csc 4504 : langages formels et applications (la méthode event-b) j paul gibson, a207

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2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.1 applications (La méthode Event-B) J Paul Gibson, A207 [email protected] http://www-public.it-sudparis.eu/ ~gibson/Teaching/Event-B/ Proof p://www-public.it-sudparis.eu/~gibson/Teaching/Event-B/Proof. Thanks to Jean-Raymond Abrial

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CSC 4504 : Langages formels et applications (La méthode Event-B) J Paul Gibson, A207 [email protected] http://www-public. it-sudparis.eu /~gibson/Teaching/Event-B/. Proof http://www-public.it-sudparis.eu/~gibson/Teaching/Event-B/Proof.pdf. Thanks to Jean-Raymond Abrial. - PowerPoint PPT Presentation

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Page 1: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.1

CSC 4504 : Langages formels et applications

(La méthode Event-B)

J Paul Gibson, A207

[email protected]

http://www-public.it-sudparis.eu/~gibson/Teaching/Event-B/

Proof

http://www-public.it-sudparis.eu/~gibson/Teaching/Event-B/Proof.pdf

Thanks to Jean-Raymond Abrial

Page 2: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

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

Language of Predicates

Page 3: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.3

Language of Predicates: Classical Results

Page 4: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

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

Language of Predicates: Classical Results

Page 5: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.5

Language of Predicates: Classical Results

Page 6: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.6

Language of Predicates: Classical Results

Page 7: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.7

Language of Predicates: Classical Results

Page 8: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.8

Language of Predicates: Refining the language

Page 9: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.9

Predicates & Expressions

•A Predicate is a formal text that can be PROVED

•An Expression DENOTES AN OBJECT.

•A Predicate denotes NOTHING.

•An Expression CANNOT BE PROVED

•Predicates and Expressions are INCOMPATIBLE.

Page 10: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.10

VARIABLES, PROPOSITIONS AND PREDICATES

Page 11: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.11

WHAT CAN WE DO WITH A PREDICATE ?

Page 12: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.12

SUBSTITUTION

Page 13: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.13

UNIVERSAL QUANTIFICATION

Page 14: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.14

Well-formedness

Each occurrence of an identifier in a formula (that is a predicate or an expression)can be either free or bound.

Intuitively, a free occurrence of an identifierrefers to a declaration of that identifier in a scope outside of the formula,

while a bound occurrence corresponds to a local declaration introduced by a quantifierin the formula itself.

For a formula to be considered well-formed, we ask that, beyond being syntacticallycorrect, it also satisfies the two following conditions:

1. Any identifier that occurs in the formula, should have only free occurrencesor bound occurrences, but not both.2. Any identifier that occurs bound in the formula, should be bound in exactlyone place (i.e., by only one quantifier).

Page 15: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.15

Well-formedness: checking automatically

There are pages of rules for checking this on the abstract syntax of Event-B expressions.

For example:

Page 16: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.16

Type Checking

Type checking consists of checking, statically, that a formula is meaningful ina certain context.

For that, we associate a type with each expression that occurs in a formula.

This type is the set of all values that the expression can take.

Then, we check that the formula abides by some type checking rules.Those rules enforce that the operators used can be meaningful.

Unfortunately, type checking, as it is a static check, cannot by itself prove that a formula is meaningful.

For some operators, like integer division, we will also need to checksome additional dynamic constraints (e.g., that the denominator is not zero).

Page 17: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.17

Type Checking

A type denotes the set of values that an expression can take.

Moreover, we want this set to be derived statically, based on the form of the expression and the context in which it appears.

As a consequence, a type can take one of the three following forms:

Page 18: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.18

Type Checking

A type variable is a meta-variable that can denote any type.

We shall use lowercase Greek letters to denote type variables.

A typing environment represents the context in which a formula is to be type checked. A typing environment is a partial function from the set of all identifiers to the set of all possible types. For instance, the typing environment

Page 19: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.19

Type Checking - Rules

There are pages of rules for checking this on the abstract syntax of Event-B expressions.

For example:

Page 20: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.20

Type Checking – Rules (Example)

Page 21: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.21

Dynamic Checking

Static checks are not enough to ensure that a formula is meaningful.

For instance, expression x÷y passes all the static checks described above, nevertheless it is meaningless if y is zero.

The aim of dynamic checking is to detect these kind of meaningless formulas.

This is done by generating (and then proving) some well-definedness lemma.

Page 22: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.22

Dynamic Checking (Well-definedness WD)

WD lemmas for predicates

Page 23: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.23

Dynamic Checking (Well-definedness WD)

WD lemmas for binary and unary expressions

Page 24: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.24

Dynamic Checking (Well-definedness WD)

WD lemmas for other expressions

Page 25: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.25

Inference Rules: for (automated) reasoning

“Ich wollte zunächst einmal einen Formalismus aufstellen, der dem wirklichen Schließen möglichst nahe kommt. So ergab sich ein „Kalkül des natürlichen Schließens”.

(“First I wished to construct a formalism that comes as close as possible to actual reasoning. Thus arose a "calculus of natural deduction".)

— Gentzen, Untersuchungen über das logische Schließen (Mathematische Zeitschrift 39, pp.176-210, 1935)

Page 26: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.26

Inference Rules: for (automated) reasoning

nom

Antécédent

Conséquent

TabularNotation

Page 27: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.27

Inference Rules: for (automated) reasoning

Page 28: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.28

Inference Rules: for (automated) reasoning

Les règles d’inférence pour ^

Page 29: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.29

Inference Rules: for (automated) reasoning

Les règles d’inférence pour ^

Page 30: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.30

Inference Rules: for (automated) reasoning

Les règles d’inférence pour ¬

règles de contradiction ( “reductio ad absurdum”)

Page 31: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.31

Inference Rules: for (automated) reasoning

Une preuve ( à la main)

Page 32: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.32

Principe général d’un prouveur de prédicats

• On procède par induction sur la syntaxe du but P d’un séquent HYP |- P.règles appliqué en arrière (backward).

• On ne monte en hypothèse (utilisation de DED) que des– prédicats simples (pas de ^, =>, . . .) ou– prédicats quantifiés universellement (∀) et normalisés

• on s’arrête avec un axiome ou sur HYP |- FAUX en cherchant une contradictiondans les hypothèses.

• sinon on relance une preuve en cherchant de nouvelles instanciations pour lesvariables des prédicats quantifiés (filtre + unification).

Page 33: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.33

Règles d’un prouveur de prédicats (génériques)

opération générique

Page 34: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.34

Règles d’un prouveur de prédicats (^)

générique

Instantiation with /\

Page 35: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.35

Règles d’un prouveur de prédicats (=>)

générique

Tactique gagnante: On utilise =>4 en dernier

=>

Page 36: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.36

Règles d’un prouveur de prédicats (not)

générique

Page 37: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.37

Règles d’un prouveur de prédicats (les axioms)

Page 38: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.38

Les prédicats quantifiés

Page 39: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.39

Floyd Hoare Logic is a method of reasoning mathematically about imperative programs.

It is the basis of most mechanized program verification systems

Tony Hoare introduced the notation {P} C {Q}, called a partial correctness specification for specifying what a program does, where:

•C is a program (code) from the programming language whose programs are being specified •P and Q are conditions on the program variables used in C

Event-B is heavily influenced by Floyd-Hoare logic

Page 40: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.40

Meaning of Hoare's Notation

{P} C {Q} is true if

whenever C is executed in a state satisfying P and if the execution of C terminates then the state in which C terminates satisfies Q

Example: {X = 1} X := X + 1 {X = 2}

P is the condition that the value of X is 1 Q is the condition that the value of X is 2 C is the assignment command X := X + 1 (i.e. `X becomes X + 1')

{X = 1} X := X + 1 {X = 2} is clearly true {X = 1} X := X + 1 {X = 3} is clearly false

BE CAREUL with partial correctness:{X = 1} WHILE true do skip {Y=3} is true

Page 41: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.41

Total Correctness

Informally: Total correctness = Termination + Partial correctness

Total correctness is the ultimate goal It is usually easier to show partial correctness and termination separately

Termination is usually straightforward to show, but there are examples where it is not, e.g.: no one knows whether the program below terminates for all values of X

WHILE X > 1 DO IF ODD(X) THEN X := (3 × X) + 1 ELSE X := X DIV 2

Where the expression X DIV 2 evaluates to the result of rounding down X/2 to a whole number

Page 42: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.42

Specification can be Tricky

"The program must set Y to the maximum of X and Y" [True] C [Y = max(X, Y)]

A suitable program (C)?: IF X >= Y THEN Y := X ELSE SKIP

Another? IF X >= Y THEN X := Y ELSE SKIP

Or even? Y := X

WARNING: Later we will be able to prove that all these programs are "correct"

WHY?: The postcondition "Y = max(X, Y)" says "Y is the maximum of X and Y in the final state"

Page 43: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.43

SKIP: possibly the simplest axiomatisation

Syntax: SKIP Semantics: the state is unchanged

The SKIP Axiom :

|- {P} SKIP {P}

It is an axiom schema P can be instantiated with arbitrary predicate calculus formulae (statements)

Instances of the SKIP axiom are:

|- {Y = 2} SKIP {Y = 2} |- {True} SKIP {True} |- {R = X + (Y × Q)} SKIP {R = X + (Y ×Q)}

Page 44: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.44

Substitution Notation and assignment axiom: the most difficult axiomatisation

Define P [E/V ] to mean the result of replacing all occurrences of V in P by E •read P [E/V ] as `P with E for V ' •for example: (X + 1 > X)[Y + Z/X] = ((Y + Z) + 1 > Y + Z)

Think of this notation as the `cancellation law': V [E/V ] = E

which is analogous to the cancellation property of fractions: v × (e/v) = e

The Assignment Axiom |- {P [E/V ]} V := E {P}

Where V is any variable, E is any expression, P is any statement and the notation P [E/V ] denotes the result of substituting the term E for all occurrences of the variable V in the statement P .

Example:|- {X + 1 = n + 1} X := X + 1 {X = n + 1}

|- can be proven

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2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.45

Precondition Strengthening is a typical development step

Recall that |- S 1 , . . . , |- Sn

|-S means |- S can be deduced from |- S 1 , . . . , |- Sn

Using this notation, the rule of precondition strengthening is:

|- P => P’ , |-{P’} C {Q} |- {P} C {Q}

Page 46: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.46

Postcondition Weakening

Just as the previous rule allows the precondition of a partial correctness specification to be strengthened, the following one allows us to weaken the postcondition:

|- {P} C {Q’}, |- Q’ => Q |- {P} C {Q}

The rules precondition strengthening and postcondition weakening are sometimes called the rules of consequence

Page 47: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.47

Existential Quantification

Page 48: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.48

Comparing the Quantification Rules

Page 49: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.49

Classical Results

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2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.50

Classical Results

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2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.51

Refining our Language: Equality (with classical results)

Page 52: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.52

Refining our Language: Set Theory

Page 53: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.53

Basic Set Operator Memberships (Axioms)

Page 54: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.54

Set Inclusion and Extensionality Axiom

Page 55: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.55

Classical Results with Relation Operators

Relations (like r, q and p)between Sets(like S) containing elements (like a and b)

Page 56: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.56

Applying a Function

Page 57: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.57

Invariant Preservation

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2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.58

Invariant Preservation

Page 59: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.59

Invariant Preservation

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2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.60

Invariant Preservation: the rules

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2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.61

Invariant Preservation: the rules

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2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.62

Invariant Preservation: the rules

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2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.63

Deadlock Freedom

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2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.64

Event interpretation for refinement proofs

The execution of this event is enabled whenever there exist some values x and y suchthat the guard P is true, then z is assigned x+y

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2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.65

Event interpretation – Example

This event is always enabled (there always exists a natural number x > 10)The result of the event is that z is assigned an arbitrary natural number greater than 10.The event is equivalet to

Page 66: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.66

Refinement is used to transform an abstract machine into aconcrete machine which does the same computation, but possibly using a different data structure and/or different internal execution

can be refined (possibly) into

Also if in the abstract machine we have a nondeterministic event, then this could be refined into a deterministic one in the concrete machines :

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2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.72

Names of context proof obligations:

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2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.73

Names of machine proof obligations:

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2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.74

Names of refinement proof obligations:

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2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.75

Names of variant proof obligations:

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2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.76

Names of Witness proof obligations:

Page 77: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.77

Names of Deadlock Freeness proof obligations:

At the moment, the deadlock freeness proof obligation generation is incomplete.

If you need it, you can generate it yourself as a theorem saying the the disjunction of the abstract guards imply the disjunction of the concrete guards.

Page 78: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.78

The Automatic Post-tactic: Rewrite rules

The following rewrite rules are applied automatically in a systematic fashion from left to right either in the goal or in the selected hypotheses.

Page 79: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.79

The Automatic Post-tactic: Rewrite rules

Page 80: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.80

The Automatic Post-tactic: Rewrite rules

Page 81: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.81

The Automatic Post-tactic: Rewrite rules

Page 82: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.82

The Automatic Post-tactic: Rewrite rules

Page 83: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.83

The Automatic Post-tactic: Rewrite rules

Page 84: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.84

The Automatic Post-tactic: Rewrite rules

Page 85: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.85

The Automatic Post-tactic: Rewrite rules

Page 86: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.86

Automatic inference rules

The following inference rules are applied automatically in a systematic fashion at the end of each proof step. They have the following possible effects:

• they discharge the goal,• they simplify the goal and add a selected hypothesis,• they simplify the goal by decomposing it into several simpler goals,• they simplify a selected hypothesis,• they simplify a selected hypothesis by decomposing it into several simpler selected hypotheses.

Page 87: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.87

Automatic inference rules

Page 88: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.88

Automatic inference rules

Page 89: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.89

Automatic inference rules

Page 90: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.90

Automatic inference rules

Page 91: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.91

Automatic inference rules

Page 92: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.92

Preferences for the Auto-proverThe auto-prover can be configured by means of a preference page, which can be obtained as follows:press the ”Window” button on the top tooolbar. On the coming menu, press the ”Preferences” button. On the coming menu, press the ”Event-B” menue, then the ”Sequent Prover’, and finally the ”Auto-Tactic” button. This yields the following window:

Page 93: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.93

Interactive inference rules: through the red buttons in prover window

Page 94: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.94

Interactive inference rules: through the red buttons in prover window

… many more

Page 95: CSC 4504 : Langages formels et applications (La méthode Event-B) J  Paul  Gibson, A207

2009: J Paul Gibson T&MSP-CSC 4504 : Langages formels et applications Event-B/Proof.95

Interactive inference rules: through the red buttons in prover window