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Parallel Postulates and Continuity Axioms: A Mechanized Study in Intuitionistic Logic Using Coq

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Abstract

In this paper we focus on the formalization of the proofs of equivalence between different versions of Euclid’s 5th postulate. Our study is performed in the context of Tarski’s neutral geometry, or equivalently in Hilbert’s geometry defined by the first three groups of axioms, and uses an intuitionistic logic, assuming excluded-middle only for point equality. Our formalization provides a clarification of the conditions under which different versions of the postulates are equivalent. Following Beeson, we study which versions of the postulate are equivalent, constructively or not. We distinguish four groups of parallel postulates. In each group, the proof of their equivalence is mechanized using intuitionistic logic without continuity assumptions. For the equivalence between the groups additional assumptions are required. The equivalence between the 34 postulates is formalized in Archimedean planar neutral geometry. We also formalize a variant of a theorem due to Szmielew. This variant states that, assuming Aristotle’s axiom, any statement which hold in the Euclidean plane and does not hold in the Hyperbolic plane is equivalent to Euclid’s 5th postulate. To obtain all these results, we have developed a large library in planar neutral geometry, including the formalization of the concept of sum of angles and the proof of the Saccheri–Legendre theorem, which states that assuming Archimedes’ axiom, the sum of the angles in a triangle is at most two right angles.

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Notes

  1. “It is no less certain that the theorem on the sum of the three angles of the triangle must be regarded as one of those fundamental truths which is impossible to dispute and which are an enduring example of mathematical certitude, which one continually pursues and which one obtains only with difficulty in the other branches of human knowledge.” The English translation is borrowed from [42].

  2. Neutral geometry designates the set of theorems which are valid in both Euclidean and hyperbolic geometry. Therefore, for any given line and any given point, there exists at least a line parallel to this line and passing through this point. This renders elliptic geometry inconsistent with neutral geometry.

  3. The meaning of equivalence does not seem to be clear for everyone. For instance, in the French version of the wikipedia page about the parallel postulate (April 2015) one can read that (our own translation) : “These propositions are roughly equivalent to the axiom of parallels. By equivalent, we mean that using some adapted vocabulary, these axioms are true in Euclidean geometry but not true in elliptic nor spherical geometry”. Thanks to a theorem of Szmielew, it is true that this definition of equivalence is equivalent to the logical one when the continuity axiom is assumed, but it is far from obvious.

  4. A Pythagorean field is a field in which every sum of two squares is a square.

  5. We will provide figures both in the Euclidean model and a non-Euclidean model, namely the Poincaré disk model. The figure on the left hand side will illustrate the validity of the axiom or lemma in Euclidean geometry. The figure on the right hand side will either depict the validity of the statement in the Poincaré disk model or exhibit a counter-example. We exhibit a counter-example for statements which are equivalent to the parallel postulate.

  6. In constrast to Euclid, we treat the words “postulate” and “axiom” as synonyms. However, we will restrict the use of the word “postulate” to statements of the parallel postulate. For the reader interested in the difference between these two words in terms of meaning we refer to [49].

  7. As mentioned in Sect. 2.1.1, we assume the decidability of the point equality, which is a tautology in classical logic.

  8. One should remark that this axiom is not named after Greenberg [34].

  9. For the sake of conciseness, we adopted the same name as Greenberg for this axiom which is also known under the name of Aristotle’s angle unboundedness axiom.

  10. We use \({{\mathrm{\mathbin {\widehat{<}}}}}\) for the strict comparison between angles.

  11. That is the reason why Tarski chose this postulate, as he wanted to avoid definitions in his axiom system.

  12. Non-degeneracy conditions are often omitted in textbook proofs.

  13. We use the expression ’proof of negation’ to describe a proof of \(\lnot A\) assuming A and obtaining a contradiction. For the reader who is not familiar with intuitionistic logic, we recall that this is simply the definition of negation and this proof rule has nothing to do with the proof by contradiction (to prove A it suffices to show that \(\lnot A\) leads to a contradiction), which is not valid in our intuitionistic setting.

  14. Up to our knowledge, the following proofs are the only ones that resemble the ones we formalized.

  15. The numbers given in parentheses are the numbers of the propositions (i.e. Satz) as given in [57].

  16. This lemma is present in [16] as it corresponds to Hilbert’s version of Pasch’s axiom.

  17. We previously referred to interior angles on the same side of a straight line as consecutive interior angles.

  18. It almost corresponds to the fact that the opposite sides of a non-degenerate quadrilateral which has its diagonals intersecting in their midpoint are parallel. To fully correspond to this fact one would need to add the hypothesis that A and D are distinct.

  19. Note that we use here the decidability of intersection of lines.

  20. Here we restrict ourselves to the postulates that we formalized.

  21. “It is true that I have come upon much which by most people would be held to constitute a proof; but in my eyes it proves as good as nothing. For example, if we could show that a rectilinear triangle whose area would be greater than any given area is possible, then I would be ready to prove the whole of [Euclidean] geometry absolutely rigorously.” The English translation is borrowed from [38].

  22. Otherwise, it would provide yet another illustration of the gravity of definitions.

  23. Quadrilaterals are usually implicitly assumed to be non-crossed.

  24. These proofs have already been presented in French [30].

  25. Usually in geometry, we give two different constructions for the perpendicular to a given line in a given point, whether the given point lies on the given line or not. If it does, we “erect” a perpendicular at this given point, and if it does not, we “drop” a perpendicular from this given point to this given line.

  26. This corresponds to the fourth axiom of Group IV from [37].

  27. This lemma represents only the part that is valid in neutral planar geometry.

  28. The comment in French Wikipedia about Amiot’s proof seems to say that the proof is valid only in Euclidean geometry because it use the construction of THE parallel to line AC trough B. To be precise, the proof does not rely on the uniqueness of this line, only on its existence, so this first step of the proof is valid also in hyperbolic geometry (but not in elliptic geometry). The Wikipedia comment fails to notice that the proof relies on Postulate 7.

  29. Here we use Greenberg’s denomination for models of Hilbert’s Axioms Group I, II, III and IV [34].

  30. We already presented this proof in French [30].

  31. This lemma is present in [33, 36] (Proposition 7.3) under the name of Crossbar Theorem. Note that in [36], the statement look different but actually is the same, because Hartshorne’s definition of a point being inside an angle is based on the two-side predicate, whereas the definition we use (borrowed from [57]) states that the ray BP intersects the segment \(\overline{AC}\).

  32. Proposition 6 corresponds to Theorem 22.24 in [44] which we mechanized. Its proof, which depends on Legendre’s first theorem (Theorem 6), will be discussed in the next subsection.

  33. One could also notice that they are also specified to form a non-degenerate acute angle. The fact it is acute plays a minor role, contrary to the non-degeneracy condition, because if one can find such an obtuse or right angle, every acute angle inside it fulfills the same properties.

  34. The \(B_i\) are not known to be collinear, but the fact \({B_i B_{i + 1}} \mathbin {\equiv } {B_0 B_1}\) and the quantity \(n B_0 B_1\) appear in this proof.

  35. One should also notice that this proof relies on Theorem 6, although it may not be obvious.

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Acknowledgements

We would like to thank Victor Pambuccian, David Braun and the anonymous referees for helpful comments and suggestions about this work.

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Authors and Affiliations

  1. ICube, UMR 7357 CNRS, University of Strasbourg, Pôle API, Bd Sébastien Brant, BP 10413, 67412, Illkirch, France

    Pierre Boutry, Charly Gries, Julien Narboux & Pascal Schreck

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  1. Pierre Boutry
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  2. Charly Gries
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  3. Julien Narboux
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  4. Pascal Schreck
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Corresponding author

Correspondence to Pierre Boutry.

Appendices

Summary of the 34 Postulates

  1. 1.

    (Tarski’s parallel postulate) Given a point D between the points B and C and a point T further away from A than D on the half line AD, one can build a line which goes through T and intersects the sides BA and BC of the angle \(\angle ABC\) respectively further away from B than A and C.

  2. 2.

    (Playfair’s postulate) There is a unique parallel to a given line going through some point.

  3. 3.

    (Triangle postulate) The sum of the angles of any triangle is two right angles.

  4. 4.

    (Bachmann’s Lotschnittaxiom) Given the lines l, m, r and s, if l and r are perpendicular, r and s are perpendicular and s and m are perpendicular, then l and m must meet.

  5. 5.

    (Postulate of transitivity of parallelism) If two lines are parallel to the same line then these lines are also parallel.

  6. 6.

    (Midpoint converse postulate) The parallel line to one side of a triangle going through the midpoint of another side cut the third side in its midpoint.

  7. 7.

    (Alternate interior angles postulate) The line falling on parallel lines makes the alternate angles equal one to one another.

  8. 8.

    (Consecutive interior angles postulate) A line falling on parallel lines makes the sum of interior angles on the same side equal to two right angles.

  9. 9.

    (Perpendicular transversal postulate) Given two parallel lines, any line perpendicular to the first line is perpendicular to the second line.

  10. 10.

    (Postulate of parallelism of perpendicular transversals) Two lines, each perpendicular to one of a pair of parallel lines, are parallel.

  11. 11.

    (Universal Posidonius’ postulate) If two lines are parallel then they are everywhere equidistant.

  12. 12.

    (Alternative Playfair’s postulate) Any line parallel to line l passing through a point P is equal to the line that passes through P and shares a common perpendicular with l going through P.

  13. 13.

    (Proclus’ postulate) If a line intersects one of two parallel lines then it intersects the other.

  14. 14.

    (Alternative Proclus’ postulate) If a line intersects in P one of two parallel lines which share a common perpendicular going through P, then it intersects the other.

  15. 15.

    (Triangle circumscription principle) For any three non-collinear points there exists a point equidistant from them.

  16. 16.

    (Inverse projection postulate) For any given acute angle, any point together with its orthogonal projection on one side of the angle form a line which intersects the other side.

  17. 17.

    (Euclid 5) Given a non-degenerated parallelogram PRQS and a point U strictly inside the angle \(\angle QPR\), there exists a point I such that Q and U are respectively strictly between S and I and strictly between P and I.

  18. 18.

    (Strong parallel postulate) Given a non-degenerated parallelogram PRQS and a point U not on line PR, the lines PU and QS intersect.

  19. 19.

    (Alternative strong parallel postulate) If a straight line falling on two straight lines make the sum of the interior angles on the same side different from two right angles, the two straight lines meet if produced indefinitely.

  20. 20.

    (Euclid’s parallel postulate) If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

  21. 21.

    (Postulate of existence of a triangle whose angles sum to two rights) There exists a triangle whose angles sum to two rights.

  22. 22.

    (Posidonius’ postulate) There exists two lines which are everywhere equidistant.

  23. 23.

    (Postulate of existence of similar but non-congruent triangles) There exists two similar but non-congruent triangles.

  24. 24.

    (Thales’ postulate) If the circumcenter of a triangle is the midpoint of a side of a triangle, then the triangle is right.

  25. 25.

    (Thales’ converse postulate) In a right triangle, the midpoint of the hypotenuse is the circumcenter.

  26. 26.

    (Existential Thales’ postulate) There is a right triangle whose circumcenter is the midpoint of the hypotenuse.

  27. 27.

    (Postulate of right Saccheri quadrilaterals) The angles of any Saccheri quadrilateral are right.

  28. 28.

    (Postulate of existence of a right Saccheri quadrilateral) There is a Saccheri quadrilateral whose angles are right.

  29. 29.

    (Postulate of right Lambert quadrilaterals) The angles of any Lambert quadrilateral are right i.e. if in a quadrilateral three angles are right, so is the fourth.

  30. 30.

    (Postulate of existence of a right Lambert quadrilateral) There exists a Lambert quadrilateral whose angles are all right.

  31. 31.

    (Weak inverse projection postulate) For any angle, that, together with itself, make a right angle, any point together with its orthogonal projection on one side of the angle form a line which intersects the other side.

  32. 32.

    (Weak Tarski’s parallel postulate) For every right angle and every point T in the interior of the angle, there is a point on each side of the angle such that T is between these two points.

  33. 33.

    (Weak triangle circumscription principle) The perpendicular bisectors of the legs of a right triangle intersect.

  34. 34.

    (Legendre’s parallel postulate) There exists an acute angle such that, for every point T in the interior of the angle, there is a point on each side of the angle such that T is between these two points.

Definitions of the Geometric Predicates

Coq

Notation

Explanation

Bet A B C

B is between A and C

Cong A B C D

\( {AB} \mathbin {\equiv } {CD} \)

The segments \(\overline{AB}\) and \(\overline{CD}\) are congruent

Col A B C

\( {{\mathrm{Col}}}\, {A} \, {B} \, {C} \)

A, B and C are collinear

Coplanar A B C D

\( {{\mathrm{Cp}}}\, {A} \,{ B} \, {C} \, {D} \)

A, B, C and D are coplanar

Par_strict A B C D

\( {A B}\mathbin {\parallel _s} {C D} \)

The lines AB and CD are strictly parallel

Par A B C D

\( {A B}\mathbin {\parallel } {X Y} \)

The lines AB and CD are parallel

CongA A B C D E F

\( {A} \, {B} \, {C} {{\mathrm{\mathbin {\widehat{=}}}}}{D} \, {E} \, {F} \)

The angles \(\angle ABC\) and \(\angle DEF\) are congruent

TS A B P Q

P and Q are on opposite sides of line AB

OS A B P Q

P and Q are on the same side of line AB

SumA A B C D E F G H I

\( {A} \, {B} \, {C} {{\mathrm{\mathbin {\widehat{+}}}}}{D} \, {E} \, {F} {{\mathrm{\mathbin {\widehat{=}}}}}{G} \, {H} \, {I} \)

The angles \(\angle ABC\) and \(\angle DEF\) sum to \(\angle GHI\)

TriSumA A B C D E F

\( {{\mathrm{\mathcal {S}}}}(\bigtriangleup {A}{B}{C}) {{\mathrm{\mathbin {\widehat{=}}}}}{D} \, {E} \, {F} \)

The angle of the triangle ABC sum to \(\angle DEF\)

Le A B C D

\( AB \le CD \)

The segment \(\overline{AB}\) is smaller or congruent to the segment \(\overline{CD}\)

Lt A B C D

\( AB < CD \)

The segment \(\overline{AB}\) is smaller than the segment \(\overline{CD}\)

Midpoint M A B

M is the midpoint of the segment \(\overline{AB}\)

Per A B C

The triangle ABC is a right triangle with the right angle at vertex B

Out P A B

B belongs to the ray PA

InAngle P A B C

\( {P} {{\mathrm{\mathbin {\widehat{\in }}}}}{A} \, {B} \, {C} \)

P belongs to the angle \(\angle ABC\)

LeA A B C D E F

\( {A} \, {B} \, {C} {{\mathrm{\mathbin {\widehat{\le }}}}}{D} \, {E} \, {F} \)

The angle \(\angle ABC\) is smaller or congruent to the angle \(\angle DEF\)

Coq

Notation

Explanation

LtA A B C D E F

\( {A} \, {B} \, {C} {{\mathrm{\mathbin {\widehat{<}}}}}{D} \, {E} \, {F} \)

The angle \(\angle ABC\) is smaller than the angle \(\angle DEF\)

Acute A B C

The angle \(\angle ABC\) is acute

Perp_at X A B C D

\({A B} \mathbin {\underset{X}{\perp }} {C D}\)

The lines AB and CD meet at a right angle in X

Perp A B C D

\( {A B} \mathbin {\perp } {C D} \)

The lines AB and CD are perpendicular

Perp2 A B C D P

The lines AB and CD have a common perpendicular which passes through P

BetS A B C

B is strictly between A and C

SAMS A B C D E F

The angles \(\angle ABC\) and \(\angle DEF\) do not make an over-obtuse angle

Saccheri A B C D

The quadrilateral ABCD is a Saccheri quadrilateral

Lambert A B C D

The quadrilateral ABCD is a Lambert quadrilateral

ReflectL P’ P A B

\(P'\) is the image of P by the reflection with respect to the line AB

Perp_bisect P Q A B

The line PQ is the perpendicular bisector of the segment \(\overline{AB}\)

Defect A B C D E F

\( {{\mathrm{\mathcal {D}}}}(\bigtriangleup {A}{B}{C}) {{\mathrm{\mathbin {\widehat{=}}}}}{D} \, {E} \, {F} \)

The angle \(\angle DEF\) is the defect of the triangle ABC

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Boutry, P., Gries, C., Narboux, J. et al. Parallel Postulates and Continuity Axioms: A Mechanized Study in Intuitionistic Logic Using Coq. J Autom Reasoning 62, 1–68 (2019). https://doi.org/10.1007/s10817-017-9422-8

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