Mechanical Theorem Proving in Tarski's geometry.

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Julien Narboux

To cite this version:

Julien Narboux. Mechanical Theorem Proving in Tarski’s geometry.. Automated Deduction in Geom- etry 2006, Francisco Botana, Aug 2006, Pontevedra, Spain. pp.139-156, �10.1007/978-3-540-77356-6�.

�inria-00118812�

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JulienNarboux

ÉquipeLogiCal, LIX

ÉolePolytehnique,91128 PalaiseauCedex,Frane

Julien.Narbouxinria.fr,

http://www.lix.polytehnique. fr/L abo/J ulien .Nar boux/

Abstrat. Thispaperdesribesthemehanizationoftheproofsofthe

rstheighthaptersofShwabäuser,SzmielewandTarski'sbook:Meta-

mathematisheMethodeninderGeometrie.Thegoalofthisdevelopment

istoprovidefoundationsforotherformalizationsofgeometryandimple-

mentations ofdeisionproedures. Weompare themehanizedproofs

with the informal proofs. We also ompare this piee of formalization

with the previous work done about Hilbert's Grundlagen der Geome-

trie.Weanalyzethedierenesbetweenthetwoaxiomsystemsfromthe

formalizationpointofview.

1 Introdution

Eulid is onsidered asthe pioneerof theaxiomati method, in theElements,

startingfromasmallnumberofself-evidenttruths,alledpostulates,orommon

notions,hederivesbypurelylogialrulesmostofthegeometrialfatsthatwere

disoveredin thetwoorthree enturiesbeforehim. Butupona loserreading

ofEulid'sElements,wendthathedoesnotadhereasstritlyasheshouldto

theaxiomatimethod.Indeed,atsomestepsinertainproofsheusesamethod

ofsuperpositionoftriangles andthis kindofjustiationsannotbederived

fromhis setofpostulates.

In1899, in der Grundlagen der Geometrie, Hilbert proposed anew axiom

systemtollthegapsin Eulid'ssystem.

Reently, the task onsisting in mehanizing Hilbert's Grundlagen der Ge-

ometrie has been partially ahieved. A rstformalization using theCoq proof

assistant[Coq04℄wasproposedbyChristopheDehlinger,Jean-FrançoisDufourd

andPasalShrek[DDS00℄.This rstapproahwasrealizedin anintuitionist

setting, and onluded that the deidability of point equality and ollinearity

is neessary to perform Hilbert's proofs. Another formalization using the Is-

abelle/Isarproofassistant[Pau℄ wasperformed by JaquesFleuriotandLaura

Meikle [MF03℄. These formalizationshaveonludedthat Hilbert proofsarein

fat not fullyformal 1

, in partiulardegenerated ases areoften impliitin the

1

NotethatinthedierenteditionsofdieGrundlagenderGeometrietheaxiomswere

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presentationofHilbert.Theproofsanbemade morerigorousby mahine as-

sistane.

Inthe early60s, WandaSzmielew and AlfredTarskistarted theprojetof

atreatyaboutthefoundationsofgeometrybasedonanotheraxiomsystemfor

geometrydesignedbyTarskiinthe20s 2

.Asystematidevelopmentofeulidean

geometrywassupposedtoonstitutetherstpartbuttheearlydeathofWanda

Szmielewputanend tothisprojet.Finally, WolframShwabhäuserontinued

the projetof Wanda Szmielew and AlfredTarski.He published thetreatyin

1983 in German: Metamathematishe Methoden in der Geometrie [SST83℄. In

[Qua89℄,ArtQuaifeusesageneralpurposetheoremprovertoautomatetheproof

ofsomelemmasinTarki'sgeometry.Inthispaperwedesribeourformalization

ofthersteighthaptersofthebookofWolframShwabhäuser,WandaSzmielew

andAlfredTarskiintheCoqproofassistant.

Wewillrst desribethedierentaxiomsof Tarski'sgeometry andgivean

historyofthedierentversionsofthis axiomsystem.Then wepresentourfor-

malizationoftheaxiomsystemandthemehanizationofoneexampletheorem.

Finally weompare ourformalizationwith existing ones and ompareTarski's

axiomatisystemwithHilbert's systemfrom themehanizationpointofview.

2 Motivations

We aimat twoappliations: the rstoneis the useof aproof assistantin the

eduationtoteahgeometry[Nar05℄,theseondoneistheproofofprogramsin

theeldomputationalgeometry.

Thesetwothemeshavealreadybeenaddressedbytheommunity.Frédérique

Guilhot has realized a largeCoq development about eulidean geometry as it

taught in frenh highshool [Gui05℄. Conerning the proof of programs in the

eld of omputational geometry we an ite the formalization of onvexhulls

algorithms by David Pihardie and Yves Bertot in Coq [PB01℄ and by Laura

Meikle and JaquesFleuriot in Isabelle [MF05℄. In[Nar04℄, we havepresented

the formalization and implementation in the Coq proof assistant of the area

deisionproedureofChou,GaoandZhang[CGZ94℄.

Formalizing geometry in a proof assistant has not only the advantage of

providingaveryhigh levelofondenein theproofgenerated,italsopermits

to ombine proofs aboutgeometry with other kindofproofssuhasthe proof

of the orretness of a program for instane. The goal whih onsist in using

thesameformaldevelopmentaboutgeometryfordierentpurposesanonlybe

ahievedifweusethesameaxiomatisystem.Thisisnottheaseforthetime

being.

Thegoalofourmehanizationistodoarststepinthis diretion.Weaim

at providing very lear foundations for other formalizations of geometry and

implementationsofdeisionproedures.

2

Thesehistorial pieesofinformation aretakenfromtheintrodutionofthepubli-

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Compared to Frédérique Guilhot formalization [Gui05℄, our development

should be onsidered low level. Our formalization has the advantageof being

based on the axiom system of Tarski whih is of an extreme simpliity: two

prediatesandelevenaxioms.Butthissimpliityhasaprie,ourformalization

isnotadaptedtotheontextofeduation.Indeed,someintuitivelysimpleprop-

ertiesarehardto provein thisontext.Forinstane,theproofoftheexistene

of the midpoint of segmentis obtained only at the end of the eighth hapter

after about150 lemmas and 4000 lines of proof. The small numberof axioms

impose a shedulingof the lemmas whih is notalwaysintuitive(some simple

propertiesanonlybeprovedlateinthedevelopment).

3 Tarski's axiom system

AlfredTarskiworkedontheaxiomatizationandmeta-mathematisofeulidean

geometryfrom1926untilhisdeathin1983.Severalaxiomsystemswereprodued

byTarskiandhisstudents.Inthissetion,werstgiveaninformaldesription

of the propositions whih appeared in the dierent versions of Tarski'saxiom

system,then weprovideanhistoryofthese versions andnallywepresentthe

versionwehaveformalized.

The axioms are based on rst order logi and two prediates: betweeness

and equidistane (or ongruene). The ternary betweeness prediate β A B C

informallystatesthat B ^lies^on^the^line AC ^betweenA ^andC^. ^The^quaternary

equidistane prediateAB≡CDînformally^means^that ^the^distane^from A^to Bîsêqual^to^the^distane^fromC^toD^.În^Târski's^geometry,ônlyâ^setôf^points

isassumed.Inpartiular,linesaredened bytwodistintpoints 3

.

3.1 Axioms

Wereprodueherethelistofpropositionswhihappearinthedierentversions

of Tarski's axiom system. We adopt the same numbering as in [TG99℄. Free

variablesareonsideredto beimpliitlyquantieduniversally.

1 ^Reexivity^forequidistane

AB≡BA 2 Pseudo-transitivityforequidistane

AB≡P Q∧AB≡RS⇒P Q≡RS 3 ^Identity^forequidistane

AB≡CC ⇒A=B

3

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4 ^Segment^onstrution

∃X, β Q A X∧AX≡BC

Thesegmentonstrutionaxiomstatesthat oneanbuild apointonaray

atagivendistane.

b

Q

b

A

b

B

b

C

b

X

Fig.1.Segmentonstrution

5 ^Five^segments

A6=B∧β A B C∧β A^′B^′C^′∧

⇒CD≡C^′D^′ AB≡A^′B^′∧BC≡B^′C^′∧AD≡A^′D^′∧BD≡B^′D^′

5¹ ^Five^segments^(variant)

A6=B∧B 6=C∧β A B C∧β A^′B^′C^′∧

⇒CD≡C^′D^′ AB≡A^′B^′∧BC≡B^′C^′∧AD≡A^′D^′∧BD≡B^′D^′

Thisseondversiondiersfrom therstoneonlybytheonditionB 6=C^. 6 ^Identity^for^betweeness

β A B A⇒A=B

TheoriginalPashaxiomstatesthatifalineintersetsonesideofatriangle

andmissesthethreevertexes,thenitmustintersetoneoftheothertwosides.

7 ^Pash^(inner^form)

β A P C∧β B Q C⇒ ∃X, β P X B∧β Q X A 7¹ ^Pash^(outer^form)

β A P C∧β Q C B⇒ ∃X, β A X Q∧β B P X

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b

A

b

B

b

C

b

Q

b

P

b

X

b

A

b

B

b

Q

b

X

b

C

b

P

b

A

b

B

b

C

b^D b

Y

b

X

b

T

Innerform Outerform Weakform

Fig.2. AxiomsofPash

72 ^Pash^(outer^form)^(variant)

β A P C∧β Q C B⇒ ∃X, β A X Q∧β X P B 7³ ^weak^Pash

β A T D∧β B D C⇒ ∃X, Y, β A X B∧β A Y C∧β Y T X

Dimensionaxiomsprovideupperand lowerbound forthedimension ofthe

spae.Notethatlowerboundaxiomsfordimensionnâre^the^negationôfûpper

boundaxiomsforthedimensionn−1^. 8(2) ^Dimension,^lower^bound²

∃ABC,¬β A B C∧ ¬β B C A∧ ¬β C A B

Therearethreenonollinearpoints.

8(n) ^Dimension,^upper^boundn

∃ABCP1P2. . . Pn−1, V

1≤i<j<nPi6=Pj∧ Vn−1

i=2 AP1≡APi∧BP1≡BPi∧CP1≡CPi∧

¬β A B C∧ ¬β B C A∧ ¬β C A B

9(1) ^Dimension,^upper^bound¹

β A B C∨β B C A∨β C A B

Threepointsarealwaysonthesameline.

9(n) ^Dimension,^upper^boundⁿ V

1≤i<j≤nPi6=Pj∧ Vn

i=2AP1≡APi∧BP1≡BPi∧CP1≡CPi

⇒β A B C∨β B C A∨β C A B

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91(2) ^Dimension,^upper^bound ²^(variant)⁴

∃Y,(ColXY A∧β B Y C)∨(ColXY B∧β C Y A)∨(ColXY C∧β A Y B) 10 ^Eulid's^axiom

β A D T ∧β B D C∧A6=D⇒ ∃X, Y β A B X∧β A C Y ∧β X T Y 10¹ ^Eulid's^axiom^(variant)

β A D T ∧β B D C∧A6=D⇒ ∃X, Y β A B X∧β A C Y ∧β Y T X 11 ^Continuity

∃a,∀xy,(x∈X∧y∈Y ⇒β a x y)⇒ ∃b,∀xy, x∈X∧y∈Y ⇒β x b y

Shema 11 ^Elementary^Continuity^(shema)

∃a,∀xy,(α∧β ⇒β a x y)⇒ ∃b,∀xy, α∧β ⇒β x b y

where αândβ âre^rstôrder ^formulas,^suh^that a^,b ândy ^do^notâppear^free

in α^;a^,b ^andx^do^not^appear^freeⁱⁿβ^.

A geometry dened by the elementary ontinuity axiomshemainstead of

thehigherorderontinuityaxiomisalledelementary.

12 ^Reexivity^ofβ

β A B B B îsâlways^betweenAândB^.

14 ^Symmetry^ofβ

β A B C⇒β C B A

IfBîs ^betweenAând C^thenB îs^betweenC ândA^. 13 Compatibilityofequalitywithβ

A=B⇒β A B A 19 Compatibilityofequalitywith ≡

A=B⇒AC ≡BC

4 ColABC ^is^dened^byβ A B C∨β B C A∨β C A B

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15 ^Transitivity(inner)ofβ

β A B D∧β B C D⇒β A B C 16 ^Transitivity(outer)ofβ

β A B C∧β B C D∧B6=C⇒β A B D 17 Connetivity(inner)ofβ

β A B D∧β A C D⇒β A B C∨β A C B 18 Connetivity(outer)ofβ

β A B C∧β A B D∧A6=B ⇒β A C D∨β A D C 20 ^T^riangle^onstrution^uniity

AC≡AC^′∧BC≡BC^′∧ β A D B∧β A D^′B∧β C D X∧ β C^′D^′X∧D6=X∧D^′6=X

⇒C=C^′

201 ^Triangle^onstrution^uniity^(variant)

A6=B∧

AC≡AC^′∧BC≡BC^′∧ β B D C^′∧(β A D C∨β A C D)

⇒C=C^′

21 ^T^riangle^onstrution^existene

AB≡A^′B^′⇒ ∃CX,AC≡A^′C^′∧BC≡B^′C^′∧

β C X P ∧(β A B X∨β B X A∨β X A B)

3.2 History

Tarskibeganto workon his axiomsystemin 1926and presentedit during his

letures at Warsaw university 5

. He submitted it for publiation in 1940 and

wasrst published in his rst form in 1967 [Tar67℄. This version ontains 20

axiomsandoneshema.Aseondversion,abitsimplerwaspublishedin[Tar51℄.

Thisrstsimpliationonsistonlyinonsideringalogiwithbuilt-inequality,

axioms 13 and 19 are then useless. This seond versionwasfurther simplied

byEvaKallin,SottTaylorandTarski intoasystemoftwelveaxioms[Tar59℄.

ThelastsimpliationwasobtainedbyGuptainitsthesis[Gup65℄,hegivesthe

proofthat twomoreaxiomsanbederivedfromtheremainingones.

Figure3givesthelist of axiomsontainedin eahof these axiomsystems.

Figure4providesthenal listofaxiomsthatweused inourformalization.

5

Weuse[TG99℄andthefootnotesin[Tar51℄togiveaquik historyofthedierent

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Year: 1940 1951 1959 1965 1983

Referene: [Tar67℄ [Tar51℄ [Tar59 ℄ [Gup65 ℄ [SST83 ℄

Axioms: 1 1 1 1 1

2 2 2 2 2

3 3 3 3 3

4 4 4 4 4

51 51 5 5 5

6 6 6 6

72 72 71 71 7

8(2) 8(2) 8(2) 8(2) 8(2)

9₁(2) 9₁(2) 9(2) 9(2) 9(2)

10 10 101 101 10

11 11 11 11 11

12 12

13

14 14

15 15 15 15

16 16

17 17

18 18 18

19

20 →20₁

21 21

Nbofaxioms: 20 18 12 10 10

+ + + + +

1shema1shema1shema1shema1shema

Fig.3.HistoryofTarski'saxiomsystems.

Identityβ A B A⇒(A=B)

Pseudo-TransitivityAB≡CD∧AB≡EF⇒CD≡EF

ReexivityAB≡BA

IdentityAB≡CC⇒A=B

Pash∃X, β A P C∧β B Q C⇒β P x B∧β Q x A

Eulid∃XY, β A D T∧β B D C∧A6=D⇒ β P x B∧β Q x A

5segments

AB≡A^′B^′∧BC≡B^′C^′∧ AD≡A^′D^′∧BD≡B^′D^′∧

β A B C∧β A^′B^′C^′∧A6=B⇒CD≡C^′D^′

Constrution∃E, β A B E∧BE≡CD

LowerDimension∃ABC,¬β A B C∧ ¬β B C A∧ ¬β C A B

UpperDimensionAP ≡AQ∧BP ≡BQ∧CP≡CQ∧P 6=Q

⇒β A B C∨β B C A∨β C A B

Continuity∀XY,(∃A,(∀xy, x∈X∧y∈Y ⇒β A x y))⇒

∃B,(∀xy, x∈X⇒y∈Y ⇒β x B y).

Fig.4.Tarski'saxiomsystem(Formalizedversion-11axioms).

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4 Formalization in Coq

Themehanizationoftheproofwehaverealizedproveformallythatthesimpli-

ationsoftherstversionofTarski'saxiomsystemareorret.Theunneessary

axiomsarederivedfromtheremainingones.

Now, we provide aquik overview of the ontent of eah hapter. We will

onlydetailanexampleproofinthenextsetion.

The rst hapter ontains the axioms and the denition of the ollinearity

prediate(noted Col).

The seond hapter ontainssomebasipropertiesoftheequidistanepred-

iate (noted Cong). It ontains also the proof of the uniity of the point

onstrutedthankstothesegmentonstrutionaxiom.

The third hapter ontainssomepropertiesofthebetweenessprediate(no-

tedBet).Itontainsin partiulartheproofoftheaxioms12,14and16.

The fourth hapter ontainstheproofofseveralpropertiesof Cong,Coland

Bet.

The fth hapter ontainssomepseudo-transitivityproperties ofbetweeness

andthedenition ofthelengthomparisonprediate(notedle)withsome

assoiated properties. It inludes in partiular the proofsof the axioms17

and18.

The sixth hapter denestheoutprediatewhihmeansthatapointlieson

alineoutofasegment.Thisprediateisusedtoprovesomeotherproperties

of Cong,ColandBetsuhastransitivitypropertiesforCol.

The seventh hapter denesthemidpointofasegmentandsymmetripoints.

Ithastobenotedthatat thissteptheexisteneofthemidpointisnotde-

rivedyet.

The eighthhapter ontainsthedenitionoftheperpendiularprediate(no-

tedPerp),andtheproofofsomerelatedpropertiessuhastheexisteneof

the foot of the perpendiular. Finally, the existene of the midpoint of a

segmentisderived.

4.1 Tworuial lemmas

OurformalizationfollowsstritlythelinesofthebookbyShwabhäuser,Szmielew

and Tarskiexeptin thefth hapter whereweintrodue tworuials lemmas

whih do notappear in the original text. These twolemmas allows to dedue

theequalityoftwopointswhihlieonasegmentunderanhypothesesinvolving

distanes.

∀ABC, β A B C∧AC ≡AB⇒C=B

b b b

A B C

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∀ABDE, β A D B∧β A E B∧AD≡AE⇒D=E.

b b b b

A D E B

4.2 A omparisonbetween the formaland informal proofs

Wereproduehereoneofthenontrivialproofs:theproofduetoGupta[Gup65℄

that axiom18an bederived fromtheremaining ones.Wetranslatetheproof

from[SST83℄andprovidein parallelthemehanizedproofasaCoqsript.

ForthereadernotfamiliarwiththeCoqproofassistant,weprovideaquik

informalexplanationoftheroleofthemaintatis weuseinthisproof.

assert isusedtostatewhatwewanttoprove.Whenitisfollowedby...this

meansthat thisassertionanbeprovedautomatially.

DeompExAnd , given an existential hypotheses, introdues the witness of the

existentialanddeomposetheknowledgeaboutit.

apply isusedtoapplyalemmaortheorem.

Tarski,sTarski,Between,... areautomatitatiswhihtrytoprovetheur-

rentgoal.Informallythisanbereadasbysimplepropertiesofbetweeness

orbydiretappliationofoneoftheaxioms.

unfold replaessomethingbyitsdenition.

ases_equality performareasoningbyasesontheequalityof twopoints.

b b

b

b b

b

A B

D C'

B' B

C D'

E

Fig.5. Proofofaxiom18

Theorem1 (Gupta). A6=B∧β A B C∧β A B D⇒β A C D∨β A D C

Preuve:LetC^′ ^andD^′ ^be^points^suh^that^:

β A D C^′∧DC^′≡CD^andβ A C D^′∧CD^′≡CD

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assert (exists C', Bet A D C' /\ Cong D C' C D)...

DeompExAnd H2 C'.

assert (exists D', Bet A C D' /\ Cong C D' C D)...

DeompExAnd H2 D'.

Wehavetoshow that C=C^′ ^orD=D^′^.

LetB ^andB^′′ ^p^oints^suh^that ^:

β A C^′B^′∧C^′B^′≡CB^andβ A D^′B^′′∧D^′B^′′≡DB

assert (exists B', Bet A C' B' /\ Cong C' B' C B)...

DeompExAnd H2 B'.

assert (exists B'', Bet A D' B'' /\ Cong D' B'' D B)...

DeompExAnd H2 B''.

Using thelemma 2.11 6

wean deduethat BC^′≡B^′′C ^and^thatBB^′ ≡B^′′B^.

assert (Cong B C' B'' C).

eapply l2_11.

3:apply ong_ommutativity.

3:apply ong_symmetry.

3:apply H11.

Between.

esTarski.

assert (Cong B B' B'' B).

eapply l2_11;try apply H2;Between.

Byuniityof the segment onstrution,we knowthatB^′′=B^′^.

assert (B''=B').

apply onstrution_uniity with

(Q:=A) (A:=B) (B:=B'') (C:=B) (x:=B'') (y:=B');Between...

smart_subst B''.

WeknowthatF SC

BCD^′C^′ B^′C^′DC

(Thepointsformavesegmentsonguration).

assert (FSC B C D' C' B' C' D C).

unfold FSC;repeat split;unfold Col;Between;sTarski.

2:eapply ong_transitivity.

2:apply H7.

2:sTarski.

apply l2_11 with (A:=B) (B:=C) (C:=D') (A':=B') (B':=C') (C':=D);

Between;sTarski;esTarski.

HeneC^′D^′≡CD ^(beauseîfB 6=C ^the ^ve^segmentsâxiom^gives ^the ônlu-

sionandif B=C ^we^an ^use^the hypotheses).

6

Thelemma 2.11 statesthat β A B C∧β A^′B^′C^′∧AB ≡A^′B^′∧BC ≡B^′C^′ ⇒ AC≡A^′C^′^.

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assert (Cong C' D' C D).

ases_equality B C.

(* First ase *)

treat_equalities.

eapply ong_transitivity.

apply ong_ommutativity.

apply H11.

Tarski.

(* Seond ase *)

eapply l4_16;try apply H3...

Using theaxiom ofPash, thereisapointE ^suh^that^: β C E C^′∧β D E D^′

assert (exists E, Bet C E C' /\ Bet D E D').

eapply inner_pash;Between.

DeompExAnd H13 E.

Wean deduethat IF S

ded^′c ded^′c^′

andIF S cec^′d

cec^′d^′

.

assert (IFSC D E D' C D E D' C').

unfold IFSC;repeat split;Between;sTarski.

apply H7.

sTarski.

assert (IFSC C E C' D C E C' D').

unfold IFSC;repeat split;Between;sTarski.

apply H5.

sTarski.

HeneEC≡EC^′ ^andED≡ED^′^.

assert (Cong E C E C').

eapply l4_2;eauto.

assert (Cong E D E D').

eapply l4_2;eauto.

Supposethat C6=C^′^.^We^have ^to^show ^that D=D^′⁷^.

7

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ases_equality C C'.

smart_subst C'.

assert (E=C).

eTarski.

smart_subst E.

unfold IFSC, FSC,Cong_3 in *;intuition.

From thehypotheses,wean inferthatC6=D^′^.

assert (C<>D').

unfold not;intro.

treat_equalities...

Using thesegmentonstrution axiom, weknowthat therearepointsP^,Q^and R ^suh^that^:

β C^′C P∧CP ≡CD^′ ^andβ D^′C R∧CR≡CE^andβ P R Q∧RQ≡RP

assert (exists P, Bet C' C P /\ Cong C P C D')...

DeompExAnd H21 P.

assert (exists R, Bet D' C R /\ Cong C R C E)...

DeompExAnd H21 R.

assert (exists Q, Bet P R Q /\ Cong R Q R P)...

DeompExAnd H21 Q.

HeneF SC

D^′CRP P CED^′

,soRP ≡ED^′ ^andRQ≡ED^.

assert (FSC D' C R P P C E D').

unfold FSC;unfold Cong_3;intuition...

eapply l2_11.

apply H25.

3:apply H26.

Between.

sTarski.

assert (Cong R P E D').

eapply l4_16.

apply H21.

auto.

assert (Cong R Q E D).

apply H28.

apply H22.

sTarski.

Wean inferthatF SC

D^′EDC P RQC

,

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assert (FSC D' E D C P R Q C).

unfold FSC;unfold Cong_3;intuition...

eapply l2_11.

3:eapply ong_ommutativity.

3:eapply ong_symmetry.

3:apply H22.

Between.

sTarski.

sousing lemma2.11 wean onlude thatD^′D ≡P Q^andCQ≡CD ^(beause

the ase D^′ 6=E îs^solvedûsing ^the ^ve ^segmentsâxiom, ândⁱⁿ ^the ôther âse

wean dedue thatD^′=D ^andP =Q^).

ases_equality D' E.

(* First ase *)

treat_equalities...

sTarski.

(* Seond ase *)

eapply l4_16;eauto.

Using the theorem 4.17 8

, as R 6=C ând R^,C ând D^′ âre ôllinear ^we ân

onlude thatD^′P ≡D^′Q^.

assert (R<>C).

unfold not;intro.

treat_equalities...

assert (Cong D' P D' Q).

apply l4_17 with (A:=R) (B:=C) (C:=D').

assumption.

3:apply H32.

unfold Col;left;Between.

sTarski.

As C6=D^′^,Col CD^′B ândCol CD^′B^′^, ^weân âlso ^deduê ^that BP ≡BQ

andB^′P ≡B^′Q^.

assert (Cong B P B Q).

eapply l4_17; try apply H20;auto.

unfold Col;right;right;Between.

(* *)

assert (Cong B' P B' Q).

eapply l4_17 with (C:=B').

apply H20.

8

Thetheorem4.17statesthatA6=B∧ColABC∧AP ≡AQ∧BP ≡BQ⇒CP≡ CQ^.

(16)

unfold Col.

Between.

assumption.

AsC6=D^′^,^we^haveB6=B^′ ^and^asCol BC^′B^′ ^we^haveC^′P ≡C^′Q^.

ases_equality B B'.

subst B'.

unfold IFSC,FSC, Cong_3 in *;intuition.

lean_dupliated_hyps.

lean_trivial_hyps.

assert (Bet A B D').

Between.

assert (B=D').

eTarski.

treat_equalities.

Tarski.

assert (Cong C' P C' Q).

eapply l4_17.

apply H37.

unfold Col;right;left;Between.

auto.

AsC6=C^′ ^andCol C^′CP ^we^haveP P ≡P Q^.

assert (Cong P P P Q).

eapply l4_17.

apply H19.

unfold Col;right;right;Between.

auto.

Using theidentity axiomfor equidistane, wean deduethat P =Q^.

assert (P=Q).

eapply ong_identity.

apply ong_symmetry.

apply H39.

As P Q≡D^′D^,^we^also ^haveD=D^′^.

subst Q.

assert (D=D').

eapply ong_identity with (A:=D) (B:=D') (C:=P).