locales ISAR IS BASED ON CONTEXTS CONTENT Slide 3 Slide 1 proof - fix x assume Ass: A. x and Ass are visible Slide 2 Slide 4 inside this context

Transcription

1 LAST TIME Syntax and semantics of IMP Hoare logic rules NICTA Advanced Course Soundness of Hoare logic Slide 1 Theorem Proving Principles, Techniques, Applications Slide 3 Verification conditions Example program proofs locales CONTENT ISAR IS BASED ON CONTEXTS Slide 2 Intro & motivation, getting started with Isabelle Foundations & Principles Lambda Calculus Higher Order Logic, natural deduction Term rewriting Proof & Specification Techniques Inductively defined sets, rule induction Datatypes, recursion, induction Slide 4 theorem x. A = C proof - fix x assume Ass: A. x and Ass are visible qed from Ass show C... inside this context More recursion, Calculational reasoning Hoare logic, proofs about programs Locales, Presentation LAST TIME 1 BEYOND ISAR CONTEXTS 2

2 BEYOND ISAR CONTEXTS DECLARING LOCALES Locales are extended contexts Declaring locale (named context) loc: Slide 5 Locales are named Fixed variables may have syntax It is possible to add and export theorems Locale expression: combine and modify locales Slide 7 locale loc = loc1 + fixes... assumes... Import Context elements CONTEXT ELEMENTS Locales consist of context elements. DECLARING LOCALES Theorems may be stated relative to a named locale. Slide 6 fixes assumes defines notes includes Parameter, with syntax Assumption Definition Record a theorem Import other locales (locale expressions) Slide 8 lemma (in loc) P [simp]: proposition proof Adds theorem P to context loc. Theorem P is in the simpset in context loc. Exported theorem loc.p visible in the entire theory. DECLARING LOCALES 3 4

3 LOCALE EXPRESSIONS Locale name: Rename: n e q 1... q n Change names of parameters in e. Merge: e 1 + e 2 Slide 9 DEMO: LOCALES 1 Slide 11 Context elements of e 1, then e 2. Syntax is lost after rename (currently). PARAMETERS MUST BE CONSISTENT! Parameters in fixes are distinct. Free variables in assumes and defines occur in preceding fixes. Defined parameters cannot occur in preceding assumes nor defines. Slide 10 Slide 12 DEMO: LOCALES 2 LOCALE EXPRESSIONS 5 NORMAL FORM OF LOCALE EXPRESSIONS 6

4 NORMAL FORM OF LOCALE EXPRESSIONS Locale expressions are converted to flattened lists of locale names. With full parameter lists Duplicates removed Slide 13 Allows for multiple inheritance! Slide 15 DEMO: LOCALES 3 INSTANTIATION Move from abstract to concrete. instantiate label: loc Slide 14 From chained fact loc t 1... t n instantiate locale loc. Imports all theorems of loc into current context. Slide 16 PRESENTATION Instantiates the parameters with t 1... t n. Interprets attributes of theorems. Prefixes theorem names with label Currently only works inside Isar contexts. 7 ISABELLE S BATCH MODE 8

5 ISABELLE S BATCH MODE GENERATING L A T E X FROM ISABELLE <..>/isatool usedir -d pdf HOL <session> used to process and check larger number of theories no interactive niceties (no sorry, no quick and dirty) controlled by file ROOT.ML and script set isatool <..>/<session>/root.ml <..>/<session>/mytheory.thy <..>/<session>/document/root.tex Slide 17 can save state for later use (images) can generate HTML and L A T E X documentation Slide 19 In ROOT.ML: no\_document use_thy "MyLibrary"; use_thy "MyTheory"; In document/root.tex: include Isabelle style packages (isabelle.sty, isabellesym.sty) include generated files session.tex (for all theories) or MyTheory.tex ISATOOL isatool <tool> <options> Get help with: isatool shows available tools Slide 18 isatool <tool> -? Interesting tools: isatool mkdir shows options for <tool> create session directory Slide 20 DEMO: EXAMPLE make/makeall run make for directory/all logics usedir batch session (documents, HTML, session graph) document/latex run L A T E X for generated sources GENERATING L A T E X FROM ISABELLE 9 LARGE DEVELOPMENTS 10

6 LARGE DEVELOPMENTS ANTIQUOTATIONS Creating Images: Inside L A T E X you can go back to Isabelle commands and syntax. <..>/<session>/isatool usedir -b HOL <session> Useful Antiquotations: <..>/<session>/root.ml t} print type τ print term t Slide 21 Slide φ} print proposition φ Processes [display] φ} print proposition φ with linebreaks Saves state after processing in [source] a} check proposition φ, print its input print fact a Makes HOL-<session> available as logic in menu Isabelle Logics Direct start of Isabelle with new logic: Isabelle -l a [no [source] s} print fact a, fixing schematic variables check availability of a, print its name print uninterpreted text s MARKUP COMMANDS WRITING ABOUT ISABELLE THEORIES document structure commands: normal text header section subsection subsubsection (meaning defined in isabelle.sty) To document definitions and proofs: put comments explanations directly in original theory keep explanations short and to the point formal definitions, lemmas, syntax should speak for themself text {... } text raw {... } Slide 22 text inside proofs Slide 24 To write a paper/thesis about a formal development txt {... } txt raw {... } use a separate theory/document on top of the development formal comments only talk about the interesting parts -- {... } use antiquoations for theorems and definitions make text invisible: ( < )... ( > ) use extra locales, definitions, syntax for polish make full proof document available separately ANTIQUOTATIONS 11 POLISH 12

7 POLISH Know your audience. Use the right notation. Change L A T E X symbol interpretations \renewcommand{\isasymlongrightarrow} {\isamath{\longrightarrow}} Slide 25 Declare special L A T E X output syntax: syntax (latex) Cons :: a a list a list ( / [66,65] 65) Slide 27 DEMO Use translations to change output syntax: syntax (latex) notex :: ( a bool) bool (binder \<notex> 10) translations \<notex>x. P <= ( x. P) in document/root.tex: \newcommand{\isasymnotex}{\isamath{\neg\exists}} USING LOCALES WE HAVE SEEN TODAY... making large developments more accessible Math textbook: Locale Declarations + Theorems in Locales Let (A,, 0) in the following be a group with x y = y x Locale Expressions + Inheritance Isabelle: Locale Instantiation Slide 26 Use locales to formalize contexts Slide 28 Generating L A T E X Antiquotations are sensitive to current locale context Writing a thesis/paper in Isabelle Example: locale agroup = group + assumes com: x y = y x... ( < ) lemma (in agroup) True ( > ) txt {... } ( < ) oops ( > ) 13 EXERCISES 14

8 EXERCISES No Exercise Today Slide 29 Theorem Proving Principles, Techniques, Applications The End EXERCISES 15