even the most expensive version of the mid-range phone still comes in at under $1000. The phone is on sale today, and every major Aussie telco is on board with this little beauty. Because opening 500 tabs to compare phone plans sucks, we have rounded up all of them for you. You're welcome. Read more But despite my battery complaints, it's important to take my own experience and background into account. I review phones for a living and use an iPhone 11 Pro Max regularly. And in recent weeks, I've been playing around with the Samsung S20 Ultra and Huawei P40. These are flagship phones with huge batteries. My prediction is that people most likely to buy an iPhone SE will be upgrading from an older Apple device, such as an iPhone 6, 7 or 8. Alternatively, they're low to moderate phone users who simply want a cheaper device without having to make a lot of compromises. If these scenarios sound like you, the battery life on the iPhone SE will be just fine. You'll probably find that it's on par with the iPhone 8 and still larger than anything older than that – especially when you take battery degradation into account. Plus, the iPhone SE charges incredibly quickly thanks to 30W fast-charge capabilities, combined with the smaller battery. I even found that the wireless charging on it was quite speedy. If you want a more robust battery, you should consider upgrading to an iPhone 11 or 11 Pro instead. But if you don't mind something a bit smaller, the iPhone SE could be likely be perfect for you. iPhone SE Bezel While this isn't a problem for me, this will probably bug some people. So let's talk about it. The 4.7-inch display has quite a thick bezel at the top and bottom of the device to house the home button and front-facing camera. This makes sense because it would cost a lot more to include an in-screen fingerprint reader and camera. However, some folks may not like how this throwback inclusion eats into the screen real estate, especially if they enjoy streaming content on their phones. While it's not my favourite feature, the bezels really don't bother me that much and after a few days of use I don't really miss Face Unlock. I also found watching Netflix and Quibi on the SE great. The smaller screen didn't bother me and the brightness and quality of the display was lovely. However, if you're a fan of having as much screen as possible, it's probably worth considering one of the Phone 11 models instead. Is the iPhone SE Worth It? Image: Tegan Jones/Gizmodo Australia I've only had the iPhone SE in my hot little hands for a couple of days, so I'll hold my hard stance recommendations until my full review. That being said, this is a great device for the price point – even if you are still paying a premium because it's Apple. While it doesn't have all the bells and whistles of the more expensive iPhones, there's more than enough of them to be perfect for some people. At the same time, it still offers some flagship features you usually have to pay more for, so it's future proofed for a few years. Qi wireless charging, eSIM functionality and water resistance are all fantastic inclusions – but the real game changer here is the A13 bionic chip. Being able to get the best phone processor on the market in a sub-$1000 phone is a real power move by Apple, and one that will probably pay off for them. The iPhone SE is as modern as it is thoughtful. It provides an option for people on a tighter budget or who don't want to stretch their fingers across the mammoth phone screens that have become the norm over the last few years. Trending Stories Right Now apple au feature iphone-se Apple iPhone SE: Australian First Look Review Tegan Jones 24 Apr 2020 2:25 PM Last year, Google turned the mid-range phone market on its head by introducing the ludicrously-priced and well-specced Pixel 3a. A few other brands have followed suit since then, but none have been quite as exciting as the new iPhone SE. Now it truly seems like flagship inclusions at lower price points are here to stay - and it's about damn time. The trend of $1,500 - $2,000 becoming the norm for new phones over the last few years has been bad for buyers. A new middle ground has been long overdue and we welcome it. But is the resurrected iPhone SE actually a good phone to buy in 2020? books chang-chen charlotte-rampling denis-villeneuve dune frank-herbert io9 jason-momoa javier-bardem josh-brolin oscar-isaac rebecca-ferguson stellan-skarsgard timothee-chalamet warner-bros zendaya A Guide To Dune's Gargantuan Cast Of Characters Beth Elderkin 24 Apr 2020 1:02 PM This could be the biggest movie event of 2020—and we mean that literally. There are so many characters in Dune that it’s hard to keep them straight. We’ve got a handy guide to the biggest characters in Denis Villeneuve’s adaptation (with as few spoilers as possible) and who’s playing them, as well as a few others we hope to see in the sequel. Want Gizmodo's email newsletter? Enter your email address Follow us, subscribe and get in touch! Latest Deals Deals: $300 Off Samsung Galaxy S20 5G And S20+ 5G Phones The Fastest NBN Providers According To Netflix Dayum, Telstra Just Doubled The Data Of Its Cheapest Plan Tell Us How You're Coping With The Coronavirus Epidemic And You Could Win A $500 Uber Eats Voucher Trending Articles 1Apple iPhone SE: Australian First Look ReviewApple iPhone SE: Australian First Look Review 231 Little-Known Settings On Your Phone And Laptop You Should Know About31 Little-Known Settings On Your Phone And Laptop You Should Know About 3A Guide To Dune's Gargantuan Cast Of CharactersA Guide To Dune's Gargantuan Cast Of Characters Here’s What You Oughta Know About Grabbing Cash Out Of Your Super During The PandemicI Looked Up Foot Fetish Porn For A Lol But Now It’s The Only Thing That Gets Me OffThese Embarrassing As Hell Work Stories’ll Make You Clench For EternityI Genuinely Fear For My Social Skills Once Iso Is Over & We Have To, You Know, Interact Battlefield V Set For Final Content Drop As Development Shifts To SequelIndie Developers Are Keeping The PlayStation Vita AliveRevenge Of The Cartels, Mayhem 2.0, Loot The Universe Now Available In Borderlands 3All The Bugs And Fish Leaving Animal Crossing: New Horizons In April Send Packages To Nearby Friends Using Uber ConnectThe Best Recipes For ANZAC Day BiscuitsReminder: No ANZAC Day Long Weekend For Most Of Us This YearWhat Is Dalgona Coffee? The 26 best players still available after the first round of NFL draftGreen Bay Packers pull off one of the biggest surprises of the draft, trading up to draft quarterback Jordan Love as a possible successor to Aaron RodgersFitness Australia says gyms should be reopened if they follow strict guidelines – but an expert says the conditions aren't right 'just yet'Singapore's richest man is getting $1 billion richer each month from selling ventilators © 2007-2020 Pedestrian GroupAboutAdvertiseContactTerms of UseGizmodo International Sign up for Gizmodo Australia's breaking news alerts and get the best stories straight away AllowClose SKIP NAVIGATION BREAKING Oil rallies more than 40% in two days as it comes back from record lows logo SIGN IN PRO WATCHLIST MAKE IT SELECT USA INTL logo MARKETS BUSINESS INVESTING TECH POLITICS CNBC TV SEARCH QUOTES HEALTH AND SCIENCE Gilead says draft of coronavirus treatment results contain ‘inappropriate characterization’ and are ‘inconclusive’ PUBLISHED THU, APR 23 20201:22 PM EDTUPDATED MOMENTS AGO Berkeley Lovelace Jr. @BERKELEYJR KEY POINTS Gilead shares fell after a report, citing a draft document, said antiviral drug remdesivir did not improve Covid-19 patients’ condition or reduce the virus’ presence in the bloodstream in a phase 3 clinical trial. Gilead said the document contained “inappropriate characterizations” and that the study’s findings were “inconclusive.” AP: Gilead Sciences 150709 In this July 9, 2015, file photo, a man walks outside the headquarters of Gilead Sciences in Foster City, Calif. Eric Risberg | AP Gilead Sciences said that a draft document showing disappointing results from a closely watched clinical trial of the company’s treatment for the coronavirus contained “inappropriate characterizations” and that the study’s findings were “inconclusive.” Shares of the biotech firm fell 4% in intraday trading Thursday after the Financial Times reported that antiviral drug remdesivir did not improve Covid-19 patients’ conditions or reduce the virus’ presence in the bloodstream in a clinical trial in China. The report, citing a draft document that was accidentally published by the World Health Organization and has since been removed, also said the drug showed significant side effects, and some patients were taken off of it. “We regret that the WHO prematurely posted information regarding the study, which has since been removed. The investigators in this study did not provide permission for publication of results,” a Gilead spokesperson said in a statement to CNBC. “Furthermore, we believe the post included inappropriate characterizations of the study. Importantly, because this study was terminated early due to low enrollment, it was underpowered to enable statistically meaningful conclusions,” according to Gilead. “As such, the study results are inconclusive, though trends in the data suggest a potential benefit for remdesivir, particularly among patients treated early in disease.” VIDEO00:30 Gilead halted after report Remdesivir flopped in coronavirus trial There are no proven therapies to treat Covid-19, which has infected more than 2.6 million worldwide and killed at least 183,820 as of Thursday, according to data compiled by Johns Hopkins University. U.S. health officials say producing a vaccine to prevent Covid-19 will take at least 12 to 18 months, making finding an effective drug treatment soon even more crucial. There a number of ongoing studies testing Gilead’s remdesivir to see if it’s effective in stopping the coronavirus from replicating. Last week, Gilead surged after details leaked of one of its clinical trials, showing what appeared to be promising results in treating the disease. The University of Chicago found that patients with Covid-19 had “rapid recoveries in fever and respiratory symptoms” and were discharged in less than a week, according to STAT News. The recent swings in Gilead shares over the past week show just how difficult it is to bet on developments for treatments for the coronavirus. Mike Bailey, director of research at FBB Capital Partners and a former health-care analyst at Stifel, told CNBC last week that retail investors are “hungry” for good news on the virus. Those investors could be taking that and channeling it into buying shares, he said. “You could argue [Gilead] buyers are hungry for good news while Moderna buyers are starving for it,” said Mike Bailey, referring to Moderna’s coronavirus vaccine currently in human trials. Remdesivir has shown some promise in treating SARS and MERS, which are also caused by coronaviruses. Some health authorities in the U.S., China and other parts of the world have also been using remdesivir, which was tested as a possible treatment for the Ebola outbreak, in hopes that the drug can reduce the duration of Covid-19 in patients. Results from a Gilead clinical trial involving patients with severe cases of the virus are expected later this month, while data on a trial testing patients with moderate symptoms is expected in May. Infectious disease experts and analysts caution the severe study is “single-arm,” meaning it will not evaluate the drug against a group of patients who didn’t receive the drug. TRENDING NOW Jamie Black-Lewis received loans from the Paycheck Protection Program for her two spas, Oasis Medspa & Salon and Amai Day Spa, in Washington state. Many of her 35 employees think they will make more money collecting unemployment benefits than from their regular paychecks. She got a forgivable Paycheck Protection Program loan. Her employees hate her for it. Remember what this felt like? People enjoy the warm spring weather at Hornstull in Stockholm on April 21, 2020, during the new coronavirus COVID-19 pandemic. Sweden resisted a lockdown, and its capital Stockholm is expected to reach ‘herd immunity’ in weeks New York Governor Andrew Cuomo (D) speaks during a press Conference at the State Capitol. Cuomo rips McConnell’s ‘really dumb’ suggestion to let states declare bankruptcy over coronavirus Treasury Secretary Steven Mnuchin speaks during the daily briefing on the novel coronavirus, COVID-19, in the Brady Briefing Room at the White House on April 2, 2020, in Washington, DC. US issues new guidance for small business loans, pressures public companies to return funds New York Governor Andrew Cuomo (D) speaking at a press Conference at the State Capitol reporting on the latest development of the Covid-19 Coronavirus situation. New York antibody study estimates 13.9% of residents have had the coronavirus, Gov. Cuomo says by TaboolaSponsored LinksFROM THE WEB Si tu aimes jouer, ce jeu de construction de ville est pour toi. Pas d’installation. 14 215 876 y jouent ! Forge Of Empires - Jeu en ligne gratuit Programme Photovoltaique - Originale Gratuit & Sans Engagement logo Subscribe to CNBC PRO Licensing & Reprints CNBC Councils Supply Chain Values Advertise With Us Join the CNBC Panel Digital Products News Releases Closed Captioning Corrections About CNBC Internships Site Map AdChoices Careers Help Contact News Tips Got a confidential news tip? We want to hear from you. 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Sign in Marketplace Subscribe Premium My Portfolio Authors Top Stocks Latest News Markets Stock Ideas Dividends ETFs Investing Strategy Podcasts Help & Support Healthcare | On the Move BioSig up 10% on encouraging data on COVID-19 antiviral Apr. 23, 2020 9:49 AM ET|About: BioSig Technologies, Inc. (BSGM)|By: Douglas W. House, SA News Editor BioSig Technologies (BSGM +10%) is up out the gate on the heels of results from laboratory studies comparing the antiviral activity of subsidiary ViralClear's merimepodib to Gilead Sciences' (GILD +2.2%) remdesivir. Cell cultures were pretreated with two concentrations of either merimepodib or remdesivir for four hours before SARS-CoV-2 virus was added. At hour 16, both concentrations of the former and the higher concentration of the latter significantly reduced viral production. At hour 24, both concentrations of both drugs significantly reduced viral production. Additional studies are underway. ViralClear intends to launch clinical trials in COVID-19 next quarter (assuming FDA sign-off). Merimepodib is a broad-spectrum antiviral that Vertex Pharmaceuticals (VRTX +1.5%) was originally developing for hepatitis C and psoriasis. See all stocks on the move » 413 people get BSGM breaking news and analysis by email alert Get email alerts on BSGM Recommended for you: DoctoRx Why Gilead's COVID Drug May Be The Magic Bullet We Need DoctoRx • Apr. 19, 2020 11:06 PM ET WG Investment Research Bank Of America: This Time Will Be Different, Seriously WG Investment Research • Apr. 23, 2020 8:10 PM ET Wall Street Breakfast Wall Street Breakfast: First Oil Major To Cut Dividend Wall Street Breakfast • Apr. 23, 2020 1:00 PM ET Robert & Sam Kovacs Johnson & Johnson: I Don't Want To Say I Told You So Robert & Sam Kovacs • Apr. 22, 2020 7:38 PM ET Eric Parnell, CFA The Bear Is On The Prowl Eric Parnell, CFA • Apr. 23, 2020 4:34 PM ET Comments 9 • Sort by Add Your Comment User 50681444 Comments27 | + Follow He is a very smart man , it would be smart fro Trump to own it. It was just proved to work better than the biggest hopeful Remdesivir in lab test! And , this is the kicker - This drug is in pill form which means that the patient can take it at home!!! This stock is going to blow up !!! Buy it while you can! 23 Apr 2020, 01:15 PMReply0Like Rudester Comments5178 | + Follow Anybody knows if Trump owns stock in this one, like he does in the largest maker of hydroxychloroquine? 23 Apr 2020, 11:32 AMReply1Like alphaseek2018 Comments1514 | + Follow Yes he has a $99 holdings in hydroxychloroquine maker. Give me a break! 23 Apr 2020, 12:55 PMReply1Like 6x7 bigelk Comments74 | + Follow Go INO go 23 Apr 2020, 11:20 AMReply0Like Oil Can Comments2739 | + Follow Good news. We need to move forward to clinical trials every drug that has potential against COVID-19. This will not be a one-and-done game; we need to have multiple candidates under evaluation until a clear winner - or winners - emerge. As I understand it, once we have a viable therapeutic, there is the possibility of accelerating vaccine trials. 23 Apr 2020, 10:46 AMReply0Like APB Comments881 | + Follow So this new treatment might be ready to go in 3 or 4 years? If 20% of the population does in fact already have antibodies for COVID 19 then it will probably be obsolete before it reaches phase 2 trial? 23 Apr 2020, 10:43 AMReply0Like Matt-Man Comments2983 | + Follow Getting safety profile would take year alone, maybe 7-10year if lucky. Also HydroChloroquine had great in-vintro results 23 Apr 2020, 10:58 AMReply1Like Todd Kenyon, CFA Comments326 | + Follow This is in a culture - so far from being approved. I could dump Clorox in that culture and it would kill the virus too. The hcq jury is still out as the negative results were just one crappy anecdotal study with many issues. As were the positive ones. We don’t know yet bottom line. 23 Apr 2020, 10:18 AMReply9Like LilKleine Comments18 | + Follow Some good news after the disappointing results from the hydroxychloroquine study 23 Apr 2020, 10:02 AMReply0Like RSS Feeds|Advertise with Us|Sitemap|About Us|Feedback|Careers|Contact Us Terms of Use|Privacy|Market Data Disclaimer|© 2020 Seeking Alpha GlobeNewswire Intrado Logo Search REGISTER SIGN IN ENGLISH FRANÇAIS CONTACT US GLOBAL DISTRIBUTION NEWSWIRE SERVICES REGULATORY FILINGS MEDIA PARTNERS Bio-Sig-Logo-horiz.jpg ViralClear Publishes Comparative In Vitro Data on Merimepodib and Remdesivir Activity Against the COVID-19 Novel Coronavirus in F1000 Research Email Print Friendly Share April 23, 2020 09:15 ET | Source: BioSig Technologies, Inc. Westport, CT, April 23, 2020 (GLOBE NEWSWIRE) -- Merimepodib is shown to decrease viral production of COVID-19 coronavirus more than remdesivir at clinically meaningful drug concentrations in pre-clinical testing. Article highlights recent work done in laboratory studies of COVID-19 with merimepodib at the Galveston National Laboratory at The University of Texas Medical Branch BioSig Technologies, Inc. (Nasdaq: BSGM) today announced that an article titled, “The IMPDH inhibitor merimepodib has similar antiviral activity against SARS-CoV-2 replication in vitro to the adenosine analogue remdesivir” was accepted by F1000 Research, an online peer-reviewed life sciences journal publishing program in biology and medicine. This manuscript is authored by Natalya Bukreyeva, Rachel A. Sattler, Emily K. Mantlo, John T. Manning, Cheng Huang and Slobodan Paessler of the UTMB Galveston National Laboratory and Dr. Jerome Zeldis of ViralClear Pharmaceuticals, Inc. (“ViralClear”) as a corresponding author. The article highlights emerging pre-clinical data generated under contract with Galveston National Laboratory at The University of Texas Medical Branch. The work was started with Trek Therapeutics and after merimepodib was acquired by ViralClear has been continued by ViralClear. “The concentrations of merimepodib used in the cell culture studies described in this article are achievable in humans by our oral solution formulation,” said Dr. Zeldis, Executive Chair and Founder of ViralClear Pharmaceuticals, Inc. “We are now exploring whether remdesivir and merimepodib are synergistic in this in vitro model. We look forward to starting our first clinical trial in COVID-19 patients upon receiving FDA clearance to commence with the proposed phase 2 trial.” Vero cells in tissue culture were pre-treated with two concentrations (2.5 and 5 µM) of either merimepodib or remdesivir for 4 hours before the SARS-CoV-2 coronavirus was added. The amount of virus released to the media was measured at baseline and 16 hours and 24 hours after infection. At 16 hours, a significant reduction in viral production was observed for both concentrations of merimepodib (1.9 log decrease in titer, p = 0.003 and 2.1 log decrease in titer, p = 0.001) but only at the higher concentration of remdesivir (1 log decrease in titer, p = 0.103 and 2.1-log decrease in titer, p = 0.001). At 24 hours both concentrations of both drugs significantly reduced viral production; however, 2.5 µM remdesivir reduced viral titer by 1.5 logs (p = 0.002) as compared to a decrease of 3.9 logs for 5 µM remdesivir (p < 0.001), whereas 2.5 and 2.7 log reductions (p = 0.001 and p < 0.001, respectively) were observed for the 2.5 and 5 µM concentrations of merimepodib. Merimepodib, a broad-spectrum anti-viral candidate, demonstrated strong activity against COVID-19 in cell cultures in laboratory testing and additional antiviral studies are underway. Merimepodib was previously in development as a treatment for chronic hepatitis C and psoriasis by Vertex Pharmaceuticals Incorporated (Vertex), with a 12 clinical trials conducted (including 315 chronic hepatitis C patients, 24 psoriasis patients, and 98 healthy volunteers) and an extensive preclinical safety package completed . ViralClear intends to pursue development of this agent for the treatment of COVID-19 through FDA-approved clinical trials in Q2 2020. About BioSig Technologies BioSig Technologies is a medical technology company commercializing a proprietary biomedical signal processing platform designed to improve signal fidelity and uncover the full range of ECG and intra-cardiac signals (www.biosig.com). The Company’s first product, PURE EP(tm) System is a computerized system intended for acquiring, digitizing, amplifying, filtering, measuring and calculating, displaying, recording and storing of electrocardiographic and intracardiac signals for patients undergoing electrophysiology (EP) procedures in an EP laboratory. About ViralClear BioSig’s subsidiary ViralClear Pharmaceuticals, Inc., is seeking to develop a novel pharmaceutical to treat advanced COVID-19. Merimepodib is a broad-spectrum anti-viral agent that has demonstrated strong activity against the COVID-19 virus in cell cultures in laboratory testing. ViralClear plans to initiate a multi-center, phase 2, randomized, double-blind, placebo-controlled study of the efficacy and safety of merimepodib administered orally every eight hours for 10 days in adult patients with advanced COVID-19 upon FDA clearance to proceed. Merimepodib has been studied in twelve clinical trials prior to this study, including five trials in patients with hepatitis C (one phase 1b, one phase 2, two phase 2a, and one phase 2b), one trial in patients with psoriasis (phase 2), and six trials in healthy volunteers (all phase 1). Forward-looking Statements This press release contains “forward-looking statements.” Such statements may be preceded by the words “intends,” “may,” “will,” “plans,” “expects,” “anticipates,” “projects,” “predicts,” “estimates,” “aims,” “believes,” “hopes,” “potential” or similar words. Forward- looking statements are not guarantees of future performance, are based on certain assumptions and are subject to various known and unknown risks and uncertainties, many of which are beyond the Company’s control, and cannot be predicted or quantified and consequently, actual results may differ materially from those expressed or implied by such forward-looking statements. Such risks and uncertainties include, without limitation, risks and uncertainties associated with (i) the geographic, social and economic impact of COVID-19 on our ability to conduct our business and raise capital in the future when needed, (ii) our inability to manufacture our products and product candidates on a commercial scale on our own, or in collaboration with third parties; (iii) difficulties in obtaining financing on commercially reasonable terms; (iv) changes in the size and nature of our competition; (v) loss of one or more key executives or scientists; and (vi) difficulties in securing regulatory approval to market our products and product candidates. More detailed information about the Company and the risk factors that may affect the realization of forward-looking statements is set forth in the Company’s filings with the Securities and Exchange Commission (SEC), including the Company’s Annual Report on Form 10-K and its Quarterly Reports on Form 10-Q. Investors and security holders are urged to read these documents free of charge on the SEC’s website at http://www.sec.gov. The Company assumes no obligation to publicly update or revise its forward-looking statements as a result of new information, future events or otherwise. Andrew Ballou BioSig Technologies, Inc. Vice President, Investor Relations 54 Wilton Road, 2nd floor Westport, CT 06880 aballou@biosigtech.com 203-409-5444, x133 Related Articles More articles issued by BioSig Technologies, Inc. More articles related to: Company Announcement Profile BioSig Technologies, Inc. Subscribe via RSS Subscribe via ATOM Javascript Westport, Connecticut, UNITED STATES Contact Data Andrew Ballou BioSig Technologies, Inc. Vice President, Investor Relations 54 Wilton Road, 2nd floor Westport, CT 06880 aballou@biosigtech.com 203-409-5444, x133 Contact Biosig Technologies (BSGM) Intraday 3 Month 6 Month 1 Year Last: $7.50 Change: +0.30 % Change: +4.17% Volume: 1M Open: $7.14 Previous Close: $7.20 Low: $7.14 High: $8.32 Market Cap: 0 Shares Outs.: 0 EPS: 0.00 P/E Ratio: - Dividend: N/A Yield: N/A 52 Week Low: $2.36 52 Week High: $9.97 Quote delayed at least 20 mins. Media Files Bio-Sig-Logo-horiz.jpg Logo Download Options Bio-Sig-Logo-horiz.jpg LOGO URL | Copy the link below http://www.globenewswire.com/attachment-download-view/2021020/564767/4/1/0/564767.jpg Tags HEALTHMEDICALelectrophysiologystocknasdaqdeviceinvestorarrhythmiaAfibVTheartEPscienceablationbioelectronic medicine Related Links https://www.biosig.com/ Newswire Distribution Network & Management Home Newsroom RSS Feeds Legal About Us GlobeNewswire is one of the world's largest newswire distribution networks, specializing in the delivery of corporate press releases financial disclosures and multimedia content to the media, investment community, individual investors and the general public. © 2020 GlobeNewswire, Inc. All Rights Reserved. Trinidad and Tobago Newsday News Latest Sports Sections N Touch NEWS Hypertension patients, diabetics at high risk for covid 19 RYAN HAMILTON-DAVIS 4 HRS AGO File photo - File photo - The elderly are not the only people especially at risk for contracting covid19. People with hypertension, diabetes and kidney disease are also at high risk. So said Dr Neil Bhagwandass, nephrologist at San Fernando Hospital, at the Wednesday's covid19 update at the Ministry of Health headquarters on Park Street in Port of Spain. He listed the categories of people especially susceptible to the disease, which also included people who have suffered strokes or who have had lung diseases. Trendingview all 1 Designer defends $75, $100 cloth masks after backlash NARISSA FRASER 2 SORT foils alleged murder plot SHANE SUPERVILLE 3 Two held in Barrackpore after chase, 2 on the run LAUREL V WILLIAMS 4 Deyalsingh: Suspected case at step-down facility SHANE SUPERVILLE 5 Judge orders couple’s release JADA LOUTOO 6 ‘Stay at home’ challenged JADA LOUTOO 7 Brother, sister gunned down at home LAUREL V WILLIAMS Comments "Hypertension patients, diabetics at high risk for covid 19" More in this section NEWS Msgr Pereira: Spread Muslim fast to other religions YVONNE WEBB NEWS Corporations get $1m each to fight covid19 YVONNE WEBB NEWS CoP calls on public: Give info to reduce murders SHANE SUPERVILLE NEWS Attorney: People may sue over roadblocks SEAN DOUGLAS Trinidad and Tobago Newsday© 2020 Trinidad and Tobago Newsday. N Touch About Us Staff Classifieds Advertise Privacy Policy Contact Us Share to Facebook Share to WhatsApp Share to Twitter Share to Email Share to Print More AddThis Share options Skip to main content Thursday 23 April, 2020 Loop Market Trinidad And Tobago Home Search T&T News World News COVID-19 Carnival Viral Entertainment Lifestyle Business Sport BREAKING NEWS | Over 300,000 with hypertension at higher COVID-19 risk, says Minister ALINA DOODNATH CREATED : 23 APRIL 2020COVID-19[iStock.com/FatCamera] [iStock.com/FatCamera] Thousands of Trinidadians with diabetes and hypertension are at a higher risk due to the novel coronavirus (COVID-19), which if it spreads, ‘could be a disaster’, according to the Health Ministry. Speaking at Thursday’s media briefing to update the public on COVID-19 measures, Health Minister Terrence Deyalsingh said 341,000 people were diagnosed with hypertension, according to Ministry statistics – 23.6 percent of the population. He added that 266,000 (20.5 percent) of the population suffer from diabetes, saying that the figure is probably an underestimation as many people may not be aware they have diabetes. Dr Neal Bhagwandass, Nephrologist and Acting Head of Department, Medicine, at the San Fernando General Hospital, said it is critical that dialysis patients do not contract or spread the virus when receiving treatment. He said there are approximately 1,400 patients receiving hemodialysis treatments through the public health care system. "A typical dialysis would have 20-30 patients dialysing at one time. Our nephrologists have instituted policies so that patients are not allowed to congregate. Patients should be kept at least six feet apart and as soon as the dialysis procedure is finished they are to leave the environment." He said dialysis patients are screened on entering and are asked about any flu-like systems. "If we have to swab, swab them because this is a very high-risk population...we do not want COVID-19 to spread among our dialysis population, that could be a disaster." Dr Bhagwandass said those patients can experience kidney failure if they were to contract the virus. Dr Bhagwandass said diabetic patients should not work if they are feeling unwell. “Any stress on a diabetic patient would predispose you to your diabetes going out of control. These people should practice ‘sick day’ guidelines. If you become unwell, you develop…flu-like symptoms, your blood sugar would spike. “The rule is not to stop your diabetic medicine…get in contact with your health professionals for further advice.” Also concerning is the fact that young people who tested positive for the virus in other countries have been seen to develop abnormal clotting due to the virus. “Even young people who do not manifest these (diseases), are being diagnosed with strokes, cardiovascular accidents, because of increased clotting that occurs when one develops a COVID-19 infection.” People with the following conditions are at higher risk of severe COVID-19 complications: - Asthmatics - Chronic lung disease - Diabetes - Serious heart conditions - Chronic kidney disease/Dialysis - Severe obesity - Elderly (65 years and older) - Carehome residents - Immunocompromised - Liver disease Deyalsingh said the Ministry has implemented several measures to reduce the prevalence of non-communicable diseases including the banning of soft drinks at public schools, which he said has reduced soft drink consumption among children by 25 percent. He added that over the past year there was a drop in visits to the Accident and Emergency departments by 1.5 to two percent for people with hypertension and diabetes. Get the latest local and international news straight to your mobile phone for free: DIALYSIS DIABETES HYPERTENSION MINISTRY HEALTH DR ROSHAN PARASRAM DR NEAL BHAGWANDASS TRINIDAD CORONAVIRUS COVID-19 Related articles Comments Share this article Share to FacebookShare to TwitterShare to LinkedInShare to WhatsAppShare to MessengerShare to EmailShare to TelegramShare to More124 Trending Recent News Loop News Theme SECTIONS T&T NewsWorld NewsCOVID-19CarnivalViralEntertainmentLifestyleBusinessSport LOOP NEWS Privacy PolicyComments PolicyAbout UsContact UsSubmit an Article NEWSLETTER Email COPYRIGHT 2017 BY TREND MEDIA | ALL RIGHTS RESERVED Electropages icon Electropages uses cookies to give you the best experience. I'm fine with thisInformation and Settings Electropages logo Home Insights Product News NPI Channels Search Term Magnifying glass icon Tetherless pulse oximeter can monitor COVID-19 patients at home 23-04-2020 | Nordic | Test & Measurement Nordic Semiconductor is meeting demand for its high-end Bluetooth SoC. This lies at the heart of an industry-leading, battery-powered wireless pulse oximeter, the Radius PPG, that was launched last year by Masimo and is now set to join the front-line battle against COVID-19 as part of that company's SafetyNet solution. As the Radius PPG is created to enable tetherless monitoring, it can be used as part of the SafetyNet platform to remotely manage COVID-19 patients at home and conceivably assist hospitals from being overwhelmed by less severe cases. "We didn't originally develop Radius PPG to combat COVID-19," says Bilal Muhsin, chief operating officer, Masimo. "We first and foremost developed it so it could monitor hospital patients accurately, even when the patient is moving, without the inconvenience of a wired connection to a monitor. But it turns out that in the current COVID-19 crisis this tetherless capability, when paired with a secure, cloud-based telehealth platform accessible from a smartphone for patients and from a web-based dashboard for providers, can provide a valuable way to manage COVID-19 patients who do not require hospitalisation. And continue to check them from their homes, allowing hospital beds to be freed up for those who need it. Masimo SafetyNet uses the same Radius PPG tetherless pulse oximeter designed for use in the hospital to provide a remote patient management solution small enough to be worn comfortably and simple enough to be set-up quickly on the patient's smartphone." "Our hearts go out to the brave front line medical and hospital personnel doing their absolute best to cope with the often overwhelming numbers of COVID-19 patients coming through their doors in COVID-19 hotspots around the world," concludes Geir Langeland, Nordic Semiconductor director of sales and marketing. Nordic's latest multiprotocol SoC is its most advanced ultra low power wireless solution. The SoC supports complex Bluetooth LE and other wireless applications that were previously not possible with a single-chip solution. Nordic logo Facebook icon Twitter icon LinkedIn icon Email icon Print icon Related product news Next generation proximity sensors for GAPS and HAPS 22-04-2020 | Test & Measurement Motion detector aids designers in adding different technologies 21-04-2020 | Test & Measurement New sensor solution for ventilators 20-04-2020 | Test & Measurement Cost-effective MEMS sensors give increased performance and reliability 17-04-2020 | Test & Measurement Information Contact Us About Archive Submit PR Advertise Manage newsletter Subscribe Manage Subscription RSS feeds Our apps Apple icon Android icon Social profiles Facebook icon Twitter icon All news & general enquiries: +44 (0)1202 237137 info@electropages.com Copyright © 2020 Electropages Media Ltd Cookie Policy | Privacy Policy Electropages Media logohttp://www.audentia-gestion.fr/0-SOMMAIRE/0-SOMMAIRE-02AISNE.htm
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