B Method Proof assistants May 16, 2017 Lucas Franceschino What is - - PowerPoint PPT Presentation

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B Method Proof assistants May 16, 2017 Lucas Franceschino What is - - PowerPoint PPT Presentation

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B Method Proof assistants May 16, 2017 Lucas Franceschino What is B method? B-method goal Actual Specifications program 3 B-method goal Actual Specifications program Machine 3 B-method goal Actual Specifications

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May 16, 2017

B Method

Proof assistants Lucas Franceschino

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What is B method?

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◎ B-method goal

Specifications Actual program

3

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◎ B-method goal

Specifications Actual program

 Machine

3

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◎ B-method goal

Specifications Actual program

 Machine  Refinement 1

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◎ B-method goal

Specifications Actual program

 Machine  Refinement 1  Refinement 2

3

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◎ B-method goal

Specifications Actual program

 Machine  Refinement 1  Refinement 2  Refinement 3

3

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◎ B-method goal

Specifications Actual program

 Machine  Refinement 1 C / ada  Refinement 2  Refinement 3

3

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◎ B-method goal

Specifications Actual program

 Machine  Refinement 1 C / ada  Refinement 2  Refinement 3

3

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◎ B-method goal

Specifications Actual program

No gap between specification specification and actual pr actual progr

  • gram

3

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♂ The initiator of B method

Jean-Raymond Abrial

1970

Specification of data structures and programs Initiated the Z-Notation (in Oxford)

  • G. Laffitte,
  • F. Mejia,
  • I. Mc Neal

B4free, Bart, ABTools…

4

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SLIDE 12

♂ The initiator of B method

Jean-Raymond Abrial

1970

Specification of data structures and programs Initiated the Z-Notation (in Oxford)

Good for formal specifications, not for development

  • G. Laffitte,
  • F. Mejia,
  • I. Mc Neal

4

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♂ The initiator of B method

Jean-Raymond Abrial

1970

Specification of data structures and programs Initiated the Z-Notation (in Oxford)

Good for formal specifications, not for development

1996

Published “The B Book” Atelier B

  • G. Laffitte,
  • F. Mejia,
  • I. Mc Neal

B4free, Bart, ABTools…

4

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SLIDE 14

♂ The initiator of B method

Jean-Raymond Abrial

1970

Specification of data structures and programs Initiated the Z-Notation (in Oxford)

Good for formal specifications, not for development

1996

Published “The B Book” Atelier B

2010

Published “Modeling in Event-B : system and software engineering” Rodin platform

  • G. Laffitte,
  • F. Mejia,
  • I. Mc Neal

B4free, Bart, ABTools…

4

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SLIDE 15

♂ The initiator of B method

Jean-Raymond Abrial

1970

Specification of data structures and programs Initiated the Z-Notation (in Oxford)

Good for formal specifications, not for development

1996

Published “The B Book” Atelier B Was in the development team of Ada

2010

Published “Modeling in Event-B : system and software engineering” Rodin platform

  • G. Laffitte,
  • F. Mejia,
  • I. Mc Neal

B4free, Bart, ABTools…

4

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Use cases

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 Train related B projects around the world

Braking system, platform screen doors…

6

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 Use case: Meteor in Paris (line 14)

110 000 lines of B  87 000 lines of Ada  29 000 lemmas 

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 Use case: Meteor in Paris (line 14)

Driverless trains  Extension in 2003  80 million passengers in 2009 

110 000 lines of B  87 000 lines of Ada  29 000 lemmas 

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 Use case: Meteor in Paris (line 14)

9.2 km . October 1998  Driverless trains  Extension in 2003  80 million passengers in 2009 

110 000 lines of B  87 000 lines of Ada  29 000 lemmas 

7

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 Use case: Meteor in Paris (line 14)

9.2 km . October 1998  No bugs discovered yet!  Still in version 1.0  Driverless trains  Extension in 2003  80 million passengers in 2009 

110 000 lines of B  87 000 lines of Ada  29 000 lemmas 

7

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Other use cases

  • Peugeot cars: formalization of sub systems (lights system,

airbags, motor) to help building diagnostic tools

  • Modeling of tasks scheduling from the software controlling the

stage separations of Ariane rocket

  • Protocol study
  • JavaCard runtime formalization

Java runtime for smartcard

Provide safe: authentication, data storage, application processing

8

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Developing in B

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B method development

Specification

Integration tests

Conception

Unit tests

Code Validation

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B method development

Specification

Integration tests

Conception

Unit tests

Code Validation

10

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SLIDE 26

B method development

Specification

Integration tests

Conception

Unit tests

Code Validation

10

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SLIDE 27

B method development

Specification

Integration tests

Conception

Unit tests

Code Validation

10

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SLIDE 28

B method development

Specification

Integration tests

Conception

Unit tests

Code Validation

10

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B method development

Specification

Integration tests

Conception

Unit tests

Code Validation

10

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How does method B works

 Module: modelling of a sub system

 Component

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How does method B works

 Module: modelling of a sub system

 Component Static part Static part Definition of: Variables, constants, sets List of invariants Dynamic part Dynamic part Initialize variables Define operations on variables Pr Proo

  • of

Static part coherence Initializing preserve invariants Operations preserve invariants

11

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How does method B works

 Module: modelling of a sub system  Abstract machine

 Component Static part Static part Definition of: Variables, constants, sets List of invariants Dynamic part Dynamic part Initialize variables Define operations on variables Pr Proo

  • of

Static part coherence Initializing preserve invariants Operations preserve invariants

11

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How does method B works

 Module: modelling of a sub system  Abstract machine  Refinements

 Component Static part Static part Definition of: Variables, constants, sets List of invariants Dynamic part Dynamic part Initialize variables Define operations on variables Pr Proo

  • of

Static part coherence Initializing preserve invariants Operations preserve invariants

11

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How does method B works

 Module: modelling of a sub system  Abstract machine  Refinements

 Component Static part Static part Definition of: Variables, constants, sets List of invariants Dynamic part Dynamic part Initialize variables Define operations on variables Pr Proo

  • of

Static part coherence Initializing preserve invariants Operations preserve invariants

 Implementation

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How does method B works

 Module: modelling of a sub system  Abstract machine  Refinements

 Component Static part Static part Definition of: Variables, constants, sets List of invariants Dynamic part Dynamic part Initialize variables Define operations on variables Pr Proo

  • of

Static part coherence Initializing preserve invariants Operations preserve invariants

 Implementation

We refine a previous component: we make it more precise and specific

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Machine, refinement, implementation

 Abstract machine  Refinements  Implementation

Substitutions

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SLIDE 37

Machine, refinement, implementation

 Abstract machine  Refinements  Implementation

Substitutions

12

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Machine, refinement, implementation

 Abstract machine  Refinements  Implementation

 Precondition  Choice  Become such that  Simultaneous operations  Sequencing  While loop  Let bindings Substitutions

12

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Machine, refinement, implementation

 Abstract machine  Refinements  Implementation

 Precondition  Choice  Become such that  Simultaneous operations  Sequencing  While loop  Let bindings

Predicated-based IF

Substitutions

12

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Machine, refinement, implementation

 Abstract machine  Refinements  Implementation

 Precondition  Choice  Become such that  Simultaneous operations  Sequencing  While loop  Let bindings

Predicated-based IF ANY x WHERE x ∈ S

Substitutions

12

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Machine, refinement, implementation

 Abstract machine  Refinements  Implementation

 Precondition  Choice  Become such that  Simultaneous operations  Sequencing  While loop  Let bindings

Predicated-based IF ANY x WHERE x ∈ S a := 1 ; ; b := 1

Substitutions

12

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Machine, refinement, implementation

 Abstract machine  Refinements  Implementation

 Precondition  Choice  Become such that  Simultaneous operations  Sequencing  While loop  Let bindings

Predicated-based IF ANY x WHERE x ∈ S a := 1 ; ; b := 1 x : (x ∈ INT ∧ x < 20) not deterministic

Substitutions

12

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Machine, refinement, implementation

 Abstract machine  Refinements  Implementation

 Precondition  Choice  Become such that  Simultaneous operations  Sequencing  While loop  Let bindings

Predicated-based IF ANY x WHERE x ∈ S a := 1 ; ; b := 1 a := 1 || || b := 1 b := 1 || || a := 1 x : (x ∈ INT ∧ x < 20) not deterministic

Substitutions

12

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Machine, refinement, implementation

 Abstract machine  Refinements  Implementation

 Precondition  Choice  Let bindings  Become such that  Simultaneous operations  Sequencing  While loop  Precondition  Choice  Become such that  Simultaneous operations  Sequencing  While loop  Let bindings

Predicated-based IF ANY x WHERE x ∈ S a := 1 ; ; b := 1 a := 1 || || b := 1 b := 1 || || a := 1 x : (x ∈ INT ∧ x < 20) not deterministic

Substitutions

12

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Machine, refinement, implementation

 Abstract machine  Refinements  Implementation

 Precondition  Choice  Let bindings  Become such that  Simultaneous operations  Sequencing  While loop  Precondition  Choice  Let bindings  Become such that  Simultaneous operations  Sequencing  While loop  Precondition  Choice  Become such that  Simultaneous operations  Sequencing  While loop  Let bindings

Predicated-based IF ANY x WHERE x ∈ S a := 1 ; ; b := 1 a := 1 || || b := 1 b := 1 || || a := 1 x : (x ∈ INT ∧ x < 20) not deterministic

Substitutions

12

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Machine, refinement, implementation

 Abstract machine  Refinements  Implementation

 Precondition  Choice  Let bindings  Become such that  Simultaneous operations  Sequencing  While loop  Precondition  Choice  Let bindings  Become such that  Simultaneous operations  Sequencing  While loop  Precondition  Choice  Become such that  Simultaneous operations  Sequencing  While loop  Let bindings

Predicated-based IF ANY x WHERE x ∈ S a := 1 ; ; b := 1 a := 1 || || b := 1 b := 1 || || a := 1 x : (x ∈ INT ∧ x < 20) not deterministic M a k i n g p r

  • g

r a m m

  • r

e c

  • n

c r e t e

Substitutions

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More substitutions

Machine Refinement Implementation Block Y Y Y Identical Y Y Y Becomes Equal Y Y Y Precondition Y Y N Assertion Y Y Y Bounded choice Y Y N IF conditional Y Y Y Conditional Bounded choice Y Y N Case Conditional Y Y Y Unbounded choice Y Y N Local Definition Y Y N Becomes Element of Y Y N Becomes such that Y Y N Local Variable N Y Y Sequencing N Y Y Operation Call Y Y Y While Loop N N Y Simultaneous Y Y N

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B language

14

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B language

State oriented

S1 S2

14

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B language

State oriented

S1 S2

Hoare logic

14

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B language

State oriented

S1 S2

Hoare logic

  • y := 3 3

2 x := x*x 4

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B language

State oriented

S1 S2

Hoare logic

  • y := 3 3

2 x := x*x 4 / ≔

ASSIGNMENT

14

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B language

State oriented

S1 S2

Hoare logic

  • y := 3 3

2 x := x*x 4 / ≔

ASSIGNMENT

, ;

COMPOSITION

14

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B language

State oriented

S1 S2

Hoare logic

  • y := 3 3

2 x := x*x 4 / ≔

ASSIGNMENT

, ;

COMPOSITION

∧ , ∧ if then else end

CONDITIONAL

14

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B language

State oriented

S1 S2

Hoare logic

  • y := 3 3

2 x := x*x 4 / ≔

ASSIGNMENT

, ;

COMPOSITION

∧ , ∧ if then else end

CONDITIONAL

∧ while do done ∧

WHILE

14

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B language

State oriented

S1 S2

Hoare logic

  • y := 3 3

2 x := x*x 4 / ≔

ASSIGNMENT

, ;

COMPOSITION

∧ , ∧ if then else end

CONDITIONAL

∧ while do done ∧

WHILE

→ , , →

CONSEQUENCE

14

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B language

15

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B language

Arithmetic

15

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B language

Arithmetic

15

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B language

Arithmetic

15

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B language

∏ ∑ Arithmetic

15

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B language

Functions and relations ∏ ∑ Arithmetic

15

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B language

Partial / total functions, surjections, lambda, domain/range manipulations, closure, inversions… Functions and relations ∏ ∑ Arithmetic

15

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B language

Partial / total functions, surjections, lambda, domain/range manipulations, closure, inversions… Functions and relations ∏ ∑ Arithmetic

15

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B language

Sets Partial / total functions, surjections, lambda, domain/range manipulations, closure, inversions… Functions and relations ∏ ∑ Arithmetic

15

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B language

Set comprehension, generalized union & intersections Sets Partial / total functions, surjections, lambda, domain/range manipulations, closure, inversions… Functions and relations ∏ ∑ Arithmetic

15

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B language

Records Set comprehension, generalized union & intersections Sets Partial / total functions, surjections, lambda, domain/range manipulations, closure, inversions… Functions and relations ∏ ∑ Arithmetic

15

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B language

Trees Records Set comprehension, generalized union & intersections Sets Partial / total functions, surjections, lambda, domain/range manipulations, closure, inversions… Functions and relations ∏ ∑ Arithmetic

15

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B language

Sequences Trees Records Set comprehension, generalized union & intersections Sets Partial / total functions, surjections, lambda, domain/range manipulations, closure, inversions… Functions and relations ∏ ∑ Arithmetic

15

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B language

No algebraic data types Sequences Trees Records Set comprehension, generalized union & intersections Sets Partial / total functions, surjections, lambda, domain/range manipulations, closure, inversions… Functions and relations ∏ ∑ Arithmetic

15

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Proofs with B

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How does B method handles proofs?

Component

 Abstract machine  Refinements  Implementation 17

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How does B method handles proofs?

Component

 Abstract machine  Refinements  Implementation

MACHINE Name(input1, input2, ...)

17

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How does B method handles proofs?

Component

 Abstract machine  Refinements  Implementation

MACHINE Name(input1, input2, ...) CONSTRAINTS input1 ∈ INT ∧ input2 ∈ INT ...

17

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SLIDE 75

How does B method handles proofs?

Component

 Abstract machine  Refinements  Implementation

MACHINE Name(input1, input2, ...) CONSTRAINTS input1 ∈ INT ∧ input2 ∈ INT ... CONSTANTS cst1, cst2, ...

17

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SLIDE 76

How does B method handles proofs?

Component

 Abstract machine  Refinements  Implementation

MACHINE Name(input1, input2, ...) CONSTRAINTS input1 ∈ INT ∧ input2 ∈ INT ... CONSTANTS cst1, cst2, ... VARIABLES var1, var2, ...

17

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How does B method handles proofs?

Component

 Abstract machine  Refinements  Implementation

MACHINE Name(input1, input2, ...) CONSTRAINTS input1 ∈ INT ∧ input2 ∈ INT ... CONSTANTS cst1, cst2, ... VARIABLES var1, var2, ... INVARIANT var1 + var2 ∈ {x . X**(1/2) ∈ } ∧ ...

17

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How does B method handles proofs?

Component

 Abstract machine  Refinements  Implementation

MACHINE Name(input1, input2, ...) CONSTRAINTS input1 ∈ INT ∧ input2 ∈ INT ... CONSTANTS cst1, cst2, ... VARIABLES var1, var2, ... INVARIANT var1 + var2 ∈ {x . X**(1/2) ∈ } ∧ ... ASSERTIONS predicate1 ∧ predicate2 ∧ ...

17

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SLIDE 79

How does B method handles proofs?

Component

 Abstract machine  Refinements  Implementation

MACHINE Name(input1, input2, ...) CONSTRAINTS input1 ∈ INT ∧ input2 ∈ INT ... CONSTANTS cst1, cst2, ... VARIABLES var1, var2, ... INVARIANT var1 + var2 ∈ {x . X**(1/2) ∈ } ∧ ... ASSERTIONS predicate1 ∧ predicate2 ∧ ...

17

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SLIDE 80

How does B method handles proofs?

Component

 Abstract machine  Refinements  Implementation

MACHINE Name(input1, input2, ...) CONSTRAINTS input1 ∈ INT ∧ input2 ∈ INT ... CONSTANTS cst1, cst2, ... VARIABLES var1, var2, ... INVARIANT var1 + var2 ∈ {x . X**(1/2) ∈ } ∧ ... ASSERTIONS predicate1 ∧ predicate2 ∧ ...

Prove all predicates

17

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SLIDE 81

How does B method handles proofs?

Component

 Abstract machine  Refinements  Implementation

MACHINE Name(input1, input2, ...) CONSTRAINTS input1 ∈ INT ∧ input2 ∈ INT ... CONSTANTS cst1, cst2, ... VARIABLES var1, var2, ... INVARIANT var1 + var2 ∈ {x . X**(1/2) ∈ } ∧ ... ASSERTIONS predicate1 ∧ predicate2 ∧ ... INITIALISATION var1 := expr || var2 := expr

Prove all predicates

17

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SLIDE 82

How does B method handles proofs?

Component

 Abstract machine  Refinements  Implementation

MACHINE Name(input1, input2, ...) CONSTRAINTS input1 ∈ INT ∧ input2 ∈ INT ... CONSTANTS cst1, cst2, ... VARIABLES var1, var2, ... INVARIANT var1 + var2 ∈ {x . X**(1/2) ∈ } ∧ ... ASSERTIONS predicate1 ∧ predicate2 ∧ ... INITIALISATION var1 := expr || var2 := expr

Prove all predicates For each initialization, prove invariant conservations

17

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SLIDE 83

How does B method handles proofs?

Component

 Abstract machine  Refinements  Implementation

MACHINE Name(input1, input2, ...) CONSTRAINTS input1 ∈ INT ∧ input2 ∈ INT ... CONSTANTS cst1, cst2, ... VARIABLES var1, var2, ... INVARIANT var1 + var2 ∈ {x . X**(1/2) ∈ } ∧ ... ASSERTIONS predicate1 ∧ predicate2 ∧ ... INITIALISATION var1 := expr || var2 := expr OPERATIONS varOutput1 ← fun1(i1, i2, ...) = ... varOutput2 ← fun2(i1, i2, ...) = ...

Prove all predicates For each initialization, prove invariant conservations

17

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SLIDE 84

How does B method handles proofs?

Component

 Abstract machine  Refinements  Implementation

MACHINE Name(input1, input2, ...) CONSTRAINTS input1 ∈ INT ∧ input2 ∈ INT ... CONSTANTS cst1, cst2, ... VARIABLES var1, var2, ... INVARIANT var1 + var2 ∈ {x . X**(1/2) ∈ } ∧ ... ASSERTIONS predicate1 ∧ predicate2 ∧ ... INITIALISATION var1 := expr || var2 := expr OPERATIONS varOutput1 ← fun1(i1, i2, ...) = ... varOutput2 ← fun2(i1, i2, ...) = ...

Prove all predicates For each initialization, prove invariant conservations Show each operation conserve invariants

17

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SLIDE 85

How does B method handles proofs?

Component

 Abstract machine  Refinements  Implementation

MACHINE Name(input1, input2, ...) CONSTRAINTS input1 ∈ INT ∧ input2 ∈ INT ... CONSTANTS cst1, cst2, ... VARIABLES var1, var2, ... INVARIANT var1 + var2 ∈ {x . X**(1/2) ∈ } ∧ ... ASSERTIONS predicate1 ∧ predicate2 ∧ ... INITIALISATION var1 := expr || var2 := expr OPERATIONS varOutput1 ← fun1(i1, i2, ...) = ... varOutput2 ← fun2(i1, i2, ...) = ...

Prove all predicates For each initialization, prove invariant conservations Show each operation conserve invariants

 Proof Obligations

17

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SLIDE 86

Proving with B method: interactive only

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Proving with B method: interactive only

18

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Proving with B method: interactive only

18

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Proving with B method: interactive only

18

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Proving with B method: interactive only

18

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SLIDE 91

Proving with B method: interactive only

18

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SLIDE 92

Proving with B method: interactive only

18

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SLIDE 93

How does B method internally prove things?

First, expressions are normalized

19

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SLIDE 94

How does B method internally prove things?

First, expressions are normalized

1

19

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SLIDE 95

How does B method internally prove things?

First, expressions are normalized

1 S ⊆ 1,2,3 ∈ 1 ∪ 2 ∪ 3

19

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SLIDE 96

How does B method internally prove things?

(btrue => a) == a ; (bvrb(x)) & (x\a) => #x.a == a ; (bvrb(x)) & (x\b) => #x.(a & b) == (#x.a & b) THEORY SimplifyX IS (i => (j => k)) == (i & j => k) ; (a => bfalse) == not(a) ; (bfalse => a) == btrue ; (a => btrue) == btrue ;

Theory files: list of rules

20

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SLIDE 97

How does B method internally prove things?

(btrue => a) == a ; (bvrb(x)) & (x\a) => #x.a == a ; (bvrb(x)) & (x\b) => #x.(a & b) == (#x.a & b) THEORY SimplifyX IS (i => (j => k)) == (i & j => k) ; (a => bfalse) == not(a) ; (bfalse => a) == btrue ; (a => btrue) == btrue ;

Theory files: list of rules 10.000 lines of rules

20

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SLIDE 98

How does B method internally prove things?

(btrue => a) == a ; (bvrb(x)) & (x\a) => #x.a == a ; (bvrb(x)) & (x\b) => #x.(a & b) == (#x.a & b) THEORY SimplifyX IS (i => (j => k)) == (i & j => k) ; (a => bfalse) == not(a) ; (bfalse => a) == btrue ; (a => btrue) == btrue ;

Theory files: list of rules 10.000 lines of rules User can add rules  No safe core

20

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SLIDE 99

How does B method internally prove things?

(btrue => a) == a ; (bvrb(x)) & (x\a) => #x.a == a ; (bvrb(x)) & (x\b) => #x.(a & b) == (#x.a & b) THEORY SimplifyX IS (i => (j => k)) == (i & j => k) ; (a => bfalse) == not(a) ; (bfalse => a) == btrue ; (a => btrue) == btrue ;

Theory files: list of rules 10.000 lines of rules User can add rules  No safe core

Still, we can prove a rule before adding it

20

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SLIDE 100

How does B method internally prove things?

(btrue => a) == a ; (bvrb(x)) & (x\a) => #x.a == a ; (bvrb(x)) & (x\b) THEORY SimplifyX IS (i => (j => k)) == (i & j => k) ; (a => bfalse) == not(a) ; (bfalse => a) == btrue ; (a => btrue) == btrue ; P == btrue ;

Theory files: list of rules 10.000 lines of rules User can add rules  No safe core

Still, we can prove a rule before adding it

20

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SLIDE 101

How does B method internally prove things?

Theory files: list of rules  No safe core 10.000 lines of rules User can add rules

THEORY SimplifyX IS (i => (j => k)) == (i & j => k) ; (a => bfalse) == not(a) ; (bfalse => a) == btrue ; (a => btrue) == btrue ;

Automatic prover

  • Applies recursively rules
  • Case proof
  • Tactics

(btrue => a) == a ; (bvrb(x)) & (x\a) => #x.a == a ; (bvrb(x)) & (x\b) P == btrue ; Still, we can prove a rule before adding it

21

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SLIDE 102

Interactive proofs

Proof construction Search information Browsing proof obligations Command repetition

22

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SLIDE 103

Interactive proofs

Proof construction Search information Browsing proof obligations Command repetition Repeat (rr), loop (bb)…

22

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SLIDE 104

Interactive proofs

Proof construction Search information Browsing proof obligations Command repetition Back (ba), Reset (re), Next (ne), Previous (pv), Goto (go)… Repeat (rr), loop (bb)…

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Interactive proofs

Proof construction Search information Browsing proof obligations Command repetition Back (ba), Reset (re), Next (ne), Previous (pv), Goto (go)… Repeat (rr), loop (bb)… Search rule / hypothesis / goal, show proof, reduce PO…

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Interactive proofs

Proof construction

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Interactive proofs

Proof construction Prover call automatic proofs & simplification

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Interactive proofs

Proof construction Prover call automatic proofs & simplification pr

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Interactive proofs

Proof construction Prover call automatic proofs & simplification pr ss

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Interactive proofs

Proof construction Prover call automatic proofs & simplification Rule applications pr ss

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Interactive proofs

Proof construction Prover call automatic proofs & simplification Rule applications pr ss ar

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Interactive proofs

Proof construction Prover call automatic proofs & simplification Rule applications Rewrite equality applications, abstract terms pr ss ar

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Interactive proofs

Proof construction Prover call automatic proofs & simplification Rule applications Rewrite equality applications, abstract terms pr ss ar eh

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Interactive proofs

Proof construction Prover call automatic proofs & simplification Rule applications Rewrite equality applications, abstract terms pr ss ar eh ae

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Interactive proofs

Proof construction Prover call automatic proofs & simplification Rule applications Rewrite equality applications, abstract terms pr ss ar eh ae 3 ∧ 3 ∈ ∧ ∈ ⇒

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Interactive proofs

Proof construction Prover call automatic proofs & simplification Rule applications Rewrite equality applications, abstract terms pr ss ar eh ae

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Interactive proofs

Proof construction Prover call automatic proofs & simplification Rule applications Rewrite equality applications, abstract terms Inference rules contradiction, false hypothesis, cases, instantiate pr ss ar eh ae

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Interactive proofs

Proof construction Prover call automatic proofs & simplification Rule applications Rewrite equality applications, abstract terms Inference rules contradiction, false hypothesis, cases, instantiate pr ss ar eh ae ct

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Interactive proofs

Proof construction Prover call automatic proofs & simplification Rule applications Rewrite equality applications, abstract terms Inference rules contradiction, false hypothesis, cases, instantiate pr ss ar eh ae ct fh

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Interactive proofs

Proof construction Prover call automatic proofs & simplification Rule applications Rewrite equality applications, abstract terms Inference rules contradiction, false hypothesis, cases, instantiate pr ss ar eh ae ct fh dc

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Interactive proofs

Proof construction Prover call automatic proofs & simplification Rule applications Rewrite equality applications, abstract terms Inference rules contradiction, false hypothesis, cases, instantiate pr ss ar eh ae ct fh dc se

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Interactive proofs

Proof construction Prover call automatic proofs & simplification Rule applications Rewrite equality applications, abstract terms Inference rules contradiction, false hypothesis, cases, instantiate Operations on hypothesis deduction, add hypothesis… pr ss ar eh ae ct fh dc se

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Interactive proofs

Proof construction Prover call automatic proofs & simplification Rule applications Rewrite equality applications, abstract terms Inference rules contradiction, false hypothesis, cases, instantiate Operations on hypothesis deduction, add hypothesis… pr ss ar eh ae ct fh dc se dd

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Interactive proofs

Proof construction Prover call automatic proofs & simplification Rule applications Rewrite equality applications, abstract terms Inference rules contradiction, false hypothesis, cases, instantiate Operations on hypothesis deduction, add hypothesis… pr ss ar eh ae ct fh dc se dd ah

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Concrete case

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Example: seat reservation system

Two operations: reserve a seat or free a seat Data : set of seats and sub set of taken seats

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Thank you for listening!

Any questions?