I'm currently doing some exercise involving proving the correctness of algorithms.
I often have to write something that resembles:
- "The element $A[i]$ is smaller than any of the element in the subarray $A[i + 1\dots n]$"
- "Every element in $A[1\dots i]$ is smaller than any element in $A[i + 1\dots n]$".
I hope to convert the statements into something more concise using mathematical symbols. So far I have thought of
- $\forall a \in A[i + 1 ... n] ~~ A[i] < a$
- $\forall a \in A[1 \dots i]~~\not\exists a' \in A[i + 1\dots n]~~ a' \ge a$
but they all seem too complicated for a supposedly Computer Science course.
Is there any notation (e.g. $A[i] < A[i + 1\dots n]$) that is perhaps more standard and less complicated?
$\qquad (1)\;\;\;$"The element $A[i]$ is smaller than any of the element in the subarray $A[i + 1\dots n]$."
$$i < t \le n \implies A[i] < A[t]$$
$\qquad (2)\;\;\;$"Every element in $A[1\dots i]$ is smaller than any element in $A[i + 1\dots n]$."
$$1 \le s \le i < t \le n \implies A[s] < A[t]$$