Formalizing Analytic Geometries · Our formalization of analytic geometry aims at establishing the...
Transcript of Formalizing Analytic Geometries · Our formalization of analytic geometry aims at establishing the...
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Formalizing Analytic Geometries
Danijela [email protected]
Department of Computer Science
Faculty of Mathematics
University of Belgrade
ADG 2012
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Outline
1 Introduction
2 Formalizing Cartesian Geometry
3 Using Isometric Transformations
4 Tarski’s geometryAxioms of Betweenness
5 Hilbert’s geometryAxioms of OrderAxioms of Continuity
6 Conclusions and Further Work
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Outline
1 Introduction
2 Formalizing Cartesian Geometry
3 Using Isometric Transformations
4 Tarski’s geometryAxioms of Betweenness
5 Hilbert’s geometryAxioms of OrderAxioms of Continuity
6 Conclusions and Further Work
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Many different geometries
Euclidean (synthetic) geometry
Descartes’s coordinate system, bridged the gap betweenalgebra and geometry
Several attempts to formalize different geometries anddifferent approaches to geometry
Using the proof assistants significantly raises the level of rigour
Isabelle/HOL
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Main applications:
- automated theorem proving in geometry- mathematical education and teaching ofgeometry
Automated theorem proving:algebraic methods vs. theorem provers based on syntheticaxiomatizations
Geometry in education
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Goals
Formalize Cartesian geometry within a proof assistant
Different definitions of basic notions of analytic geometry allturn out to be equivalent
The standard Cartesian plane geometry represents a model ofseveral geometry axiomatizations
Formally analyze model-theoretic properties of differentaxiomatic systems
Formally analyze axiomatizations and models ofnon-Euclidean geometries and their properties
Formally establish connections of the Cartesian planegeometry with algebraic methods
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Outline
1 Introduction
2 Formalizing Cartesian Geometry
3 Using Isometric Transformations
4 Tarski’s geometryAxioms of Betweenness
5 Hilbert’s geometryAxioms of OrderAxioms of Continuity
6 Conclusions and Further Work
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Our formalization of analytic geometry aims at establishingthe connection with synthetic geometries so it followsprimitive notions given in synthetic approaches
objects: points, lines
relations: incidence, betweenness, congruence
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Points
pairs of real numbers (R2)
easily formalized in Isabelle/HOL by type synonym
pointag = ”real × real”
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
The Order of Points
Betweenness relation: B(A,B,C)
Some axiomatizations, e.g. Tarski’s, allow the case when B isequal to A or C
BagT (xa, ya) (xb, yb) (xc, yc)←→(∃(k :: real). 0 ≤ k ∧ k ≤ 1 ∧
(xb− xa) = k · (xc− xa) ∧ (yb− ya) = k · (yc− ya))
Hilbert’s axiomatizations does not allow points to be equal
BagH (xa, ya) (xb, yb) (xc, yc)←→(∃(k :: real). 0 < k ∧ k < 1 ∧
(xb− xa) = k · (xc− xa) ∧ (yb− ya) = k · (yc− ya))
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Congruence
Square distance
d2ag (x1, y1) (x2, y2) = (x2 − x1) · (x2 − x1) + (y2 − y1) · (y2 − y1)
Congruence
A1B1∼=ag A2B2 ←→ d2ag A1 B1 = d2ag A2 B2
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
typedef
Type definition
Any non-empty subset of an existing type can be turned intoa new type
Example:typedef three = ”{0 :: nat, 1, 2}”
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Constants declared:- three :: nat set- Rep three :: three −→ nat- Abs three :: nat −→ three
Some features:- Rep three surjective: Rep three x ∈ three- Rep three inverse: Abs three (Rep three x) = x- Abs three inverse:y ∈ three ⇒ Rep three (Abs three y) = y
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
LinesLine equations
A · x+B · y + C = 0 where A 6= 0 ∨B 6= 0
same line:
- A · x+B · y + C = 0- k ·A · x+ k ·B · y + k · C = 0, for a real k 6= 0
Lines
typedef line coeffsag =
{((A :: real), (B :: real), (C :: real)). A 6= 0 ∨B 6= 0}
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
LinesLine equations
Equivalent triplets
l1 ≈ag l2 ←→(∃ A1B1C1A2B2C2.(Rep line coeffs l1 = (A1, B1, C1)) ∧Rep line coeffs l2 = (A2, B2, C2) ∧(∃k. k 6= 0 ∧ A2 = k ·A1 ∧ B2 = k ·B1 ∧ C2 = k · C1))
(x, y) ∈agH l
ag in h (x, y) l←→(∃ A B C. Rep line coeffs l = (A, B, C)∧(A · x+B · y + C = 0))
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
LinesAffine definition
vecag = ”real × real”
typedef line point vecag = {(p :: pointag, v :: vecag). v 6= (0, 0)}
Equivalent triplets
l1 ≈ag l2 ←→ (∃ p1 v1 p2 v2.Rep line point vec l1 = (p1, v1)∧Rep line point vec l2 = (p2, v2)∧(∃km. v1 = k · v2 ∧ p2 = p1 +m · v1))
Incidence
ag in h p l ←→ (∃ p0 v0.Rep line point vec l = (p0, v0)∧(∃k. p = p0 + k · v0))
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Isometries
Translation
transpag (v1, v2) (x1, x2) = (v1 + x1, v2 + x2)
Rotation
rotpag α (x, y) = ((cosα) · x− (sinα) · y, (sinα) · x+ (cosα) · y)
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Isometries
All isometries are invariance properties
BagT A B C ←→BagT (transpag v A) (transpag v B) (transpag v C)
AB ∼=ag CD ←→(transpag v A)(transpag v B) ∼=ag (transpag v C)(transpag v D)
BagT A B C ←→ Bag
T (rotpag α A) (rotpag α B) (rotpag α C)
AB ∼=ag CD ←→(rotpag α A)(rotpag α B) ∼=ag (rotpag α C)(rotpag α D)
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Isometries
used only to transform points to canonical position
∃v. transpag v P = (0, 0)
∃α. rotpag α P = (0, p)
BagT (0, 0) P1 P2 −→ ∃α p1 p2. rotpag α P1 = (0, p1)
∧rotpag α P2 = (0, p2)
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Outline
1 Introduction
2 Formalizing Cartesian Geometry
3 Using Isometric Transformations
4 Tarski’s geometryAxioms of Betweenness
5 Hilbert’s geometryAxioms of OrderAxioms of Continuity
6 Conclusions and Further Work
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Motivation example
BagT A X B
AX
B
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Motivation example
BagT A X B ∧ BagT A B Y
XA
YB
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Motivation example
BagT A X B ∧ BagT A B Y −→ BagT X B Y
AX
YB
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Motivation example
BagT A X B ∧ BagT A B Y −→ BagT X B Y
AX
YB
A = (xA, yA), B = (xB, yB), and X = (xX , yX)
X = B holds trivially
from BagT A X B there is a real number k1, 0 ≤ k1 ≤ 1 suchthat
- (xX − xA) = k1 · (xB − xA)- (yX − yA) = k1 · (yB − yA)
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Motivation example
AX
YB
from BagT A B Y there is a real number k2, 0 ≤ k2 ≤ 1-(xB − xA) = k2 · (xY − xA) - (yB − yA) = k2 · (yY − yA)
k = (k2 − k2 · k1)/(1− k2 · k1)
if X 6= B, it is shown that- 0 ≤ k ≤ 1- (xB − xX) = k · (xY − xX) (yB − yX) = k · (yY − yX)
and thus BagT X B Y holds
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Motivation example
A B YX
we can assume: A = (0, 0), B = (0, yB), and X = (0, yX),and that 0 ≤ yX ≤ yB.
yB = 0 holds trivially
Otherwise, xY = 0 and 0 ≤ yB ≤ yY . Hence yX ≤ yB ≤ yY
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Invariant property
inv P t←→ (∀ A B C. P A B C ←→ P (tA) (tB) (tC))
Used to reduce the statement to any three collinear points to thepositive part of the y-axis
lemma
assumes "∀ yB yC . 0 ≤ yB ∧ yB ≤ yC −→P (0, 0) (0, yB) (0, yC)""∀ v. inv P (transpag v )""∀α. inv P (rotpag α )"
shows "∀ABC. BagT A B C −→ P A B C"
John Harrison, Without loss of generality
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of Betweenness
Outline
1 Introduction
2 Formalizing Cartesian Geometry
3 Using Isometric Transformations
4 Tarski’s geometryAxioms of Betweenness
5 Hilbert’s geometryAxioms of OrderAxioms of Continuity
6 Conclusions and Further Work
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of Betweenness
AB ∼=t BAAB ∼=t CC −→ A = BAB ∼=t CD ∧ AB ∼=t EF −→ CD ∼=t EF
Bt(A,B,A) −→ A = B
Bt(A,P,C) ∧ Bt(B,Q,C) −→ (∃X. (Bt(P,X,B) ∧ Bt(Q,X,A)))
∃ A B C. ¬ Ct(A,B,C)
(∃a. ∀x. ∀y. φ x ∧ ψ y −→ Bt(a, x, y)) −→(∃b. ∀x. ∀y. φ x ∧ ψ y −→ Bt(x, b, y))
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of Betweenness
AP ∼=t AQ ∧ BP ∼=t BQ ∧ CP ∼=t CQ ∧ P 6= Q −→ Ct(A,B,C)
∃E. Bt(A,B,E) ∧ BE ∼=t CD
AB ∼=t A′B′ ∧ BC ∼=t B
′C ′ ∧ AD ∼=t A′D′ ∧ BD ∼=t B
′D′ ∧Bt(A,B,C) ∧ Bt(A
′, B′, C ′) ∧ A 6= B −→ CD ∼=t C′D′
Bt(A,D, T ) ∧ Bt(B,D,C) ∧ A 6= D −→(∃XY. (Bt(A,B,X) ∧ Bt(A,C, Y ) ∧ Bt(X,T, Y )))
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of Betweenness
The axiom of Pasch
The axiom of Pasch
Bt(A,P,C) ∧ Bt(B,Q,C) −→ (∃X. (Bt(P,X,B) ∧ Bt(Q,X,A)))
A
B
P C
Q
X
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of Betweenness
The axiom of Pasch
Properties used in proof of Pasch axiom
BagT A A B
BagT A B C −→ BagT C B A
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of Betweenness
The axiom of Pasch
Properties used in proof of Pasch axiom
BagT A A B
BagT A B C −→ BagT C B A
BagT A X B
A X B
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of Betweenness
The axiom of Pasch
Properties used in proof of Pasch axiom
BagT A A B
BagT A B C −→ BagT C B A
BagT A X B ∧ BagT A B Y
XA B Y
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of Betweenness
The axiom of Pasch
Properties used in proof of Pasch axiom
BagT A A B
BagT A B C −→ BagT C B A
BagT A X B ∧ BagT A B Y −→ BagT X B Y
A X B Y
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of Betweenness
The axiom of Pasch
Properties used in proof of Pasch axiom
BagT A A B
BagT A B C −→ BagT C B A
BagT A X B ∧ BagT A B Y −→ BagT X B Y
A X B Y
BagT A X B
A X B
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of Betweenness
The axiom of Pasch
Properties used in proof of Pasch axiom
BagT A A B
BagT A B C −→ BagT C B A
BagT A X B ∧ BagT A B Y −→ BagT X B Y
A X B Y
BagT A X B ∧ BagT A B Y
XA B Y
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of Betweenness
The axiom of Pasch
Properties used in proof of Pasch axiom
BagT A A B
BagT A B C −→ BagT C B A
BagT A X B ∧ BagT A B Y −→ BagT X B Y
A X B Y
BagT A X B ∧ BagT A B Y −→ BagT A X Y
BXA Y
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of Betweenness
The axiom of Pasch
∃X. (Bt(P,X,B) ∧ Bt(Q,X,A))
If P = C, then Q is thepoint sought
A
B
C= P
Q
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of Betweenness
The axiom of Pasch
∃X. (Bt(P,X,B) ∧ Bt(Q,X,A))
If P = C, then Q is thepoint sought
A
B
C= P
Q
Bt(A,B,C), B is the pointsought
A P Q CB
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of Betweenness
The axiom of Pasch
use isometric transformations so that:A = (0, 0), P = (0, yP ), C = (0, yC)B = (xB, yB), Q = (xQ, yQ),X = (xX , yX)
from Bt(A,P,C), there is a real numberk3 such that
- 0 ≤ k3 ≤ 1 yP = k3 · yC
from Bt(B,Q,C), there is a real numberk4 such that
- 0 ≤ k4 ≤ 1- xQ − xB = −k4 ∗ xB- yQ − yB = k4 ∗ (yC − yB)
A P C
Q
B
X
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of Betweenness
The axiom of Pasch
define a real number k1 =k3·(1−k4)
k4+k3−k3·k4and show
- 0 ≤ k1 ≤ 1- xX = k1 · xB- yX − yP = k1 · (yB − yP )
thus Bt(P,X,B) holds
define a real number k2 =k4·(1−k3)
k4+k3−k3·k4and show
- 0 ≤ k2 ≤ 1- xX − xQ = −k2 · xQ- yX − yQ = −k2 · yQ
thus Bt(Q,X,A) holds
A P C
Q
B
X
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of Betweenness
Axiom (Schema) of Continuity
(∃a. ∀x. ∀y. φ x ∧ ψ y −→ Bt(a, x, y)) −→(∃b. ∀x. ∀y. φ x ∧ ψ y −→ Bt(x, b, y))
A x yB
Isabelle/HOL does not restrict predicate φ and ψ to be FOLpredicates
One of the sets is empty, the statement trivially hold
The sets have a point in common, that point is the pointsought
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of Betweenness
Axiom (Schema) of Continuity
A = (0, 0)
P QB
Isometry transformations are applied so that all points fromboth sets lie on the positive part of the y-axis.
lemma
assumes
"P = {x. x ≥ 0 ∧ φ(0, x)}" "Q = {y. y ≥ 0 ∧ ψ(0, y)}""¬(∃b. b ∈ P ∧ b ∈ Q)" "∃x0. x0 ∈ P" "∃y0. y0 ∈ Q""∀x ∈ P. ∀y ∈ Q. BagT (0, 0) (0, x) (0, y)"shows
"∃b. ∀x ∈ P. ∀y ∈ Q. BagT (0, x) (0, b) (0, y)"
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of Betweenness
Axiom (Schema) of Continuity
A = (0, 0)
P QB
Completeness of reals
(∃x. x ∈ P ) ∧ (∃y. ∀x ∈ P. x < y) −→∃S. (∀y. (∃x ∈ P. y < x)↔ y < S)
P satisfies the supremum property
∀x ∈ P. ∀y ∈ Q. x < y
there is b such that:∀x ∈ P. x ≤ b and ∀y ∈ Q. b ≤ y
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of OrderAxioms of Continuity
Outline
1 Introduction
2 Formalizing Cartesian Geometry
3 Using Isometric Transformations
4 Tarski’s geometryAxioms of Betweenness
5 Hilbert’s geometryAxioms of OrderAxioms of Continuity
6 Conclusions and Further Work
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of OrderAxioms of Continuity
A 6= B −→ ∃! l. A ∈h l ∧ B ∈h l
∃AB. A 6= B ∧ A ∈h l ∧ B ∈h l
∃ABC. ¬ Ch(A,B,C)
Bh(A,B,C) −→ A 6= B ∧ A 6= C ∧ B 6= C ∧Ch(A,B,C) ∧ Bh(C,B,A)
A 6= C −→ ∃B. Bh(A,C,B)
A ∈h l ∧ B ∈h l ∧ C ∈h l ∧ A 6= B ∧ B 6= C ∧ A 6= C −→(Bh(A,B,C) ∧ ¬Bh(B,C,A) ∧ ¬Bh(C,A,B)) ∨(¬Bh(A,B,C) ∧ Bh(B,C,A) ∧ ¬Bh(C,A,B)) ∨(¬Bh(A,B,C) ∧ ¬Bh(B,C,A) ∧ Bh(C,A,B))
A 6= B ∧ B 6= C ∧ C 6= A ∧ Bh(A,P,B)∧P ∈h l ∧ ¬C ∈h l ∧ ¬A ∈h l ∧ ¬B ∈h lh −→∃Q. (Bh(A,Q,C) ∧ Q ∈h l) ∨ (Bh(B,Q,C) ∧ Q ∈h l)
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
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Axioms of OrderAxioms of Continuity
A 6= B ∧ A ∈h l ∧ B ∈h l ∧ A′ ∈h l
′ −→∃B′ C ′. B′ ∈h l
′ ∧ C ′ ∈h l′ ∧ Bh(C
′, A′, B′)∧AB ∼=h A
′B′ ∧ AB ∼=h A′C ′
AB ∼=h A′B′ ∧ AB ∼=h A
′′B′′ −→ A′B′ ∼=h A′′B′′
Bh(A,B,C) ∧ Bh(A′, B′, C ′) ∧ AB ∼=h A
′B′ ∧ BC ∼=h B′C ′ −→
AC ∼=h A′C ′
¬P ∈h l −→ ∃! l′. P ∈h l′ ∧ ¬(∃ P1. P1 ∈h l ∧ P1 ∈h l
′)
Let A1 be any point upon a straight line between the arbitrarilychosen points A and B. Choose the points A2, A3, A4, . . . so that A1
lies between A and A2, A2 between A1 and A3, A3 between A2 andA4 etc. Moreover, let the segments AA1, A1A2, A2A3, A3A4, . . . beequal to one another. Then, among this series of points, therealways exists a point An such that B lies between A and An.
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of OrderAxioms of Continuity
Axiom of Pasch
A 6= B ∧ B 6= C ∧ C 6= A ∧ Bh(A,P,B)∧P ∈h l ∧ ¬C ∈h l ∧ ¬A ∈h l ∧ ¬B ∈h l −→∃Q. (Bh(A,Q,C) ∧ Q ∈h l) ∨ (Bh(B,Q,C) ∧ Q ∈h l)
A
B
P
C
Q
The statement holds trivially if A, B and C are collinear
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of OrderAxioms of Continuity
A = (0, 0), B = (xB, 0), P = (xP , 0),C = (xC , yC) andRep line coeffs l = (lA, lB, lC)
Bh(A,P,B) −→ lA · yB 6= 0
k1 =−lClA·yB
k2 =lA·yB+lClA·yB
0 < k1 < 1, Q = (xQ, yQ) is determed by- xQ = k1 · xC yQ = k1 · yC- Bh(A,Q,C)
0 < k2 < 1, Q = (xQ, yQ) is determed by- xQ = k2 · (xC − xB) + xB- yQ = k2 · yC- Bt(B,Q,C)
A
B
P
C
Q
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of OrderAxioms of Continuity
Axiom of Archimedes
Let A1 be any point upon a straight line between the arbitrarilychosen points A and B. Choose the points A2, A3, A4, . . . so thatA1 lies between A and A2, A2 between A1 and A3, A3 betweenA2 and A4 etc. Moreover, let the segmentsAA1, A1A2, A2A3, A3A4, . . . be equal to one another. Then,among this series of points, there always exists a point An suchthat B lies between A and An.
A A1
B
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of OrderAxioms of Continuity
Axiom of Archimedes
Let A1 be any point upon a straight line between the arbitrarilychosen points A and B. Choose the points A2, A3, A4, . . . so thatA1 lies between A and A2, A2 between A1 and A3, A3 betweenA2 and A4 etc. Moreover, let the segmentsAA1, A1A2, A2A3, A3A4, . . . be equal to one another. Then,among this series of points, there always exists a point An suchthat B lies between A and An.
B
A A1 A2 A3 A4 A5
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of OrderAxioms of Continuity
Axiom of Archimedes
Let A1 be any point upon a straight line between the arbitrarilychosen points A and B. Choose the points A2, A3, A4, . . . so thatA1 lies between A and A2, A2 between A1 and A3, A3 betweenA2 and A4 etc. Moreover, let the segmentsAA1, A1A2, A2A3, A3A4, . . . be equal to one another. Then,among this series of points, there always exists a point An suchthat B lies between A and An.
A1 A2 A3 A5
B
A A4
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of OrderAxioms of Continuity
Axiom of Archimedes
definition
congruentl l −→ length l ≥ 3 ∧∀i. 0 ≤ i ∧ i+ 2 < length l −→(l ! i)(l ! (i+ 1)) ∼=h (l ! (i+ 1))(l ! (i+ 2)) ∧Bh((l ! i), (l ! (i+ 1)), (l ! (i+ 2)))
Axiom of Archimedes
Bh(A,A1, B) −→(∃l. congruentl(A # A1 # l) ∧ (∃i. Bh(A,B, (l ! i))))
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Axioms of OrderAxioms of Continuity
Axiom of Archimedes
d2ag A A′ > d2ag A B and d2ag A A′ = t · d2ag A A1
t · d2ag A A1 > d2ag A B (applying Archimedes’ rule for realnumbers)
inductively proved that there exists a list l such that:
- congruentl l,- longer then t,- first two elements are A and A1
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Outline
1 Introduction
2 Formalizing Cartesian Geometry
3 Using Isometric Transformations
4 Tarski’s geometryAxioms of Betweenness
5 Hilbert’s geometryAxioms of OrderAxioms of Continuity
6 Conclusions and Further Work
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Conclusions
Developed a formalization of Cartesian plane geometry withinIsabelle/HOL
Formally shown that the Cartesian plane satisfies all Tarski’saxioms and most of the Hilbert’s axioms
Proof that analytic geometry models geometric axioms arerather complex
Applying isometry transformations
Our formalization of the analytic geometry relies on theaxioms of real numbers and properties of reals are usedthroughout our proofs
Danijela Petrovic [email protected] Formalizing Analytic Geometries
IntroductionFormalizing Cartesian GeometryUsing Isometric Transformations
Tarski’s geometryHilbert’s geometry
Conclusions and Further Work
Further Work
Formalizing analytic models of non-Euclidean geometries
Connect our formal developments to the implementation ofalgebraic methods for automated deduction in geometry
Danijela Petrovic [email protected] Formalizing Analytic Geometries