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Toward the use of a proof assistant to teach mathematics. Julien - - PowerPoint PPT Presentation

Apr 30, 2023 288 likes •713 views

Toward the use of a proof assistant to teach mathematics. Julien Narboux under the supervision of Hugo Herbelin LIX, Ecole Polytechnique ICTMT7, 26 July 2005, Bristol, UK Julien Narboux Introduction What is a proof assistant ?

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Toward the use of a proof assistant to teach mathematics.

Julien Narboux

under the supervision of Hugo Herbelin

LIX, ´ Ecole Polytechnique

ICTMT7, 26 July 2005, Bristol, UK

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion

Outline

1

What is a proof assistant ?

2

Introduction to the Coq proof assistant

3

Some examples

4

Motivation for its use in the classroom

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion

The impact of the use of software on the proving activity is a well addressed issue in the litterature. CAS, DGS, . There are software whose sole purpose is to produce proofs : the proof assistants.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion

The impact of the use of software on the proving activity is a well addressed issue in the litterature. CAS, DGS, . There are software whose sole purpose is to produce proofs : the proof assistants.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion

The impact of the use of software on the proving activity is a well addressed issue in the litterature. CAS, DGS, PA. There are software whose sole purpose is to produce proofs : the proof assistants.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion What ? Why ?

CAS (Maple, MuPAD, Mathematica . . . ) Definitions, Questions → Results Computation Theorem Prover (Otter, Vampire . . . ) Definitions, Axioms, Statement → True, False, I don’t know, Nothing. Automatic proof Proof assistant (Coq (84), Isabelle, HOL, PVS (90’s). . . ) Axioms, Statement, Interractive Proof → Correct or not Interractive proof

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion What ? Why ?

Proof oriented software For example :

logic oriented (Hyperproof . . . ) geometry oriented (Geometrix, Baghera . . . ) algebra oriented (MathXpert . . . )

They are :

User friendly Give hints to the student

Proof assistants Not specialized A very large span of applications A very high level of confidence Real mathematics

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion What ? Why ?

What can we prove ?

Programs (Line 14 of Paris’ subway . . . ) Mathematical statements (The fondamental theorem of algebra (Henk Barendregt’s group), The four colours theorem (Gonthier, Werner). . . )

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion What ? Why ?

What can we prove ?

Programs (Line 14 of Paris’ subway . . . ) Mathematical statements (The fondamental theorem of algebra (Henk Barendregt’s group), The four colours theorem (Gonthier, Werner). . . )

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion What ? Why ?

Why ?

To understand what a proof is. To ensure correctness of the proof (The four colours theorem again). To generate proofs that could not be done by hand, either

Proof of programs (often long but straightforward proofs with too many cases for an exhaustive search). Proof of mathematical statements (The four colours theorem).

For teaching.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion What ? Why ?

Why ?

To understand what a proof is. To ensure correctness of the proof (The four colours theorem again). To generate proofs that could not be done by hand, either

Proof of programs (often long but straightforward proofs with too many cases for an exhaustive search). Proof of mathematical statements (The four colours theorem).

For teaching.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion What ? Why ?

Why ?

To understand what a proof is. To ensure correctness of the proof (The four colours theorem again). To generate proofs that could not be done by hand, either

Proof of programs (often long but straightforward proofs with too many cases for an exhaustive search). Proof of mathematical statements (The four colours theorem).

For teaching.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion What ? Why ?

Why ?

To understand what a proof is. To ensure correctness of the proof (The four colours theorem again). To generate proofs that could not be done by hand, either

Proof of programs (often long but straightforward proofs with too many cases for an exhaustive search). Proof of mathematical statements (The four colours theorem).

For teaching.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion What ? Why ?

Why ?

To understand what a proof is. To ensure correctness of the proof (The four colours theorem again). To generate proofs that could not be done by hand, either

Proof of programs (often long but straightforward proofs with too many cases for an exhaustive search). Proof of mathematical statements (The four colours theorem).

For teaching.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion What ? Why ?

Why ?

To understand what a proof is. To ensure correctness of the proof (The four colours theorem again). To generate proofs that could not be done by hand, either

Proof of programs (often long but straightforward proofs with too many cases for an exhaustive search). Proof of mathematical statements (The four colours theorem).

For teaching.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion Can we trust a proof checked by the Coq proof assistant ?

Coq

Coq (http://coq.inria.fr/) a free software (GPL2), based on the Calculus of Inductive Constructions, developped at INRIA in the LogiCal team, since 1984.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion Can we trust a proof checked by the Coq proof assistant ?

Coq

Coq (http://coq.inria.fr/) a free software (GPL2), based on the Calculus of Inductive Constructions, developped at INRIA in the LogiCal team, since 1984.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion Can we trust a proof checked by the Coq proof assistant ?

Coq

Coq (http://coq.inria.fr/) a free software (GPL2), based on the Calculus of Inductive Constructions, developped at INRIA in the LogiCal team, since 1984.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion Can we trust a proof checked by the Coq proof assistant ?

Coq

Coq (http://coq.inria.fr/) a free software (GPL2), based on the Calculus of Inductive Constructions, developped at INRIA in the LogiCal team, since 1984.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion Can we trust a proof checked by the Coq proof assistant ?

What you need to trust :

The theory behind Coq. The Coq kernel implementation match the theory. Coq : > 130000 lines of code The kernel : < 11000 lines of code Your hardware, operating system and Ocaml compiler. Yours axioms.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion Can we trust a proof checked by the Coq proof assistant ?

What you need to trust :

The theory behind Coq. The Coq kernel implementation match the theory. Coq : > 130000 lines of code The kernel : < 11000 lines of code Your hardware, operating system and Ocaml compiler. Yours axioms.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion Can we trust a proof checked by the Coq proof assistant ?

What you need to trust :

The theory behind Coq. The Coq kernel implementation match the theory. Coq : > 130000 lines of code The kernel : < 11000 lines of code Your hardware, operating system and Ocaml compiler. Yours axioms.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion Can we trust a proof checked by the Coq proof assistant ?

What you need to trust :

The theory behind Coq. The Coq kernel implementation match the theory. Coq : > 130000 lines of code The kernel : < 11000 lines of code Your hardware, operating system and Ocaml compiler. Yours axioms.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion 1

∀xyz, x = y ∧ y = z → x = z

2

k=n

  • k=0

2k + 1 = (n + 1)2 A few difficulties : f(x, y) is noted (f x y). A → B → C is used to express A ∧ B → C. ¬A is defined by A → False.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion

Let’s start :

1 subgoal x : nat y : nat z : nat H : x = y /\ y = z __________________(1/1) x = z Examples: We know that : x, y and z are natural numbers and x = y ∧ y = z. We need to show that : x = z.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion

How can I prove something ?

The proof can be described step-by-step using : case distinction absurd induction application of a theorem computation rewriting and sometimes automation . . .

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion

Some examples now.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion

It clarifies what we know, what we want to prove, what are the theorems, lemmas, axioms, definitions. . . It is rigourous. It helps to understand the logic. It clarifies what the logical rules are. It is fair: the proof is correct iff it is accepted by the system.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion

It clarifies what we know, what we want to prove, what are the theorems, lemmas, axioms, definitions. . . It is rigourous. It helps to understand the logic. It clarifies what the logical rules are. It is fair: the proof is correct iff it is accepted by the system.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion

It clarifies what we know, what we want to prove, what are the theorems, lemmas, axioms, definitions. . . It is rigourous. It helps to understand the logic. It clarifies what the logical rules are. It is fair: the proof is correct iff it is accepted by the system.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion

It clarifies what we know, what we want to prove, what are the theorems, lemmas, axioms, definitions. . . It is rigourous. It helps to understand the logic. It clarifies what the logical rules are. It is fair: the proof is correct iff it is accepted by the system.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion

It clarifies what we know, what we want to prove, what are the theorems, lemmas, axioms, definitions. . . It is rigourous. It helps to understand the logic. It clarifies what the logical rules are. It is fair: the proof is correct iff it is accepted by the system.

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion

Limitations

Notations Error messages Interface Associativity-Commutativity Not enough automation Too much automation

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion

Limitations

Notations Error messages Interface Associativity-Commutativity Not enough automation Too much automation

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion

Limitations

Notations Error messages Interface Associativity-Commutativity Not enough automation Too much automation

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion

Limitations

Notations Error messages Interface Associativity-Commutativity Not enough automation Too much automation

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion

Limitations

Notations Error messages Interface Associativity-Commutativity Not enough automation Too much automation

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion

Limitations

Notations Error messages Interface Associativity-Commutativity Not enough automation Too much automation

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion

Work done or in progress

PCoq A gui to ease the usage of Coq. F . Guilhot’s work A formalization of high school geometry in Coq. CoqWeb An interface for solving exercises online using Coq (Work in progress). DrGeoCaml A gui for interactive proof in geometry using Coq (Work in progress).

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion

DGS to conjecture CAS to compute PA to prove Thank you !

Julien Narboux

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Introduction What is a proof assistant ? Introduction to the Coq proof assistant Some examples Motivation for its use in the classroom Conclusion

DGS to conjecture CAS to compute PA to prove Thank you !

Julien Narboux