2009: j paul gibsont&msp-csc 4504 : langages formels et applications event-b/oddeven.1 csc 4504...
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2009: J Paul Gibson T&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
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 Gibson T&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 Gibson T&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 Gibson T&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 Gibson T&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 Gibson T&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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1. The addition of two even numbers is even
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2009: J Paul Gibson T&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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1. The addition of two even numbers is even
repeat sequence for instantiation of b
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1. The addition of two even numbers is even
Don’t forget to save if you are making progress
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1. The addition of two even numbers is even
Add hypothesisand select for proofUsing autoprover
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1. The addition of two even numbers is even
Repeat the same sequence for b
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1. The addition of two even numbers is even
Free existential variables
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2009: J Paul Gibson T&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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1. The addition of two even numbers is even
Try autoprover on original theorem
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1. The addition of two even numbers is even
Instantiate with y+y+y0+y0Remove unnecessary hypothesesAdd new hypothesis:
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2009: J Paul Gibson T&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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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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1. The addition of two even numbers is even
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1. The addition of two even numbers is even
Save and explore proof tree
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2009: J Paul Gibson T&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 Gibson T&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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3. The multiplication of an even number with an odd number is even
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2009: J Paul Gibson T&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/DisablePost-tactic
Reset persepective
context1
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3. The multiplication of an even number with an odd number is even
Proof by contradiction(of goal)
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3. The multiplication of an even number with an odd number is even
Pruneand run Post-tactics
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3. The multiplication of an even number with an odd number is even
Proof by contradiction(of hypothesis)
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3. The multiplication of an even number with an odd number is even
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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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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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For Next Week: send to me (by email) before 20th March
Prove theorem 2 or theorem 4 for context 2