MAT 7003 : Mathematical Foundations (for Software Engineering) J Paul Gibson, A207

MAT 7003 : Mathematical Foundations (for Software Engineering) J Paul Gibson, A207 paul.gibson@it-sudparis.eu http://www-public.it-sudparis.eu/~gibson/Teaching/MAT7003/ ProofsWith RODIN http://www-public.it-sudparis.eu/~gibson/Teaching/MAT7003/L8-ProofsWithRodin.pdf TSP: MSC SAI Mathematical Foundations

Working with RODIN: different proof techniques Proof by exhaustion, establishes the conclusion by dividing it into a finite number of cases and proving each one separately. Proof by contradiction (reductio ad absurdum) - it is shown that if some statement were true then a logical contradiction occurs, hence the statement must be false. Proof by transposition (contrapositive) establishes the conclusion "if p then q" by proving the equivalent statement "if not q then not p". Proof by mathematical induction establishes a "base case" and then an "induction rule" is used to prove a series of, possibly infinite, other cases Proof by construction, or proof by example, is the construction of a concrete example with a property to show that something having that property exists A nonconstructive proof establishes that a certain mathematical object must exist without explaining how such an object can be found. Often, this uses a proof by contradiction in which the nonexistence of the object is proven to be impossible. TSP: MSC SAI Mathematical Foundations

The proving perspective (Rodin User Manual) http://wiki.event-b.org/index.php/The_Proving_Perspective_(Rodin_User_Manual) TSP: MSC SAI Mathematical Foundations

The proving perspective (Rodin User Manual) http://wiki.event-b.org/index.php/The_Proving_Perspective_(Rodin_User_Manual) Decoration The leaves of the tree are decoratedwith one of threeicons: • meansthatthisleafisdischarged, • meansthatthisleafis not discharged, • meansthatthisleaf has been reviewed. TSP: MSC SAI Mathematical Foundations

The proving perspective (Rodin User Manual) http://wiki.event-b.org/index.php/The_Proving_Perspective_(Rodin_User_Manual) Proof Control View TSP: MSC SAI Mathematical Foundations

The proving perspective (Rodin User Manual) http://wiki.event-b.org/index.php/The_Proving_Perspective_(Rodin_User_Manual) SearchHypothesesView TSP: MSC SAI Mathematical Foundations

Example 1: odd and evenintegers How wouldyouspecify the sets of odd and evenintegers? Whatinterestingpropertiesshouldwebe able to prove? Does the structure of the specification help/hinder the proof process? Wecan examine how to do thisusing Rodin TSP: MSC SAI Mathematical Foundations

OddEven : proposed solution 1 Q: Can youexplain the axioms and theorems ? TSP: MSC SAI Mathematical Foundations

OddEven 1: proving 2 iseven Whycan’t the tool do thisautomatically? Interactive proof – the red bits provide interaction points TSP: MSC SAI Mathematical Foundations

OddEven 1: proving 2 iseven A good startis to simplify by removing the axiomsthat are not relevant in the proof TSP: MSC SAI Mathematical Foundations

OddEven 1: proving 2 iseven We know 2 isevenbecause 2 = 1 + 1 … soweneed to tell the tool by using the forallaxiom. But wecanseparate the <=> as weonlyneedit in 1 direction. This rewrites the equivalenceas 2 implications TSP: MSC SAI Mathematical Foundations

OddEven 1: proving 2 iseven NOTE: The proof treeisupdated Which of twoforallaxioms do we no longer need? TSP: MSC SAI Mathematical Foundations

OddEven 1: proving 2 iseven Now, wewant to instantiate x with the value 2 and apply modus ponens (by clicking on the =>) This gives a goal whichisimmediatelyprovable by instantiation of y to 1 TSP: MSC SAI Mathematical Foundations

OddEven 1: proving 2 iseven Now, dont forget to save the proof TSP: MSC SAI Mathematical Foundations

OddEven 1: proving 4 iseven Follow the samereasoning as for proving 2 iseven TSP: MSC SAI Mathematical Foundations

OddEven 1: proving 3 isodd The goal seemsobvious, but whyisit not provenautomatically? In order not to waste time wecan mark it as reviewed TSP: MSC SAI Mathematical Foundations

OddEven 1: proving 3 isodd TSP: MSC SAI Mathematical Foundations

OddEven 1: proving 5 isodd Wecan do the same for 5 TSP: MSC SAI Mathematical Foundations

OddEven 1: provingeven+even = even Can you do the proof yourselves? TSP: MSC SAI Mathematical Foundations

OddEven : proposed solution 2 Q: Can youexplain the axioms and theorems ? Think about why certain are more easilyproventhanothers … try to prove axm5 and review axiom7 TSP: MSC SAI Mathematical Foundations

OddEven : proposed solution 3 Q: Can youexplain the axioms and theorems ? Think about why certain are more easilyproventhanothers … try to prove axm10 TSP: MSC SAI Mathematical Foundations

OddEven : proposed solution 3 Westart the proof by considering the simplest cases where a=0 or b = 0 … dc a = 0 dc b = 0 TSP: MSC SAI Mathematical Foundations

OddEven : proposed solution 3 Wecanthenaddhypotheses to help in the proof QUESTION: But, are wemissingsomethingcritical? TSP: MSC SAI Mathematical Foundations

Arrays in Event-B Some of youasked about specifyingarrays. These are simply a functionfrominteger indexes to arrayelement values TSP: MSC SAI Mathematical Foundations

Another Event-B Example : PurseBehaviour TSP: MSC SAI Mathematical Foundations

Another Event-B Example : PurseBehaviour Modelling a change of state to a Purse: adding a coin Question: canyou model the removal of a coin? TSP: MSC SAI Mathematical Foundations

MAT 7003 : Mathematical Foundations (for Software Engineering) J Paul Gibson, A207