(La méthode Event-B) OddEven. Express as theorems and use RODIN to prove: thm4. The multiplication of two odd numbers is odd

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1 CSC 4504 : Langages formels et applications (La méthode Event-B) J Paul Gibson, A207 paul.gibson@it-sudparis.eu OddEven : J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.1 Simple proofs on properties of odd and even numbers Express as theorems and use RODIN to prove: thm 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 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.2

2 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven : J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.4

3 Autoprovers 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.5 instantiate 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.6

4 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.7 Remove selected hypothesis 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.8

5 repeat sequence for instantiation of b 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.9 Don t forget to save if you are making progress 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.10

6 Add hypothesis and select for proof Using autoprover 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.11 Repeat the same sequence for b 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.12

7 Free existential variables 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.13 Remove hypotheses (no longer needed) 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.14

8 Try autoprover on original theorem 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.15 Instantiate with y+y+y0+y0 Remove unnecessary hypotheses Add new hypothesis: 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.16

9 Add abstract expression: y+y0 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.17 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 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.18

10 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.19 Save and explore proof tree 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.20

11 The theorem has been marked as proved 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.21 What if we change specification of EVEN? What if we change theorem to prove? lets try theorem 3 with context1 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.22

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

13 3. The multiplication of an even number with an odd number is even Proof by contradiction (of goal) 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven The multiplication of an even number with an odd number is even Prune and run Post-tactics 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.26

14 3. The multiplication of an even number with an odd number is even Proof by contradiction (of hypothesis) 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven The multiplication of an even number with an odd number is even 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.28

15 3. The multiplication of an even number with an odd number is even Circular proof steps: Try to simplify proof tree, where possible 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven The multiplication of an even number with an odd number is even Proof for context2 try to complete it yourselves 2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/OddEven.30

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