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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.1 CSC 4504 : Langages formels et applications (La méthode Event-B) J Paul Gibson, A207 paul.gibson@it-sudparis.eu http://www-public.it-sudparis.eu/~gibson/Teaching/Event-B/ OddEven http://www-public.it-sudparis.eu/~gibson/Teaching/Event-B/OddEven.pdf

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.2 Simple proofs on properties of odd and even numbers Express as theorems and use RODIN to prove: thm1. The addition of two even numbers is even thm2. The difference between two odd numbers is even thm3. The multiplication of an even number with an odd number is even thm4. The multiplication of two odd numbers is odd

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.3 1. The addition of two even numbers is even

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.4 1. The addition of two even numbers is even

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.5 1. The addition of two even numbers is even Autoprovers

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.6 1. The addition of two even numbers is even instantiate

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.7 1. The addition of two even numbers is even

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.8 1. The addition of two even numbers is even Remove selected hypothesis

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.9 1. The addition of two even numbers is even repeat sequence for instantiation of b

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.10 1. The addition of two even numbers is even Don’t forget to save if you are making progress

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.11 1. The addition of two even numbers is even Add hypothesis and select for proof Using autoprover

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.12 1. The addition of two even numbers is even Repeat the same sequence for b

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.13 1. The addition of two even numbers is even Free existential variables

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.14 1. The addition of two even numbers is even Remove hypotheses (no longer needed)

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.15 1. The addition of two even numbers is even Try autoprover on original theorem

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.16 1. The addition of two even numbers is even Instantiate with y+y+y0+y0 Remove unnecessary hypotheses Add new hypothesis:

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.17 1. The addition of two even numbers is even Add abstract expression: y+y0

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.18 1. The addition of two even numbers is even This sequence leads to the final proof step Setting y1 to ae proves the theorem: can the automated prover do this? p0 fails but p1 manages it

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.19 1. The addition of two even numbers is even

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.20 1. The addition of two even numbers is even Save and explore proof tree

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.21 1. The addition of two even numbers is even The theorem has been marked as proved

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.22 1. The addition of two even numbers is even What if we change specification of EVEN? What if we change theorem to prove? … lets try theorem 3 with context1

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.23 3. The multiplication of an even number with an odd number is even

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.24 3. The multiplication of an even number with an odd number is even Enable/ Disable Post-tactic Reset persepective context1

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.25 3. The multiplication of an even number with an odd number is even Proof by contradiction (of goal)

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.26 3. The multiplication of an even number with an odd number is even Prune and run Post-tactics

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.27 3. The multiplication of an even number with an odd number is even Proof by contradiction (of hypothesis)

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.28 3. The multiplication of an even number with an odd number is even

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.29 3. The multiplication of an even number with an odd number is even Circular proof steps: Try to simplify proof tree, where possible

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.30 3. The multiplication of an even number with an odd number is even Proof for context2 … try to complete it yourselves

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2009: J Paul GibsonT&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.31 For Next Week: send to me (by email) before 20th March Prove theorem 2 or theorem 4 for context 2

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