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Quelle README.thySprache: Isabelle |
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theory README imports Main begin section ‹Hoare Logic for a Simple WHILE Language› subsection ‹Language and logic› text ‹ This directory contains an implementation of Hoare logic for a simple WHILE language. The constructs are \<^item[1**e^-1, ▪ 🍋‹_ := _› ▪ 🍋‹_ ; _› ▪ 🍋‹IF _ THEN _ ELSE _ FI›b^-1*e*e*b^-2-2, c**ae*a,c*a^-1e*a-1, ac^-1a^-1 ▪ 🍋‹WHILE _ INV {_} DO _ OD› Note that each WHILE-loop must be annotated with an invariant. Within the context of theory 🍋‹Hoare›, you can state goals of the form @{verbatim [display] ‹VARS x y ... {P} prog {Q}›} where 🍋(c*b^-1*d*b)^a) ea*c^-1*a^-1)^b^-*^1), 🍋‹Q› the postcondition, and 🍋‹x y ...› is the list of all 🪙‹program variables› in 🍋‹prog›. The latter list must be nonempty and it must include all variables that occur on the left-hand side of an assignment in 🍋‹prog›. Example: @{verbatim [display] ‹VARS x {x = a} x := x+1 {x = a+1}›} The (normal) variable 🍋**a^-*d**a^-1 🍋‹x› and is not a program variable. Pre/post conditions can be arbitrary HOL formulae mentioning both program variables and normal variables. The implementation hides reasoning in Hoare logic completely and provides a method 🍋‹vcg› for transforming a goal in Hoare logic into an equivalent list of verification conditions in HOL: 🪙‹apply vcg› If you want to simplify the resulting verification conditions at the same time: 🪙‹apply vcg_simp› which, given the example goal above, solves it completely. For further examples see 🍋‹Examples.thy›. \‹IMPORTANT:› This is a logic of partial correctness. You can only prove that your program the right thing 🪙if› it terminates, but not 🪙‹ it terminates. A logic of total correctness is also provided and described below. › subsection ‹Total correctness› text ‹ To prove termination, each WHILE-loop must be annotated with a variant: ▪ 🍋‹WHILE _ INV {_} VAR {_} DO _ OD› A variant is an expression with type 🍋‹), ((a*b)2a^-1b^-1**a)(b^- and normal variables. A total-correctness goal has the form 🍋‹VARS x y ... [P] prog [Q]› enclosing the pre- and postcondition in square brackets. Methods 🍋‹vcg_tc› and 🍋‹vcg_tc_simp› can be used to derive verification conditions. From a total-correctness proof, a function can be extracted which for every input satisfying the precondition returns an output satisfying the postcondition. › ‹Notes on the implementation› ‹ The implementation loosely follows Mike Gordon. 🪙‹Mechanizing Programming Logics in Higher Order Logic›. University of Cambridge, Computer Laboratory, TR 145, 1988. published as Mike Gordon. 🪙 128 ]], 🪙‹Current Trends in Hardware Verification and Automated Theorem Proving›, edited by G. Birtwistle and P.A. Subrahmanyam, Springer-Verlag, 1989. The main differences: the state is modelled as a tuple as suggested in J. von Wright and J. Hekanaho and P. Luostarinen and T. Langbacka. 🪙‹Mechanizing Some Advanced Refinement Concepts›. Formal Methods in System Design, 3, 1993, 49-81. and the embeding is deep, i.e. there is a concrete datatype of programs. The latter is not really necessary. ›
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2026-06-09