by S.Berardi and S. Valentini. The tutorial is by Enrico Tassi.
+The tutorial spends a considerable amount of effort in defining
+notations that resemble the ones used in the original paper. We believe
+this a important part of every formalization, not only for the estetic
+point of view, but also from the practical point of view. Being
+consistent allows to follow the paper in a pedantic way.
+
Orientering
-----------
- A ∪ B `A \cup B`
- A ∩ B `A \cap B`
+- A ≬ B `A \between B`
- x ∈ A `x \in A`
- Ω^A, that is the type of the subsets of A, `\Omega ^ A`
The first (technical) definition
--------------------------------
+Before defining the cover relation as an inductive predicate, one
+has to notice that the infinity rule uses, in its hypotheses, the
+cover relation between two subsets, while the inductive predicate
+we are going to define relates an element and a subset.
+
+An option would be to unfold the definition of cover between subsets,
+but we prefer to define the abstract notion of cover between subsets
+(so that we can attach a (ambiguous) notation to it).
+
+Anyway, to ease the understaing of the definition of the cover relation
+between subsets, we first define the inductive predicate unfolding the
+definition, and we later refine it with.
+
+DOCEND*)
+
+ninductive xcover (A : Ax) (U : Ω^A) : A → CProp[0] ≝
+| xcreflexivity : ∀a:A. a ∈ U → xcover A U a
+| xcinfinity : ∀a:A.∀i:𝐈 a. (∀y.y ∈ 𝐂 a i → xcover A U y) → xcover A U a.
+
+(*DOCBEGIN
+
+We defined the xcover (x will be removed in the final version of the
+definition) as an inductive predicate. The arity of the inductive
+predicate has to be carefully analyzed:
+
+> (A : Ax) (U : Ω^A) : A → CProp[0]
+The syntax separates with `:` abstractions that are fixed for every
+constructor (introduction rule) and abstractions that can change. In that
+case the parameter `U` is abstracted once and forall in front of every
+constructor, and every occurrence of the inductive predicate is applied to
+`U` in a consistent way. Arguments abstracted on the right of `:` are not
+constant, for example the xcinfinity constructor introduces `a ◃ U`,
+but under the assumption that (for every y) `y ◃ U`. In that rule, the left
+had side of the predicate changes, thus it has to be abstrated (in the arity
+of the inductive predicate) on the right of `:`.
+
+DOCEND*)
+
+(* ncheck xcreflexivity. *)
+
+(*DOCBEGIN
+
+We want now to abstract out `(∀y.y ∈ 𝐂 a i → xcover A U y)` and define
+a notion `cover_set` to which a notation `𝐂 a i ◃ U` can be attached.
+
+This notion has to be abstracted over the cover relation (whose
+type is the arity of the inductive `xcover` predicate just defined).
+
+Then it has to be abstracted over the arguments of that cover relation,
+i.e. the axiom set and the set U, and the subset (in that case `𝐂 a i`)
+sitting on the left hand side of `◃`.
DOCEND*)
ndefinition cover_set :
- ∀c: ∀A:Ax.Ω^A → A → CProp[0]. ∀A:Ax.∀C,U:Ω^A. CProp[0]
+ ∀cover: ∀A:Ax.Ω^A → A → CProp[0]. ∀A:Ax.∀C,U:Ω^A. CProp[0]
≝
- λc: ∀A:Ax.Ω^A → A → CProp[0]. λA,C,U.∀y.y ∈ C → c A U y.
+ λcover. λA, C,U. ∀y.y ∈ C → cover A U y.
-ndefinition cover_set_interactive :
- ∀c: ∀A:Ax.Ω^A → A → CProp[0]. ∀A:Ax.∀C,U:Ω^A. CProp[0].
-#cover; #A; #C; #U; napply (∀y:A.y ∈ C → ?); napply cover;
-##[ napply A;
-##| napply U;
-##| napply y;
-##]
-nqed.
+(*DOCBEGIN
+
+The `ndefinition` command takes a name, a type and body (of that type).
+The type can be omitted, and in that case it is inferred by the system.
+If the type is given, the system uses it to infer implicit arguments
+of the body. In that case all types are left implicit in the body.
+
+We now define the notation `a ◃ b`. Here the keywork `hvbox`
+and `break` tell the system how to wrap text when it does not
+fit the screen (they can be safely ignore for the scope of
+this tutorial). we also add an interpretation for that notation,
+where the (abstracted) cover relation is implicit. The system
+will not be able to infer it from the other arguments `C` and `U`
+and will thus prompt the user for it. This is also why we named this
+interpretation `covers set temp`: we will later define another
+interpretation in which the cover relation is the one we are going to
+define.
+
+DOCEND*)
-(* a \ltri b *)
notation "hvbox(a break ◃ b)" non associative with precedence 45
for @{ 'covers $a $b }.
interpretation "covers set temp" 'covers C U = (cover_set ?? C U).
+(*DOCBEGIN
+
+We can now define the cover relation using the `◃` notation for
+the premise of infinity.
+
+DOCEND*)
+
ninductive cover (A : Ax) (U : Ω^A) : A → CProp[0] ≝
-| creflexivity : ∀a:A. a ∈ U → cover ? U a
-| cinfinity : ∀a:A.∀i:𝐈 a. 𝐂 a i ◃ U → cover ? U a.
+| creflexivity : ∀a. a ∈ U → cover ? U a
+| cinfinity : ∀a. ∀i. 𝐂 a i ◃ U → cover ? U a.
+(** screenshot "cover". *)
napply cover;
nqed.
+(*DOCBEGIN
+
+Note that the system accepts the definition
+but prompts the user for the relation the `cover_set` notion is
+abstracted on.
+
+![The system asks for a cover relation][cover]
+
+The orizontal line separates the hypotheses from the conclusion.
+The `napply cover` command tells the system that the relation
+it is looking for is exactly our first context entry (i.e. the inductive
+predicate we are defining, up to α-conversion); while the `nqed` command
+ends a definition or proof.
+
+We can now define the interpretation for the cover relation between an
+element and a subset fist, then between two subsets (but this time
+we fixed the relation `cover_set` is abstracted on).
+
+DOCEND*)
+
interpretation "covers" 'covers a U = (cover ? U a).
-(* interpretation "covers set" 'covers a U = (cover_set cover ? a U). *)
+interpretation "covers set" 'covers a U = (cover_set cover ? a U).
+
+(*DOCBEGIN
+
+We will proceed similarly for the fish relation, but before going
+on it is better to give a short introduction to the proof mode of Matita.
+We define again the `cover_set` term, but this time we will build
+its body interactively. In λ-calculus Matita is based on, CIC, proofs
+and terms share the same syntax, and it thus possible to use the
+commands devoted to build proof term to build regular definitions.
+
+DOCEND*)
+
+
+ndefinition xcover_set :
+ ∀c: ∀A:Ax.Ω^A → A → CProp[0]. ∀A:Ax.∀C,U:Ω^A. CProp[0].
+(** screenshot "xcover-set-1". *)
+(*DOCBEGIN
+The system asks for a proof of the full statement, in an empty context.
+![xcover_set proof step ][xcover-set-1]
+The `#` command in the ∀-introduction rule, it gives a name to an
+assumption putting it in the context, and generates a λ-abstraction
+in the proof term.
+DOCEND*)
+#cover; #A; #C; #U; (** screenshot "xcover-set-2". *)
+(*DOCBEGIN
+![xcover_set proof step ][xcover-set-2]
+We have now to provide a proposition, and we exhibit it. We left
+a part of it implicit; since the system cannot infer it it will
+ask it later. Note that the type of `∀y:A.y ∈ C → ?` is a proposition
+whenever `?` is.
+DOCEND*)
+napply (∀y:A.y ∈ C → ?); (** screenshot "xcover-set-3". *)
+(*DOCBEGIN
+![xcover_set proof step ][xcover-set-3]
+The proposition we want to provide is an application of the
+cover relation we have abstracted in the context. The command
+`napply`, if the given term has not the expected type (in that
+case it is a product versus a proposition) it applies it to as many
+implicit arguments as necessary (in that case `? ? ?`).
+DOCEND*)
+napply cover; (** screenshot "xcover-set-4". *)
+(*DOCBEGIN
+![xcover_set proof step ][xcover-set-4]
+The system will now ask in turn the three implicit arguments
+passed to cover. The syntax `##[` allows to start a branching
+to tackle every sub proof individually, otherwise every command
+is applied to every subrpoof. The command `##|` switches to the next
+subproof and `##]` ends the branching.
+DOCEND*)
+##[ napply A;
+##| napply U;
+##| napply y;
+##]
+nqed.
+
+(*DOCBEGIN
+The definition of fish works exactly the same way as for cover, except
+that it is defined as a coinductive proposition.
+DOCEND*)
ndefinition fish_set ≝ λf:∀A:Ax.Ω^A → A → CProp[0].
λA,U,V.
interpretation "fish set" 'fish A U = (fish_set fish ? U A).
interpretation "fish" 'fish a U = (fish ? U a).
+(*DOCBEGIN
+
+Matita is able to generate elimination rules for inductive types,
+but not introduction rules for the coinductive case.
+
+DOCEND*)
+
+(* ncheck cover_rect_CProp0. *)
+
+(*DOCBEGIN
+
+We thus have to define the introduction rule for fish by corecursion.
+Here we again use the proof mode of Matita to exhibit the body of the
+corecursive function.
+
+DOCEND*)
+
nlet corec fish_rec (A:Ax) (U: Ω^A)
(P: Ω^A) (H1: P ⊆ U)
- (H2: ∀a:A. a ∈ P → ∀j: 𝐈 a. 𝐂 a j ≬ P):
- ∀a:A. ∀p: a ∈ P. a ⋉ U ≝ ?.
-#a; #p; napply cfish; (** screenshot "def-fish-rec". *)
-##[ napply H1; napply p;
-##| #i; ncases (H2 a p i); #x; *; #xC; #xP; @; ##[napply x]
- @; ##[ napply xC ] napply (fish_rec ? U P); nassumption;
+ (H2: ∀a:A. a ∈ P → ∀j: 𝐈 a. 𝐂 a j ≬ P): ∀a:A. ∀p: a ∈ P. a ⋉ U ≝ ?.
+(** screenshot "def-fish-rec-1". *)
+#a; #p; napply cfish;
+##[ (** screenshot "def-fish-rec-2". *) napply H1;
+ (** screenshot "def-fish-rec-3". *) napply p;
+##| (** screenshot "def-fish-rec-4". *) #i; ncases (H2 a p i);
+ (** screenshot "def-fish-rec-5". *) #x; *; #xC; #xP;
+ (** screenshot "def-fish-rec-5". *) @;
+ ##[ (** screenshot "def-fish-rec-6". *) napply x
+ ##| (** screenshot "def-fish-rec-7". *)
+ @; ##[ napply xC;
+ ##| (** screenshot "def-fish-rec-8". *)
+ napply (fish_rec ? U P);
+ nassumption;
+ ##]
+ ##]
##]
nqed.
-notation "◃U" non associative with precedence 55
-for @{ 'coverage $U }.
+(*DOCBEGIN
+![fish proof step][def-fish-rec-1]
+![fish proof step][def-fish-rec-2]
+![fish proof step][def-fish-rec-3]
+![fish proof step][def-fish-rec-4]
+![fish proof step][def-fish-rec-5]
+![fish proof step][def-fish-rec-6]
+![fish proof step][def-fish-rec-7]
+![fish proof step][def-fish-rec-8]
+DOCEND*)
ndefinition coverage : ∀A:Ax.∀U:Ω^A.Ω^A ≝ λA,U.{ a | a ◃ U }.
+notation "◃U" non associative with precedence 55 for @{ 'coverage $U }.
+
interpretation "coverage cover" 'coverage U = (coverage ? U).
ndefinition cover_equation : ∀A:Ax.∀U,X:Ω^A.CProp[0] ≝ λA,U,X.
Bla Bla,
-<div id="figure1" class="img" ><img src="figure1.png"/>foo</div>
DOCEND*)
--- /dev/null
+pre.sh_sourceCode {\r
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--- /dev/null
+/*\r
+SHJS - Syntax Highlighting in JavaScript\r
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+ if (name.toLowerCase() === htmlClasses[i].toLowerCase()) {\r
+ return;\r
+ }\r
+ }\r
+ htmlClasses.push(name);\r
+ element.className = htmlClasses.join(' ');\r
+}\r
+\r
+/**\r
+Extracts the tags from an HTML DOM NodeList.\r
+@param nodeList a DOM NodeList\r
+@param result an object with text, tags and pos properties\r
+*/\r
+function sh_extractTagsFromNodeList(nodeList, result) {\r
+ var length = nodeList.length;\r
+ for (var i = 0; i < length; i++) {\r
+ var node = nodeList.item(i);\r
+ switch (node.nodeType) {\r
+ case 1:\r
+ if (node.nodeName.toLowerCase() === 'br') {\r
+ var terminator;\r
+ if (/MSIE/.test(navigator.userAgent)) {\r
+ terminator = '\r';\r
+ }\r
+ else {\r
+ terminator = '\n';\r
+ }\r
+ result.text.push(terminator);\r
+ result.pos++;\r
+ }\r
+ else {\r
+ result.tags.push({node: node.cloneNode(false), pos: result.pos});\r
+ sh_extractTagsFromNodeList(node.childNodes, result);\r
+ result.tags.push({pos: result.pos});\r
+ }\r
+ break;\r
+ case 3:\r
+ case 4:\r
+ result.text.push(node.data);\r
+ result.pos += node.length;\r
+ break;\r
+ }\r
+ }\r
+}\r
+\r
+/**\r
+Extracts the tags from the text of an HTML element. The extracted tags will be\r
+returned as an array of tag objects. See sh_highlightString for the format of\r
+the tag objects.\r
+@param element a DOM element\r
+@param tags an empty array; the extracted tag objects will be returned in it\r
+@return the text of the element\r
+@see sh_highlightString\r
+*/\r
+function sh_extractTags(element, tags) {\r
+ var result = {};\r
+ result.text = [];\r
+ result.tags = tags;\r
+ result.pos = 0;\r
+ sh_extractTagsFromNodeList(element.childNodes, result);\r
+ return result.text.join('');\r
+}\r
+\r
+/**\r
+Merges the original tags from an element with the tags produced by highlighting.\r
+@param originalTags an array containing the original tags\r
+@param highlightTags an array containing the highlighting tags - these must not overlap\r
+@result an array containing the merged tags\r
+*/\r
+function sh_mergeTags(originalTags, highlightTags) {\r
+ var numOriginalTags = originalTags.length;\r
+ if (numOriginalTags === 0) {\r
+ return highlightTags;\r
+ }\r
+\r
+ var numHighlightTags = highlightTags.length;\r
+ if (numHighlightTags === 0) {\r
+ return originalTags;\r
+ }\r
+\r
+ var result = [];\r
+ var originalIndex = 0;\r
+ var highlightIndex = 0;\r
+\r
+ while (originalIndex < numOriginalTags && highlightIndex < numHighlightTags) {\r
+ var originalTag = originalTags[originalIndex];\r
+ var highlightTag = highlightTags[highlightIndex];\r
+\r
+ if (originalTag.pos <= highlightTag.pos) {\r
+ result.push(originalTag);\r
+ originalIndex++;\r
+ }\r
+ else {\r
+ result.push(highlightTag);\r
+ if (highlightTags[highlightIndex + 1].pos <= originalTag.pos) {\r
+ highlightIndex++;\r
+ result.push(highlightTags[highlightIndex]);\r
+ highlightIndex++;\r
+ }\r
+ else {\r
+ // new end tag\r
+ result.push({pos: originalTag.pos});\r
+\r
+ // new start tag\r
+ highlightTags[highlightIndex] = {node: highlightTag.node.cloneNode(false), pos: originalTag.pos};\r
+ }\r
+ }\r
+ }\r
+\r
+ while (originalIndex < numOriginalTags) {\r
+ result.push(originalTags[originalIndex]);\r
+ originalIndex++;\r
+ }\r
+\r
+ while (highlightIndex < numHighlightTags) {\r
+ result.push(highlightTags[highlightIndex]);\r
+ highlightIndex++;\r
+ }\r
+\r
+ return result;\r
+}\r
+\r
+/**\r
+Inserts tags into text.\r
+@param tags an array of tag objects\r
+@param text a string representing the text\r
+@return a DOM DocumentFragment representing the resulting HTML\r
+*/\r
+function sh_insertTags(tags, text) {\r
+ var doc = document;\r
+\r
+ var result = document.createDocumentFragment();\r
+ var tagIndex = 0;\r
+ var numTags = tags.length;\r
+ var textPos = 0;\r
+ var textLength = text.length;\r
+ var currentNode = result;\r
+\r
+ // output one tag or text node every iteration\r
+ while (textPos < textLength || tagIndex < numTags) {\r
+ var tag;\r
+ var tagPos;\r
+ if (tagIndex < numTags) {\r
+ tag = tags[tagIndex];\r
+ tagPos = tag.pos;\r
+ }\r
+ else {\r
+ tagPos = textLength;\r
+ }\r
+\r
+ if (tagPos <= textPos) {\r
+ // output the tag\r
+ if (tag.node) {\r
+ // start tag\r
+ var newNode = tag.node;\r
+ currentNode.appendChild(newNode);\r
+ currentNode = newNode;\r
+ }\r
+ else {\r
+ // end tag\r
+ currentNode = currentNode.parentNode;\r
+ }\r
+ tagIndex++;\r
+ }\r
+ else {\r
+ // output text\r
+ currentNode.appendChild(doc.createTextNode(text.substring(textPos, tagPos)));\r
+ textPos = tagPos;\r
+ }\r
+ }\r
+\r
+ return result;\r
+}\r
+\r
+/**\r
+Highlights an element containing source code. Upon completion of this function,\r
+the element will have been placed in the "sh_sourceCode" class.\r
+@param element a DOM <pre> element containing the source code to be highlighted\r
+@param language a language definition object\r
+*/\r
+function sh_highlightElement(element, language) {\r
+ sh_addClass(element, 'sh_sourceCode');\r
+ var originalTags = [];\r
+ var inputString = sh_extractTags(element, originalTags);\r
+ var highlightTags = sh_highlightString(inputString, language);\r
+ var tags = sh_mergeTags(originalTags, highlightTags);\r
+ var documentFragment = sh_insertTags(tags, inputString);\r
+ while (element.hasChildNodes()) {\r
+ element.removeChild(element.firstChild);\r
+ }\r
+ element.appendChild(documentFragment);\r
+}\r
+\r
+function sh_getXMLHttpRequest() {\r
+ if (window.ActiveXObject) {\r
+ return new ActiveXObject('Msxml2.XMLHTTP');\r
+ }\r
+ else if (window.XMLHttpRequest) {\r
+ return new XMLHttpRequest();\r
+ }\r
+ throw 'No XMLHttpRequest implementation available';\r
+}\r
+\r
+function sh_load(language, element, prefix, suffix) {\r
+ if (language in sh_requests) {\r
+ sh_requests[language].push(element);\r
+ return;\r
+ }\r
+ sh_requests[language] = [element];\r
+ var request = sh_getXMLHttpRequest();\r
+ var url = prefix + 'sh_' + language + suffix;\r
+ request.open('GET', url, true);\r
+ request.onreadystatechange = function () {\r
+ if (request.readyState === 4) {\r
+ try {\r
+ if (! request.status || request.status === 200) {\r
+ eval(request.responseText);\r
+ var elements = sh_requests[language];\r
+ for (var i = 0; i < elements.length; i++) {\r
+ sh_highlightElement(elements[i], sh_languages[language]);\r
+ }\r
+ }\r
+ else {\r
+ throw 'HTTP error: status ' + request.status;\r
+ }\r
+ }\r
+ finally {\r
+ request = null;\r
+ }\r
+ }\r
+ };\r
+ request.send(null);\r
+}\r
+\r
+/**\r
+Highlights all elements containing source code on the current page. Elements\r
+containing source code must be "pre" elements with a "class" attribute of\r
+"sh_LANGUAGE", where LANGUAGE is a valid language identifier; e.g., "sh_java"\r
+identifies the element as containing "java" language source code.\r
+*/\r
+function sh_highlightDocument(prefix, suffix) {\r
+ var nodeList = document.getElementsByTagName('pre');\r
+ for (var i = 0; i < nodeList.length; i++) {\r
+ var element = nodeList.item(i);\r
+ var htmlClasses = sh_getClasses(element);\r
+ for (var j = 0; j < htmlClasses.length; j++) {\r
+ var htmlClass = htmlClasses[j].toLowerCase();\r
+ if (htmlClass === 'sh_sourcecode') {\r
+ continue;\r
+ }\r
+ if (htmlClass.substr(0, 3) === 'sh_') {\r
+ var language = htmlClass.substring(3);\r
+ if (language in sh_languages) {\r
+ sh_highlightElement(element, sh_languages[language]);\r
+ }\r
+ else if (typeof(prefix) === 'string' && typeof(suffix) === 'string') {\r
+ sh_load(language, element, prefix, suffix);\r
+ }\r
+ else {\r
+ throw 'Found <pre> element with class="' + htmlClass + '", but no such language exists';\r
+ }\r
+ break;\r
+ }\r
+ }\r
+ }\r
+}\r
projects / helm.git / commitdiff