Formalizing the Qualitative Superposition of Rectangles in

Proof Assistant Isabelle/HOL

Fadoua Ghourabi and Kazuko Takahashi

Graduate School of Science and Technology, Kwansei Gakuin University, Nishinomiya, Japan

Keywords:

Knowledge Representation and Reasoning, Qualitative Spatial Reasoning, Theorem Proving, Isabelle/HOL,

Rectangle Superposition.

Abstract:

We formalize and verify the superposition of rectangles in Isabelle/HOL. The superposition is associated with

the arrangement of rectangular software windows while keeping some regions visible and other hidden. We

adopt a qualitative spatial reasoning approach to represent these rectangles and the relations between their

regions. The properties of the model are formally proved and show some characteristics of superposition

operation. Although, this work is limited to 29 structures of rectangles, the superpositions produce hundreds

of cases that are tedious to tackle in Isabelle/HOL. We also explain our strategy to optimize the proofs.

1 INTRODUCTION

Qualitative spatial reasoning (QSR) methods describe

the objects of the space qualitatively. Such approach

provides a low demand in numerical computation

and, furthermore, contains enough expressiveness to

teach a program to reason about the spatial objects.

QSR methods target various applications, such as ge-

ographical information systems and robotics (Liu and

Daneshmend, 2004; Bennett, 1996). The importance

of QSR methods leads us to address their reliability

and correctness. While various research works focus

on developing QSR methods, little attention is given

to their formal veriﬁcation.

On the other hand, proof assistants are increas-

ingly used to formalize mathematical models (Avi-

gad and Harrison, 2014). With the help of proof

assistants, mathematical proofs are formally veriﬁed

where each step is transparent and, thus, the involved

logical reasoning can be checked. In the case of the

ﬁeld of QSR, there are two major approaches: the

topological knowledge about objects (e.g. RCC calcu-

lus (Randell et al., 1992b)), and the positional knowl-

edge about objects (e.g. direction relations (Frank,

1991; Clementini et al., 1997)). Both of RCC and di-

rection relations are formalized using ﬁrst-order logic

and relation algebra, which motivate us to use them in

automated reasoning by a proof assistant such as Is-

abelle/HOL (Nipkowet al., 2002). The only work that

tackle the formal veriﬁcation in QSR of which we are

aware dates back to early 90s. The OTTER proof sys-

tem for ﬁrst-order logic has been used to assist pen-

and-paper proofs of theorems in RCC (Randell et al.,

1992a). As far as we know, there is no (full) formal

presentation of QSR methods using proof assistant.

The purpose of this research is to extend the use of

proof assistants to the ﬁeld of QSR. We focus on win-

dow allocation problem. When using software, we

often rearrange windows by dragging, resizing, su-

perposing, etc. until obtaining better visibility. A su-

perﬂuous information in a window can be hidden and

superposed by a relevant information in another win-

dow. A systematic method of superposing windows is

proposed in (Konishi and Takahashi, 2012). The win-

dow parts that should be visible are pre-speciﬁed by

the user. The window and its parts form a rectangu-

lar structure. The problem of arranging software win-

dows is regarded as the problem of superposing rect-

angles while keeping some pre-speciﬁed parts visible.

The rectangles are regarded as spatial objects, and the

superposition is examined from a QSR point of view.

This paper is based on the method presented

in (Konishi and Takahashi, 2012), and the contribu-

tion that we seek is twofold.

1. The original qualitative representation of rectan-

gles is simple but not expressive enough. When

checking properties about superposition, extra

conditions are deﬁned to detect degenerate situ-

ations. Due to the numerous cases of superpo-

sition, the question is whether all the degenerate

situations are covered. We therefore propose a

more expressive representation based on matrices

530

Ghourabi F. and Takahashi K..

Formalizing the Qualitative Superposition of Rectangles in Proof Assistant Isabelle/HOL.

DOI: 10.5220/0005280005300539

In Proceedings of the International Conference on Agents and Artiﬁcial Intelligence (ICAART-2015), pages 530-539

ISBN: 978-989-758-074-1

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

of direction relations. The matrix representation

is more natural as the correspondence between

the parts of a rectangular structure and the ele-

ments of its matrix representation is straightfor-

ward. Furthermore, the extra conditions are not

required and a single set membership test is sufﬁ-

cient.

2. We formalize our revised method of superposition

in Isabelle/HOL. Proof assistants are interactive

systems. The formal proofs are, consequently, te-

dious to do if the model present hundreds of cases.

We therefore take into consideration the practical-

ity of proving in Isabelle/HOL despite the numer-

ous cases of superposition. Our proof strategy re-

lies on establishing equivalence relations between

the qualitative representations of rectangles and

grouping them into classes.

The structure of the rest of the paper is as follows. In

Section 2, we summarize the notations that we use.

In Section 3, we deﬁne matrices in Isabelle/HOL. In

Section 4, we explain the operations over matrices. In

Section 5, we give matrix representation to rectangu-

lar units. In Section 6, we apply matrix superposition

to compute the superposition of units. In Section 7,

we introduce properties about superposition of units

and explain how we proceed in order to prove them.

In Section 8, we conclude with remarks on future di-

rections of research.

2 NOTATION

A matrix

of dimension

has

rows and

columns, where

∈ N. We write for short “

matrix

”. The expression

(

) denotes the ele-

ment of

at the

-th column and the

-th row, where

0 ≤

and 0 ≤

. Let S be the set of matrices,

be a matrix in S and

:S → S be an unary function,

we use

to mean the term

(

) and

to denote

the n(∈ N) compositions of

. We recurrently use the

notation

for

(

The formalization presented in this paper is done

in the proof assistant Isabelle/HOL (Nipkow et al.,

2002). The choice of the tool is made principally due

to the availability of powerful libraries for reasoning

with equivalence classes (Paulson, 2006) and mathe-

matical operations on matrices (Sternagel and Thie-

mann, 2010) on which the proposed qualitative rep-

resentation depends. In the following, we present the

elements of syntax used in this paper. Isabelle/HOL

provides a rich collection of formalized theories, use-

ful proof tactics (e.g. natural and structural induction,

and case splitting), elaborate techniques for pattern

matching and term rewriting, etc. Furthermore, Is-

abelle/HOL is a strongly typed system. The expres-

sion “

m::

′

list

” is a type constraint over a variable

. The variable

is of type list whose elements are

of variable type

′

. The type of a function

is writ-

ten “

f::

⇒ . . . ⇒ τ

⇒ τ

n+1

”. In this paper, we use

Isabelle/HOL expression “

f x

. . .

” to denote the

term

(

, .. .,

). A list is represented by a sequence

of elements between square brackets, i.e. [e

, e

, . . .,

]. The constructor “#” adds an element to a list, i.e.

v#[e

, e

, . . ., e

] = [v, e

, e

, . . ., e

]. The i-th element

of a list m is given by “

! i”. For an equivalence rela-

tion r, the expression r‘‘{

} in Isabelle/HOL denotes

the equivalence class [

]

The formulas that we prove are written using com-

mon mathematical symbols. We explain some of the

proof in Isabelle/HOL using natural language. We

use Isar, which is an extension of Isabelle/HOL, to

write structured and human-readable proofs (Wenzel,

1999). The proofs in the classical Isabelle/HOL proof

style intertwines with Isar proofs.

3 FORMALIZATION OF

MATRICES

In Isabelle/HOL, no type is deﬁned for matrix. We

use the types “

’a vec

” and “

’a mat

” provided by the

Matrix

Arith theory (Sternagel and Thiemann, 2010).

type

synonym ’a vec = ’a list

type

synonym ’a mat = ’a vec list

A matrix is implemented as a list of lists, i.e. a

list of columns. For example, list [[

],[

]]

is of type “

′

a mat

”, and represents the 2×3 matrix



a c e

b d f



3.1 Dimension

The following issues arise when working with matri-

ces as list of lists. First, a matrix of type “’a mat”

has a dimension

, i.e. the length of the whole list

, and each column list has

elements. To make

sure that we work with mathematically valid matrices,

we use predicate

mat::nat

⇒

nat

⇒

’a mat

⇒

bool

(Ster-

nagel and Thiemann, 2010).

mat p q m

(

length m

) ∧ (∀ x ∈

set m

length x

(1)

For a closer look at the proofs, our Isabelle/HOL

theory ﬁles are available at http://ist.ksc.kwansei.ac.jp/

∼ktaka/SuperpositionTheory/.

FormalizingtheQualitativeSuperpositionofRectanglesinProofAssistantIsabelle/HOL

531

where

length

and

set

are Isabelle/HOL functions that

compute the length of a list and the set of elements of

a list, respectively.

Second, empty matrix (i.e. [ ]) and matrices with

empty columns (i.e. [[ ]], [[ ],[ ]], etc.) are of type

“

′

a mat

”. We want to exclude these matrices for the

following reasons. In the case of empty matrix, the

second part of the conjunction in deﬁnition (1) causes

vacuous truth that we wish to avoid (i.e. “

mat p

0 [ ]”

holds for any

). Moreover, the transpose of a matrix

with empty columns gives rise to the empty matrix,

hence we wish to avoid these cases too. We there-

fore restrict ourselves to a set

of type “

′

a mat set

”

whose elements satisfy the following condition.

∈

, ∃

p q::nat

mat

(

Suc p

) (

Suc q

)

The theorems that we prove in this work are deﬁned

for matrices

∈

. In the premises, we impose the

condition

∈

from which we can deduce the di-

mension

by adding the following Isar line.

from ‘m ∈ M‘ obtain p q

where mat p q m and 0 < p and 0 < q

by (rule M.cases, simp)

The command “by (rule M.cases, simp)” proves

that the dimension of m is

and that

and

are

strictly positive.

3.2 Matrix Equality

The equality is established between two matrices of

the same dimension. In order to assert that two matri-

ces are equal, we either check that all their respective

elements are equal or that their respectivecolumn lists

are equal, depending on the proof strategy. Lemmas

about the equality of matrices are provided by the Ma-

trix

Arith theory.

4 OPERATIONS ON MATRICES

We explain two operations on matrices, namely rota-

tion

and superposition.

4.1 Rotation

Before giving a formal deﬁnition of rotation, we ﬁrst

examine what does rotation of a matrix intuitively

mean. An operation of rotation has a spatial con-

notation. It involves a circular movement of objects

Here, operation of rotation over matrices is not to be

confused with “rotation matrix” used to compute the new

coordinates of a geometrical object that undergoes rotation

in Euclidean space.

deﬁned by parameters such as the center of rotation,

angle of rotation, etc. If we picture a matrix as an ob-

ject (not only as data-structure), we can perform a

counter-clockwise rotation. For instance, if we rotate

the matrix

a c e

b d f

in a counter-clockwise

direction, we obtain







e f

c d

a b







From a computational point of view, function ρ =

◦

represents the rotation operation over matrices.

Functions

and

denote the transpose and the re-

verse. For a matrix

∈

◦

is obtained by

reversing the columns of

. In the previous exam-

ple, the transpose is







a b

c d

e f







. Then, the result of

reversing the columns of the transpose is







e f

c d

a b







Functions

transpose

reverse

and ρ implement

and ρ in Isabelle/HOL. We use function

transpose

nat

⇒

′

a mat

⇒

′

a mat

given in the Matrix Arith

theory, and we deﬁne functions

reverse

and ρ (both

of type

′

a mat

⇒

′

a mat

) as follows.

reverse m

map rev m

reverse (transpose (nr mat m) m)

Function

rev

reverses a list, and function

reverse

applies

rev

to all the column lists. Note that function

transpose

has two arguments. The ﬁrst argument is

a natural number that corresponds to the number of

rows, which is computed by the function call “

nr mat

”.

Furthermore, we can perform n successive rota-

tions of matrix

by computing

, where n ∈ N. In

Isabelle/HOL, we use the notation for function power,

i.e. “ρ ˆˆn”, to implement successive rotations. We

prove a collection of useful properties about rotation

operation that are stated as inference rules in (2) − (7)

depicted in Fig. 1.

The variables in the rules (2) − (7) are universally

quantiﬁed, and their types are understood from the

context. Rule (5) expresses the transitivity of ρ

. A

matrix is invariant by a number of rotations multiple

of 4 (c.f. rules (4) and (6)). If matrix

is obtained

by n rotations of matrix

, then

can be obtained

back by 3× n rotations of

(c.f. rule (7)).

Based on rotation, we establish a relation

rot rel

In other words, if a matrix

is obtained by n rota-

tions of matrix

, then we write (

) ∈

rot rel

,{(

)|∀

.∃

n::nat

∈

∧

∈

∧

}

(8)

We furthermore prove that the relation

rot rel

is an

equivalence relation over the set

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532

∈

M mat p q m even n

mat p q m

(2)

∈

M mat p q m

even n

mat q p m

(3)

∈

n mod 4

(4)

∈

◦ρ

(5)

∈

4×n

(6)

∈

M m

∈

M m

3×n

(7)

Figure 1: Rules on rotations of matrices.

lemma equiv M rot_rel

Relation

rot rel

is obviously reﬂexive over

since

4×

. From property (5), we deduce that

rot rel

is transitive, and from property (7) that

rot rel

is symmetric.

4.2 Superposition

The other operation that we consider is superposition

of matrices. Intuitively, we superpose two matrices

and

by “putting”

onto

. The super-

position requires that both of

and

are of the

same dimension

. The superposition amounts to

“putting” elements of

onto elements of

Let

and

be two functions of type

′

a mat

⇒

′

mat

⇒

′

a mat

and

′

⇒

′

⇒

′

, respectively. We

deﬁne predicate

is superposition

as follows.

same dimension p q m

mat p q m

∧

mat p q m

is superposition f s

∀

∈

. ∀

p q i j::nat

(

same dimension p q m

) →

< q →

< p →

(

f m

)(

) =

s m

(

)

(

)

If “

is superposition f s

” holds then we write that

is a superposition with respect to

. For instance,

addition over matrices is a superposition with respect

to addition operation.

lemma is_superposition mat_plus plus

The implementation of

that we use in this paper

is the function

map thread

deﬁned as follows.

fun map_thread::(’a ⇒’a ⇒’a)⇒’a mat⇒

’a mat⇒ ’a mat

where map_thread s m1 m2 =

map (λ(v1,v2). map (λ(a1,a2). s a1 a2)

(zip v1 v2)) (zip m1 m2)

Since matrices are implemented as lists, the super-

position “

map thread

” applies function

to pair-wise

combinations of elements of

and

. Function

is applied to elements of

and

that have the

same position (i.e. same column and row numbers).

For instance, calling “(

map thread s

) [[a

, a

], . . .]

[[b

, b

], . . .]” gives rise to the matrix “[[s a

, s

, s a

], . . .]”. We show that “

map thread s

” is

a superposition with respect to

lemma is_superposition (map_thread s) s

Furthermore, we prove properties about superpo-

sition of rotation of matrices shown in rules (9) −

(11). In the left-side of the equation in (9), we com-

pute the superposition of matrices

and

rotated

by the same number n. We can take ρ

outside as

shown in the right-side of the equation in (9), i.e. we

ﬁrst superpose

and

then rotate the result by n.

More relevant to our formalization is showing how

superposition behaves when we rotate

and

distinct numbers

and

, respectively. Depending

on whether

is less than

, we distinguish the two

cases in (10) and (11). In case

≤

, we can take

outside as shown in the right-side of (10). There-

fore, matrix

is rotated by

−

. Rules (10) and

(11) are used as substitution rules to replace the occur-

rence of left-side of the equality by the right-side of

the equality. The two rules are important to optimize

the proofs which we will explain later in Sect. 7.2.

Recall that from properties (2) and (3), the dimen-

sion of matrix

depends on whether

is even.

Hence,

and

are not necessary of the same

dimension, and therefore we cannot always perform

superposition. A sufﬁcient condition, but not neces-

sary, is to consider only

square matrices, where

. So far, all the lemmas that we have proved are

for arbitrary

matrices, but, as we see in next sec-

tions, only square matrices are used to formalize the

superposition of rectangles.

The qualitative spatial reasoning approach to the

problem of superposition of software windows is

based on operations of rotation and superposition of

matrices. First, a window is given a qualitative rep-

resentation using matrices, which is the subject of the

next section.

The type of function

is not speciﬁed in the statement

of the lemma since it is inferred by Isabelle/HOL.

FormalizingtheQualitativeSuperpositionofRectanglesinProofAssistantIsabelle/HOL

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∈

M m

∈

(

map thread s

)

= ((

map thread s

)

(9)

∈

M m

∈

M n

≤

mat p p m

(

map thread s

)

= ((

map thread s

)

−n

)

(10)

∈

M m

∈

M n

mat p p m

(

map thread s

)

= ((

map thread s

)

3×(n

−n

)

(11)

Figure 2: Inference rules on matrix superposition.

5 QUALITATIVE

REPRESENTATION OF UNIT

The spatial object that we investigate is the software

window modelled as a rectangle, called unit. Some

parts of a unit are required to be visible, which are

modelled as white rectangular plates. The parts that

can be hidden are black rectangular plates. The size

of a unit is unﬁxed, and changes in a fashion similar

to the way software windows are shrunk or expanded.

We explain the qualitative representation of a unit that

reﬂects its structure, i.e. locations of its black and

white plates.

5.1 Black Plates

Let U be a rectangular unit of unﬁxed length l and un-

ﬁxed height h, i.e. of size l × h. In this paper, we use

units with at most two black plates, and we consider

the following two assumptions about the black plates

of a unit U.

1. If U has one black plate p, then the size of p is

either l × v or u× h, where u ≤ l and v ≤ h. Each

of the units in Fig. 3(a) and 3(b) has one black

plate that stretches along the length/height of the

unit.

2. If U has two black plates p

and p

, then they are

perpendicular, and at least one of them is of size

u×h or l ×v, where u ≤ l and v ≤ h. Furthermore,

and p

must be overlapping. The overlapping

part forms a rectangular shared area denoted by

⊓ p

(e.g. the units in Fig. 3(c) and 3(d)).

5.2 Regions of Unit

We work with units that are located on the 2D plane.

The 2D plane is the grey region, simply denoted by

From the white and black plates, we compute regions

(a) (b)

Figure 3: Examples of units with one black plate ((a)

and (b)) and with two perpendicular and overlapping black

plates ((c) and (d)).

of unit, namely white and black regions that we de-

note

and

, respectively. We deﬁne the datatype

as follows.

datatype rg = w | b | g | N

The region N is called undeﬁned region that is rel-

evant for checking the success of superposition which

we explain in Sect. 6. The regions

and

of a unit

U are determined as follows.

(a) A white plate of U is a

region.

(b) If p is the only black plate of U, then p is also the

only

region of U.

and p

are two distinct black plates of U,

then together they generate 3, 4 or 5

regions de-

pending on their placements (cf. Fig. 4). Note that

the area p

⊓ p

is one of the

regions generated

by p

and p

Next, we deﬁne the core of a unit. The core region

of unit U, denoted by C

, is a

region. If U has only

one black plate p, then p is the coreC

. Otherwise, let

and p

be the two black plates of U, then the core

of U is the shared area between p

and p

, i.e. C

⊓ p

. Since a unit has at most two black plates as

established in the assumptions in the previous section,

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b b

(a)

b b

(b)

b b b

(c)

Figure 4: Black plates,

regions and the core regions.

then a unit has one and only one core. In our ﬁgures,

the cores are highlighted in darker black (cf. Figs. 3

and 6).

5.3 Representation

One common approach to represent positional know-

eledge is the use of matrix representation. In particu-

lar, the object interaction matrix (OIM) (Chen et al.,

2010) encodes the direction relations between spa-

tial objects. In this paper, we use OIM matrix to

represent the positions of the unit regions w.r.t. the

core region. Recall that the core C

is a rectangu-

lar black area, thus it has 4 edges and 4 vertices.

Extending the edges of C

divides the plane into 9

tiles where the central tile is bounded and 8 are un-

bounded (e.g. the tiling in Fig. 5). The tiles decom-

pose the plane into the 9 regions of direction rela-

tions, i.e.

up left

left

bottom left

center

bottom

up right

right

bottom right

. The core region occu-

pies the central bounded tile. The intersection of the

tiles and the rectangular unit determines the direction

relations between the coreC

and the 8 regions of the

unit that are connected to the core.

The reference of direction system, being the core,

is the center of the matrix representation. The remain-

ing elements of the matrix are the values of the inter-

sections of the unit and the 8 unbound tiles. We use

the following 3×3 matrix to represent the positions of

the 8 regions that surround the core.





le ft up up right

le ft core right

bottom

le ft bottom bottom right





We place a unit in the 2D plane and we substitute

the values

for the elements of the above ma-

trix. The obtained matrix is the qualitative represen-

tation of the unit. For instance, the qualitative rep-

resentations of the units in Fig. 3 are







g w g

g b g

g g g













g g g

w b w

g g g













w b g

b b g

g g g







and







w b w

b b b

w w w







, re-

spectively.

Regions that share a point are considered connected.





w b w

b b b

w w w





Figure 5: Tiling of the plane and the OIM matrix of a unit.

In Isabelle/HOL, we provide a set

rgM

of all the

3×3 matrices with

core region, and prove that

rgM

is a subset of

deﬁned in Sect. 3.1.

deﬁnition core::rg mat ⇒ rg

where core m = m!1!1

deﬁnition rgM ::rg mat set where

rgM = {m. (mat 3 3 m)∧(core m = b)}

lemma rgM ⊂ M

5.4 Valid Units

There are 29 possible cases of ﬁtting one or two black

plates in a rectangular unit. They are depicted in

Fig. 6. We call them valid units. Since we deal with

qualitative representation of units where size does not

matter, we observe that some units are obtained by

rotating others.

6 SUPERPOSITION OF UNITS

Superposition of units is a non-symmetric binary op-

eration. Superposing two units U

and U

means that

we put U

onto U

while keeping the

regions of

visible. The superposition, that we deﬁne in this

paper, operates by putting the core C

onto the core

. The cores C

and C

are not necessarily of

the same size. Operations of shrinking, expanding,

lengthening are performed on U

and U

so that their

cores ﬁt. Consequently, the sizes of the units and their

regions change. Such operations may also affect the

amount of visible information. We assume that the

operating system or the software application automat-

ically generates scrollbars for the viewing. For the

simplicity of our examples, we only modify the size

of U

and its core. The changes in the size do not af-

fect the matrix representation since it is a qualitative

representation.

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(a) Class B (b) Class W

(d) Class I2

(e) Class L1

(f) Class L2

(g) Class L3

(h) Class T1

(i) Class T2

(j) Class PLUS

Figure 6: Valid units grouped into 10 equivalence classes.

The result of superposition is a unitU

whose core

is C

= C

. The superposition of U

and U

is computed from the superposition of their respective

matrix representations. In Sect. 4.2, we introduced

superposition of matrices where function

has not yet

been deﬁned. In the following section we discuss an

implementation of

(a) Unit L1

(b) Unit I1

Figure 7: Superposition of unit I1 onto unit L1.

6.1 Function

We deﬁne a function

that computes superposition

of regions.

fun on::rg ⇒ rg ⇒ rg

where

on g u = u | on u g = u |

on N _ = N | on _ N = N |

on w _ = N | on _ w = w | on _ b = b

The equation “

on x y = z

” means that region

is to

be put on region

, and the result is region

. Function

is deﬁned using Isabelle/HOL function deﬁnition

fun. The order of appearance of the equations “

on x y

= z

” matters. The pattern matching of “

on w b

” suc-

ceeds ﬁrst with the 5th equation and thus gives rise to

, i.e. the superposition should not be allowed since

region is hidden.

The superposition of two matrices

and

achieved by calling “

(map thread on) m

”.

Example. We take the two units of type L1 and

I1 in Fig. 7(a) and 7(b) with matrix representations







w b g

b b g

g g g







and







g w g

g b g

g g g







, respec-

tively. We resize I1, then we put C

onto C

. The

computation of “(

map thread on

)

” gives rise







w w g

b b g

g g g







which corresponds to the unit in

Fig. 7(c).

The unit in Fig. 7(c) has one more

regionbesides

the core. The

region is adjacent to the core to the

left. In other words, it shares an edge with the core.

We merge it with the core to form one

region. The

conditions for the merge as well as its impact on the

matrix representation of a unit are discussed in the

next section.

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(a) (b)

b b

(c)

Figure 8: The result of putting I2 type unit on L1 type unit.

6.2 Merging Black Regions

The merge is necessary only if the following two con-

ditions apply.

1. There is only one

region that is connected to the

core by an edge. In other words, there is a unique

region located either up or left or bottom or right

to the core. Fig. 7(c) illustrates such situation. Be-

sides the core, only left region is black.

2. If the above condition holds, then we check

whether the

region and the core are connected to

the same regions. Referring to Fig. 7(c) with ma-

trix representation







w w g

b b g

g g g







, both

region

and core are connected to the same upper

re-

gion, and to the same bottom

region. In that

case we can merge the left, up left and bottom

left regions into one

region. Hence, the matrix

representation becomes







g w g

g b g

g g g







. In the ex-

ample illustrated by Fig. 8, we put I2 unit on L1

unit. The result of superposition (cf. Fig. 8(c))

has the matrix representation







g w w

g b b

g w g







. The

right region is the only

region besides the core.

The core is connected to a

bottom region while

the right region is connected to a

bottom region.

We, therefore, cannot merge the right, up right and

bottom right regions into one

region.

The ﬁnal result is merging the superposition (if appli-

cable) which is given by function

puton

merge

◦

(

map thread on

7 ON PROVING PROPERTIES

ABOUT

puton

7.1 The Properties

We deﬁne properties to answer questions about the

ﬁnal result of superposing two units. Let

and

be matrices in

rgM

, the properties of applying “

puton

” that we want to check are the following.

Success. Are all the

regions visible?

Effectiveness. Does the ﬁnal result of superposition

has only one

region?

Validity. Is the ﬁnal result of the superposition a

valid unit?

The superposition proceeds by putting the core of a

unit onto the core of another one. This does not guar-

antee that all the

regions are visible. In order to

judge whether a superposition is successful, we com-

pute the occurrence of

in the matrix representation

obtained by

puton

deﬁnition successful::rg mat ⇒ bool where

successful m ≡ (occurs N (concat m))=0

Naturally, effectiveness and validity are checked

for successful

puton

. “

puton m

” is said effective

if its result has only one

region.

deﬁnition effective::rg mat ⇒ bool where

effective m ≡ (occurs b (concat m)) = 1

“

puton m

” is valid if its result is one of the

29 units in Fig. 6.

deﬁnition valid::rg mat ⇒ bool where

valid m ≡ ∃V::rg mat. V ∈ U0 ∧

(m ∈ rot_rel‘‘{V})

7.2 Proof Strategy

Given the units in Fig. 6, the number of all the

pairwise combinations is 29 × (29 − 1) + 29 = 841.

Checking properties of success, effectiveness and va-

lidity for all possible superpositions involves numer-

ous computations of

puton

. It is crucial to design

an efﬁcient proof strategy to prove a large number of

properties.

Earlier in Sect. 4.2, we explained the results (10)

and (11) about superposition of rotations of matrices.

Namely, if

and

are two square matrices of the

same dimension, and given that “

map thread on

” is

a superposition with respect to function

, then we

FormalizingtheQualitativeSuperpositionofRectanglesinProofAssistantIsabelle/HOL

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Table 1: Computation of “

map thread on

” for elements of the classes C

and C

f m

Note: Function

stands for “

map thread on

”.

Table 2: Results of applying (12) on the entries of Table 1.

f m

(

f m

)

(

f m

)

(

f m

)

(

f m

)

(

f m

)

(

f m

)

(

f m

)

(

f m

)

(

f m

)

(

f m

)

(

f m

)

(

f m

)

Note: Function

stands for “

map thread on

”.

have the following.

(

map thread on

)

(

((

map thread on

)

−n

)

, if n

≤ n

((

map thread on

)

3×(n

−n

)

, otherwise

(12)

Suppose that we want to check properties of all

the possible applications of

puton

on elements of two

equivalence classes C

and C

. We need to compute

“(

map thread on

)

” for all

∈ C

and

∈ C

as shown in Table 1. By applying the result in (12),

the entries of Table 1 are equivalent to those depicted

in Table 2.

The 16 entries in Table 2 can be grouped into

equivalence classes using relation

rot rel

. Namely, we

have the following 4 equivalence classes [

f m

[

f m

], [

f m

] and [

f m

]. Note that

due to rule (4), the entries (

f m

)

and (

f m

)

in Table 2 are equal to (

f m

)

and

(

f m

)

, respectively, and therefore elements of

the classes [

f m

] and [

f m

], respectively.

Now to ﬁnish computing

puton

, we need to ap-

ply function

merge

. To that end, we introduce (in-

ﬁx) predicate “

preserves

”. A function

preserves an

equivalence relation r if the image of all the elements

in [x]

are in [

g preserves

r , ∀

x y

. (

) ∈ r → (

g x

g y

) ∈ r

We prove that

merge

preserves relation

rot rel

lemma merge preserves rot_rel

Applying

merge

gives rise to the 4 equivalence

classes [

merge

(

f m

)], [

merge

(

f m

)],

[

merge

(

f m

)] and [

merge

(

f m

)].

Next, we use (inﬁx) predicate “

respects

” deﬁned

in the Equiv Relations theory (Paulson, 2006). A

function

respects an equivalence relation r if

re-

turns the same value for all the elements of an equiv-

alence class generated by r.

g respects

r , ∀

x y

. (

) ∈ r →

g x

g y

The properties that we want to check, namely

successful

effective

and

valid

can be regarded as

boolean functions. We prove that they respect

rot rel

If a property P(∈{

successful, effective, valid

}) holds

for one element of a class then it holds for all the re-

maining elements, otherwise it gives False for all the

remaining elements.

lemma successful respects rot_rel

lemma effective respects rot_rel

lemma valid respects rot_rel

In order to check properties of

puton

applied to

all the 16 pairwise combinations of two equivalence

classes C

and C

, it is sufﬁcient to check them for 4

puton

computations that are the

putons

applied to a

representative of C

and all the elements of C

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8 CONCLUSION

We covered the formalization of superposition of rect-

angular units in Isabelle/HOL. A unit is given a quali-

tative matrix representation. The superposition of two

units is regarded as superposition function applied to

their respective matrix representations. Function

pu-

ton

is deﬁned to reﬁne the result of superposition

by merging black regions. We deﬁned properties of

success, effectiveness and validity, and explained our

proof strategy to tackle the numerous cases of super-

position. We can think of two directions for future

work of this research.

First, we plan to relax all the assumptions on the

unit structure that are enumerated in Sect. 5.1. We

would like to include more complex unit structures

such as the ones depicted in Fig. 9. To that end, we

need to extend the set of direction relations to ex-

press the locations of regions that are not directly con-

nected. The reﬁnement to higher granularity level al-

lows representing any unit structure. We also plan to

formalize the generalization in Isabelle/HOL.

(a) (b) (c)

Figure 9: Unit structures whose qualitative representations

require extending direction relations.

Second, QSR theories, such as direction relations,

rely on composition tables that are relevant for de-

ciding whether QSR constraints are consistent (Renz,

2002; Frank, 1991). In order to construct the com-

position tables, the work presented in (Randell et al.,

1992a) uses OTTER proof system for ﬁrst-order logic

to assist pen-and-paper proofs. The authors have enu-

merated the challenge of such proofs because of the

number of the entries of a composition table. We plan

to investigate the proof of the correctness of the com-

position tables using proof assistant.

ACKNOWLEDGEMENTS

This work is supported by JSPS KAKENHI Grant

No. 25330274.

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