2. Assume a declarative interface where n and max are constant integers, and A is an array of integers of size max. Consider the following correctness statement: ASSERT ( 1 < n <= max int i; i = n-1; A [n-1] = 1; while( i >= 1) { A[i-1] = A[i]ni + 1; i = i-1%; } ASSERT ForAll(k = 0%; k < n) A[k] == (n-k)* (n-k+1)/2) 1 (a) Give a complete proof tableau for the above correctness statement by adding all the intermediate assertions. State all the mathematical facts that are used in the proof. Hints: As our discussion in class will also emphasize the following two assertions are obvious tautologies: Forall (k=i; k Forall (k=i; k P(i) && Forall (k=i+1; k

Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
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2. Assume a declarative interface where n and max are constant integers, and A is an array of
integers of size max. Consider the following correctness statement:
ASSERT ( 1 < n <= max
int i;
i = n-1;
A [n-1] = 1;
while( i >= 1) {
A[i-1]
= A[i]ni + 1;
i = i-1%;
}
ASSERT ForAll(k = 0%; k < n) A[k]
==
(n-k)* (n-k+1)/2)
1
(a) Give a complete proof tableau for the above correctness statement by adding all the
intermediate assertions. State all the mathematical facts that are used in the proof.
Hints: As our discussion in class will also emphasize the following two assertions are
obvious tautologies:
Forall (k=i; k<j) P(k) <=>
Forall (k=i; k<j) P(k) <=>
P(i) && Forall (k=i+1; k<j) P(k)
P(j-1) && Forall (k=i; k<j-1) P(k)
That is, we can always separate as an extra conjunctive term an “end" of the range in a
universally quantified property. Also note that this time the loop goes backward. This
might be intimidating but actually that does not make much of a difference.
(b) Provide a formal argument for the total correctness of the statement.
Transcribed Image Text:2. Assume a declarative interface where n and max are constant integers, and A is an array of integers of size max. Consider the following correctness statement: ASSERT ( 1 < n <= max int i; i = n-1; A [n-1] = 1; while( i >= 1) { A[i-1] = A[i]ni + 1; i = i-1%; } ASSERT ForAll(k = 0%; k < n) A[k] == (n-k)* (n-k+1)/2) 1 (a) Give a complete proof tableau for the above correctness statement by adding all the intermediate assertions. State all the mathematical facts that are used in the proof. Hints: As our discussion in class will also emphasize the following two assertions are obvious tautologies: Forall (k=i; k<j) P(k) <=> Forall (k=i; k<j) P(k) <=> P(i) && Forall (k=i+1; k<j) P(k) P(j-1) && Forall (k=i; k<j-1) P(k) That is, we can always separate as an extra conjunctive term an “end" of the range in a universally quantified property. Also note that this time the loop goes backward. This might be intimidating but actually that does not make much of a difference. (b) Provide a formal argument for the total correctness of the statement.
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