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Automated Theorem Proving in GeometryInteractive Theorem Proving in Geometry

Construction Problems

Automated Reasoning in Geometry

Predrag Janicic, Faculty of Mathematics,University of Belgrade, Serbia

Logic and Applications 2018 (LAP 2018)Dubrovnik, September 25, 2018.

Predrag Janicic Automated Reasoning in Geometry

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The talk is partly based on a book chapter:

Julien Narboux (Univ of Strasbourg),Predrag Janicic (Univ of Belgrade),Jacques Fleuriot (Univ of Edinburgh):Computer-Assisted Theorem Proving inSynthetic GeometryfromHandbook of Geometric Constraint SystemsPrinciples (Eds: M. Sitharam, A. John,J. Sidman), Chapman and Hall/CRC, 2018.


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Computer-assisted theorem proving:Automated theorem proving – theorem are proved completelyautomatically, answer is often just yes or noInteractive theorem proving – interactive proving under controlof a proof assistantCombinations of the previous two

Geometric constraints solvingConstruction problems – streighedge & compass constructionproblemsAlso, robotic movements, origami, folding solar panels forsatelites ...


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Automated Reasoning in Geometry (2)

Many challenges in different fields:algorithmslogicgeometry

Some questions concerning logic and proofs:„What we have really proved?“„How reliable is our proof?“


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Synthetic Methods: Early and „AI“ SystemsSynthetic Methods: Automated Proving in Coherent LogicSynthetic Methods: Resolution MethodSemisynthetic Methods: Area MethodSemisynthetic Methods: Full Angle Method and Vector MethodAlgebraic MethodsImplementations of Provers and Integration

Automated Theorem Proving in Geometry

First systems already in 1950s.Early got status of a “classical AI discipline”Important because of:

paradigmatic rigorous reasoningwidely applicable techniquesapplications in mathematical educationindustrial applications in robotics, computer vision, CADsystems, etc....

The domain is typically Euclidean planimetry


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Automated Theorem Proving in Geometry (2)

Three sorts of methods:Synthetical methodsSemisynthetical methodsAlgebraic methods

Early systems: simple and of very limited powerModern systems can prove very complex theorems and areused in development of monumental machine-verifiable proofs(e.g. a proof od Kepler’s conjecture)...... but, there are still many challenges...


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Issues: Axioms and Non-degeneracy conditions (NDG)

Choice of axiomsTheorems being proved are often of the form:

∀ ∗ (C ⇒ G )

The formula C corresponds to a configuration, somegeometrical figure (in a procedural, constructive or in adeclarative way).Often some conditions have to be added to C , typically somenon-degeneracy conditions (e.g. some points are distinct).For such conditions ndg , the following formulae have to bevalid: ∃ ∗ (C ∧ ndg) i ∀ ∗ (C ∧ ndg ⇒ G ).Some methods can produce NDG conditions automatically(not necessarily the weakest ones).


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Synthetic Methods: Early and „AI“ Systems

Geometry Machine, author: Gelernter et.al.“In early spring, 1959, an IBM 704 computer, with theassistance of a program comprising some 20000 individualinstructions, proved its first theorem in elementary Euclideanplane geometry”Among other techniques, it introduced:

backward reasoningusing diagrams (for semantic checks)dealing with symmetriesrules anticipating modern rewrite rules

Basic „rules“ are lemmas about congruence of triangles (e.g.,two line segments are congruent if they are corresponding sidesof two congruent triangles).


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Geometry Machine (2)

Example. If ∠ABD ∼= ∠DBC , AD ⊥ AB , DC ⊥ BC , thenAD ∼= CD.

A

B C

D

1 ∠ABD ∼= ∠DBC (Premise)2 ∠DAB is right angle (by Definition of perpendicular)3 ∠DCB is right angle (by Definition of perpendicular)4 ∠BAD ∼= ∠BCD (by All right angles are congruent)5 BD ∼= BD (by Identity)6 4ADB ∼=4CDB (by Side-Angle-Angle)7 AD ∼= CD (by Corresponding elements of

congruent triangles are congruent)


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Other early and related systems

Gilmore (1970), Goldstein (1973), Nevis (1975), Elcock(1977), Greeno (1979), Coelho (1979), Anderson (1986),Koedinger (1990),...They use a number of heuristics (hence the name „AI systems“)Mainly of very limited powerDo not consider NDG conditionsThey use only lemmas and conjectures of the form (Ai , B areatomic formulae)

A1(~x) ∧ . . . ∧ An(~x)⇒ B(~x)


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DD method, GRAMY, iGeoTutor

Deductive database (DD) method (Chou et al, 2000), Gramy(Matsuda i Vanlehn, 2004), iGeoTutor (Wang i Su, 2015) as„rules“ they uses lemmas and prove theorems of the form :

A1(~x) ∧ . . . ∧ An(~x)⇒ B1(~x)

where Bi are atomic, Aj are atomic of conjunctions of atomicformulae.Some lemmas have the form:

A1(~x) ∧ . . . ∧ An(~x)⇒ ∃~y(B1(~x , ~y) ∧ . . . ∧ Bm(~x , ~y))

These method may use „auxiliary” points


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DD metoda, GRAMY, iGeoTutor (2)

„Auxiliary” points substantially extends a set of provabletheorems...... but complicate proof search (because of combinatorialexplosion)For controlling the number of inferred facts and introducedpoints they use a number of heuristicsDD considers NDG conditions


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DD example

Example. Altitudes of triangle intersect in one point: Col E AC,BE ⊥ AC, Col F B C, AF ⊥ BC, Col H AF, Col H B E,Col G AB, Col G C H ⇒ CG ⊥ AB

A B

C

G

F

E

H

1 cyclic(C , F ,E ,H) (by D42, (hyp) FH ⊥ FC , (hyp) EH ⊥ EC)2 cyclic(A, F ,B,E) (by D42, (hyp) FB ⊥ FA, (hyp) EB ⊥ EA)3 ∠[HF ,HC ] = ∠[EF ,EC ] (by D41, (1))4 ∠[BF ,BA] = ∠[EF ,EA] (by D41, (2))5 ∠[AF ,CH] = ∠[FE ,AC ] (by (hyp) Col A H F , (hyp) Col A E C , (3))6 ∠[BC ,AB] = ∠[FE ,AC ] (by (hyp) Col C B F , (hyp) Col C E A, (4))7 ∠[BC ,AB] = ∠[AF ,CH] (by D22, (6), (5))8 ∠[BC ,AF ] = ∠[AB,CH] (by D21, (7))9 CG ⊥ AB (by D74, (hyp) BC ⊥ AF , (hyp) Col G C H, (8))


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iGeoTutor example

Example. If P belongs to interior of a square ABCD such thatAP ∼= PD and ∠PAD = 15◦, then 4PBC is equilateral.

P

Q

A

B C

D 1 4AQB ∼= 4APD (by Side-Angle-Side)2 ∠BAQ = 15◦, ∠BQA = 150◦ (by Corresponding parts of

congruent triangles are congruent)3 ∠QAP = 60◦ (∠QAP = 90◦ − ∠BAQ − ∠PAD)4 ∠AQP = 60◦, ∠APQ = 60◦ (by AQ ∼= AP, Isosceles triangle)5 AQ ∼= PQ (by 4AQP is equilateral)6 ∠BQP = 150◦ (∠BQP = 360◦ − ∠BQA− ∠AQP)7 4AQB ∼= 4PQB (by BQ ∼= BQ, ∠BQA ∼= ∠BQP

AQ ∼= PQ, Side-Angle-Side)8 AB ∼= BP (by Corresponding parts of

congruent triangles are congruent)9 BC ∼= BP, BC ∼= CP (by AB ∼= BC , AB ∼= BP, BP ∼= CP,

Transitivity of congruence)


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Synthetic Methods: Automated Proving in Coherent Logic

A FOL formula is said to be coherent if it is of the form:

A1(~x) ∧ . . . ∧ An(~x)⇒ ∃~y(B1(~x , ~y) ∨ . . . ∨ Bm(~x , ~y))

where universal closure is assumed, 0 ≤ n, 0 ≤ m, Ai denoteatomic formulae, Bi denote conjunctions of atomic formulaeNo function symbols and no negationCL formulae are sometimes called geometric (but more often itis assumed that geometric formulae allow infinitarydisjunctions)


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Properties of Coherent Logic (CL)

CL is simple, allows simple forward chaining proofsCL is suitable for automated deduction in geometry:

for producing human-readable proofs,for producing machine verifiable proofs,for proving statements of the form ∀∃,

CL provers are not very powerful on their own, but can bepowerful in synergy with other tools


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Coherent Logic – Toy Example

Axioms:p(x)⇒ ∃y(r(x , y) ∨ q(x , y))r(x , y)⇒ s(x , y)q(x , y)⇒ s(x , y)

Theorem: p(x)⇒ ∃y s(x , y)

Proof sketch:p(a), for fresh ap(a) implies there is b such that r(a, b) ∨ q(a, b)if r(a, b) then s(a, b)if q(a, b) then s(a, b)QED


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Coherentisation/Geometrisation

Many theories can be simply expressed in CL.Coherentisation/geometrisation: any FOL theory can betranslated into CL (possibly with additional predicate symbols)One FOL formula may give several CL formulaeIn translations, negations are pushed down to atomic formulaeand for every predicate symbol R , a new symbol R stands for¬R , with ∀~x(R(~x) ∧ R(~x)⇒ ⊥), ∀~x(R(~x) ∨ R(~x))

If a CL formula can be classically proved from a set of CLformulae, then it can be also constructively provedTranslation from FOL to CL is not necessarily constructive


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Provability and Proof System

The problem of provability in coherent logic is semi-decidableCL admits a simple proof system, such as:

Γ, ax,A1(~a), . . . ,An(~a),B1(~a,~b) ∨ . . . ∨ Bm(~a,~b) ` P

Γ, ax,A1(~a), . . . ,An(~a) ` Pemp (extended mp)

where ax = A1(~x) ∧ . . . ∧ An(~x)⇒ ∃~y(B1(~x,~y) ∨ . . . ∨ Bm(~x,~y))

Γ,B1(~c) ` P . . . Γ,Bn(~c) ` P

Γ,B1(~c) ∨ . . . ∨ Bm(~c) ` Pcs (case split)

Γ,Bi (~a,~b) ` ∃~y(B1(~a,~y) ∨ . . . ∨ Bm(~a,~y))as (assumption)

Γ,⊥ ` Pefq (ex falso quodlibet)

Any CL proof can be represented asproof ::= emp∗

(cs

(proof ≥2

)| as | efq

)Predrag Janicic Automated Reasoning in Geometry

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Proof System – Toy Example

AX:p(x)⇒ ∃y(r(x , y) ∨ q(x , y))r(x , y)⇒ s(x , y)q(x , y)⇒ s(x , y)

Theorem: p(x)⇒ ∃y s(x , y)

AX , p(a), r(a, b), s(a, b) ` ∃y s(a, y)AX , p(a), r(a, b) ` ∃y s(a, y)

AX , p(a), q(a, b), s(a, b) ` ∃y s(a, y)AX , p(a), q(a, b) ` ∃y s(a, y)

AX , p(a), r(a, b) ∨ q(a, b) ` ∃y s(a, y)

AX , p(a) ` ∃y s(a, y)


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Automated provers for CL

Euclid (Janicic and Kordic, 1992), Prolog, specialized forEuclidean geometry; export to natural languageArgoCLP (Stojanovic, Pavlovic, Janicic, 2010), C++, exportto natural language and Isabelle, automated simplification ofproofsCL (Bezem), export to Coq; used for proving Hessenberg’stheorem: Pappus’ axiom implies Desargues’ axiomGeologUI (Fisher), with graphical interfacecoherent (Berghofer), ML, within IsabelleGeo (De Nivelle), with learning lemmasCalypso (Nikolic and Janicic, 2015), based on CDCL SATsolving


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ArgoCLP Example Proof1. It holds that B−A−A (by th 3 1).2. From the facts that A−B−C , it holds that Col C A B (by ax 4 10 3).3. From the facts that (A, B) ∼= (A, D), it holds that (A, D) ∼= (A, B) (by th 2 2).4. It holds that A = B or A 6= B (by ax g1).

5. Assume that A = B.6. From the facts that (A, D) ∼= (A, B) and A = B it holds that (A, D) ∼= (A, A).7. From the facts that (A, D) ∼= (A, A), it holds that A = D (by ax 3).8. From the facts that A = B and A = D it holds that B = D.

This proves the conjecture.9. Assume that A 6= B.

Let us prove that A 6= C by reductio ad absurdum.10. Assume that A = C .

11. From the facts that A−B−C and A = C it holds that A−B−A.12. From the facts that A−B−A, and B−A−A, it holds that A = B (by th 3 4).13. From the facts that A 6= B, and A = B we get a contradiction.

Contradiction.Therefore, it holds that A 6= C .

14. From the fact that A 6= C , it holds that C 6= A (by the equality axioms).15. From the facts that C 6= A, Col C A B, (C , B) ∼= (C , D), and (A, B) ∼= (A, D), it holds that B

= D (by th 4 18).This proves the conjecture.


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Synthetic Methods: Resolution Method

Well-known, widely used uniform proof methodQuaife (1989) proved a number of Tarski’s theorem usingOTTERBeeson and Wos (2015) proved ≈ 200 Tarski’s theorems usinga modern version of OTTER


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Resolution Method: Example Proof

Example. If C is between B and D, and each of them is betweenA and E , then C is between A and E :

37 U−V−W ⇒W−V−U (by Axiom)45 U−V−X ,V−W−X ⇒ U−W−X (by Lemma B8)46 U−V−W ,U−W−X ⇒ U−V−X (by Lemma B9)74 U−V−X ,U−W−X ⇒ U−V−W ,U−W−V (by Lemma T9)77 A−B−E (by Hyp)78 B−C−D (by Hyp)79 A−D−E (by Hyp)80 ¬A−C−E (by negated goal)91 ¬A−C−D (by 80, 46, 79)92 ¬A−C−B (by 80, 46, 77)

109 ¬A−B−D (by 91, 45, 78)127 ¬B−C−A (by 92, 37)184 A−D−B (by 109, 74, 79, 77)253 ¬B−D−A (by 127, 46, 78)309 Contradiction! (by 184, 37, 253)


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Semisynthetic Methods: Area Method

Authors: Chou, Gao and Zhang (1992)Geometrical properties expressed in terms of expressions oversuitable „quantities“:

signed segment ratio ABCD

signed area of triangle SABC

Pythagora’s difference PABC = AB2

+ CB2 − AC

2).

It holds:A = B iff PABA = 0;Col A B C iff SABC = 0;AB ⊥ CD iff PABA 6= 0 ∧ PCDC 6= 0 ∧ PACD = PBCD ;AB ‖ CD iff PABA 6= 0 ∧ PCDC 6= 0 ∧ SACD = SBCD , etc.


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Area Method (2)

There is a number of construction steps (used for describing aconfiguration)For instance, Inter X A B C D constructs a point X as theintersection of the lines AB and CDFor each type of construction and each type of expression thereis a lemma that enables elimination of the constructed point


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Area Method example

Example (Ceva’s theorem). A,B,C ,P are free points and Inter D A P B C ,Inter E B P A C , Inter F C P A B,...

A 6= P ∧ B 6= C ∧ AP ∦ BC ∧ SAPD = 0 ∧ SBCD = 0 ∧B 6= P ∧ A 6= C ∧ BP ∦ AC ∧ SBPE = 0 ∧ SACE = 0 ∧C 6= P ∧ A 6= B ∧ CP ∦ AB ∧ SCPF = 0 ∧ SABF = 0 ∧

F 6= B ∧ D 6= C ∧ E 6= A ∧ AF ‖ FB ∧ BD ‖ DC ∧ CE ‖ EA ⇒ AFFB

BDDC

CEEA

= 1

A

B CD

F

EP

AFFB

BDDC

CEEA

=

= SAPCSBCP

BDDC

CEEA

(the point F is eliminated)

= SAPCSBCP

SBPASCAP

CEEA

(the point D is eliminated)

= SAPCSBCP

SBPASCAP

SCPBSABP

(the point E is eliminated)= 1


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Area Method Axioms

1 AB = 0 if and only if the points A and B are identical2 SABC = SCAB3 SABC = −SBAC4 If SABC = 0 then AB + BC = AC5 There are points A, B, C such that SABC 6= 0 (dimension; not all points are

collinear)6 SABC = SDBC + SADC + SABD (dimension; all points are in the same plane)7 For each element r of F , there exists a point P, such that SABP = 0 and

AP = rAB (construction of a point on the line)8 If A 6= B,SABP = 0,AP = rAB,SABP′ = 0 and AP′ = rAB, then P = P′

(unicity)9 If PQ ‖ CD and PQ

CD= 1 then DQ ‖ PC (parallelogram)

10 If SPAC 6= 0 and SABC = 0 then ABAC

= SPABSPAC

(proportions)11 If C 6= D and AB ⊥ CD and EF ⊥ CD then AB ‖ EF12 If A 6= B and AB ⊥ CD and AB ‖ EF then EF ⊥ CD13 If FA ⊥ BC and SFBC = 0 then 4S2

ABC = AF 2BC2 (area of a triangle)


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Semisynthetic Methods: Full Angle Method and VectorMethod

Authors: Chou, Gao and Zhang (1990s)Similar in spirit to the area method:

For each type of construction and each type of expression thereis a lemma that enables elimination of the constructed point


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Algebraic Methods

Grobner bases method (by Buchberger)Wu’s method (one of "the four new great Chineseinventions"!)Work by algebrization and, essentially, by checking if apolynomial (corresponding to the goal) belongs to an ideal(corresponding to the configuration)Can produce NDG conditions (in algebraic terms)Extremely powerful, but do not produce readable proofs


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Implementations of provers and integration

Only a few dozens implementations of proversOurs:

Goran Predovic (and Predrag Janicic): GB method and Wu’smethod (C++)Ivan Petrovic (and Predrag Janicic): reimplemented in JavaC++ version (plus area method, Janicic): used within GCLCJava version used within GeoGebra (educational software withtens of millions of users)


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Integration - GCLC


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Ingegration - GeoGebra


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Hilbert’s axiomsHilbert’s axiomsCL Vernacular

Formalization of axiomatic systems: Hilbert

Early XX century, five groups of axiomsSome parts formalized

within Isabelle (Scott/Fleuriot)within Coq (Narboux et al)using a CL prover (Stojanovic-Djurdjevic/Marinkovic)


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Formalization of axiomatic systems: Tarski (SST)

Metamathematische Methoden in der Geometrie, by WolframSchwabhauser, Wanda Szmielew, and Alfred Tarski (SST)Axioms were developed for decades and are super elegant,there are just eleven of them, belong to FOL, the axioms donot use defined notions.Geometry is expressed in terms of FOL with equality, withpoints as only primitive objects and with two primitivepredicate symbols – cong and bet (for betweeness)SST book has been a subject of several automation andformalization projects


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Tarski’s axioms (SST)

A1 Symmetry AB ∼= BAA2 Pseudo-Transitivity AB ∼= CD ∧ AB ∼= EF ⇒ CD ∼= EFA3 Cong Identity AB ∼= CC ⇒ A = BA4 Segment construction ∃E ,A−B−E ∧ BE ∼= CDA5 Five-segment AB ∼= A′B′ ∧ BC ∼= B′C ′∧

AD ∼= A′D′ ∧ BD ∼= B′D′∧A−B−C ∧ A′−B′−C ′ ∧ A 6= B ⇒ CD ∼= C ′D′

A6 Between Identity A−B−A⇒ A = BA7 Inner Pasch A−P−C ∧ B−Q−C ⇒ ∃X ,P−X−B ∧ Q−X−AA8 Lower Dimension ∃ABC ,¬A−B−C ∧ ¬B−C−A ∧ ¬C−A−BA9 Upper Dimension AP ∼= AQ ∧ BP ∼= BQ ∧ CP ∼= CQ ∧ P 6= Q

⇒ A−B−C ∨ B−C−A ∨ C−A−B.A10 Parallel postulate ∃XY (A−D−T ∧ B−D−C ∧ A 6= D ⇒

A−B−X ∧ A−C−Y ∧ X−T−Y )A11 Continuity ∀Φ ∀Ψ ∃A ∀X ∀Y ((X ∈ Φ ∧ Y ∈ Ψ⇒ A−X−Y )

⇒ ∃B ∀X ∀Y (X ∈ Φ ∧ Y ∈ Ψ⇒ X−B−Y )


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Formalization of axiomatic systems: SST

Large part formalizedin Coq (Narboux et al)using resolution (Beeson and Wos)formalized using CL (Stojanovic-Djurdjevic/Narboux/Janicic)”A1-A10 and Hilbert’s four groups are biinterpretable“(Braun/Narboux)Continuity still not formalized


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CL Vernacular

Vernacular is the everyday, ordinary language of the people ofsome regionDe Bruijn used the term, trying to “put a substantial part ofthe mathematical vernacular into the formal system”Wiedijk: “Apparently there is a canonical style of presentingmathematics that people discover independently: somethinglike a natural mathematical vernacular.”The language discussed by Wiedijk is closely related to the CLproof language


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CL Vernacular support

A proof representation called “coherent logic vernacular” canlink different proof formats and toolsThe proposed proof representation is accompanied by acorresponding XML format, specified by a DTD

Automated theorem provers

ArgoCLP

XML DTD

Interactive theorem provers

Isar Coq ... LATEX HTML


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Theorem th 4 19 : ∀ (A:point) (B:point) (C :point) (D:point), (bet A B C ∧ cong A B A D ∧ cong CB C D) → B = D.Proof.intros.assert (bet B A A) by applying (th 3 1 B A ) .assert (col C A B) by applying (ax 4 10 3 A B C ) .assert (cong A D A B) by applying (th 2 2 A B A D ) .assert (A = B ∨ A 6= B) by applying (ax g1 A B ) .by cases on (A = B ∨ A 6= B).- {

assert (cong A D A A) by (substitution).assert (A = D) by applying (ax 3 A D A ) .assert (B = D) by (substitution).conclude.}

- {assert (A = C ∨ A 6= C) by applying (ax g1 A C ) .by cases on (A = C ∨ A 6= C).- {

assert (bet A B A) by (substitution).assert (A = B) by applying (th 3 4 A B A ) .assert (False) by (substitution).contradict.}

- {assert (C 6= A) by (substitution).assert (B = D) by applying (th 4 18 C A B D ) .conclude.}

}Qed. Predrag Janicic Automated Reasoning in Geometry

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Coherentisation and Computer Supported Proving: SSTbook

The process of coherentisation of the axioms and theoremswas straightforwardFrom the original 179 theorems (from chapters 1 to 12 of thebook), the process gave 238 CL formulae (while 5 schematictheorems involving n-tuples were considered only for n = 2)Proving process for one theorem:

All axioms and theorems that precede the theorem are passedto resolution provers (Vampire, E, and SPASS)If one of the resolution provers proves the conjecture, the list ofused axioms/theorems is used for proving the conjecture againuntil the list of used axioms/theorems remains unchangedWith the obtained list of axioms/theorems, ArgoCLP prover isinvoked, and (if successful) the proof is exported in the CLvernacular XML formatPredrag Janicic Automated Reasoning in Geometry

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Results and Outputs

ArgoCLP (supported by resolutionprovers) proved 37% of the theoremsautomaticallyGiven preceeding theorems, thepercentage rises to 42%One of the outputs of the study: a digitalversion of the book, with all axioms,definitions, theorems, and generatedproofs filled-in, all in the natural languageform


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CL and Construction Problems

Statements of the form ∀∃ naturally arise in geometry, forinstance – in construction problemsFor a construction problem, roughly said, the task is to proveconstructively a theorem of the form:

∀~x∃~y Ψ(~x ;~y)

Sometimes even more (solution exists iff Φ(~x)):

∀~x(Φ(~x)⇒ ∃~y Ψ(~x ;~y) ∧ ¬Φ(~x)⇒ ¬∃~y Ψ(~x ;~y))


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CL and Construction Problems – Example

Example (Wernick’s problem 4): Given points A, B, and G,construct a triangle ABC, such that G is the centroid of ABC.A careful analysis leads to the theorem that gives a fullcharacterization of solvability:

∀A,B,G (¬collinear(A,B,G )⇔

∃C (¬collinear(A,B,C ) ∧ centroid(G ,A,B,C )))

With the help of our solver for construction problemsArgoTriCS, the above conjectures can be proved by ArgoCLP


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Compendium of Construction Problems

Vesna Marinkovic’s Compendium


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Conclusions

A lot has been achieved ...... but we still cannot solve some primary school geometryproblems!Provers with wider scopeReadable proofsGradual development of geometryCompletely formalized school geometryMeta-theorems on geometry


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After Conclusions

And, after the conclusions, a memorable quote by Larry Wos:

The beauty of a theorem from mathematics, thepreciseness of an inference rule in logic, the intrigue of apuzzle, and the challenge of a game – all are present inthe field of automated reasoning.

And, let’s add – especially in automated reasoning in geometry!


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