Why new axiomatic rules: example 2

Given, M is a two-dimensional matrix and e(k, i) is an expression

depending on matrix indices k and i, consider the following triple:

{Q(M ← rep(M, mat(e

1

, e

2

, e

3

, e

4

), e(s, t)))}

for(k = e

1

; k <= e

2

; k++)

for(i = e

3

; i <= e

4

; i++)

M[k][i] = e(k, i);

{Q}

where matrix rep(M, mat(e

1

, e

2

, e

3

, e

4

), e(s, t))) results from replacement

of all elements corresponding to sub-matrix mat(e

1

, e

2

, e

3

, e

4

) by

expression e.

All these logical functions (rep, mat, etc) are dened by a set of

axioms. For example

rep(rep(M, S

1

, e(s, t)), S

2

, e(s, t)) = rep(M, S

1

∪ S

2

, e(s, t))