The Maximal Value

### Program Information

Name: The Maximal Value
Domain: Numerical program
Functionality: Calculates the maximal value from a set of input data
Input: Suppose that there is a set of real numbers, $\mathbf{X}=\{x_1,x_2,\cdots ,x_n\}$.And there is another set $\mathbf{X}'$.
Output: The maximal value of $\mathbf{X}$ is $m$,and the maximal value in $\mathbf{X}'$ is $m'$.

#### Reference

Teaching an End-User Testing Methodology https://doi.org/10.1109/CSEET.2010.28

### MR Information

#### MR1------

Description:
Property: $\mathbf{X}'=\mathbf{X}\cup \{a\}$ where $a>m$ $\Rightarrow$ $m'>m$
Source input: $\mathbf{X}$
Source output: $m$
Follow-up input: $\mathbf{X}'$
Follow-up output: $m'$
Input relation: $\mathbf{X} \Rightarrow \mathbf{X}'$ where $\mathbf{X}'=\mathbf{X}\cup \{a\}$ and $a>m$
Output relation: $m'>m$
Pattern:

#### MR2------

Description:
Property: $\mathbf{X}'=\mathbf{X}\cup \{b\}$ where $b<m$ $\Rightarrow$ $m'=m$
Source input: $\mathbf{X}$
Source output: $m$
Follow-up input: $\mathbf{X}'$
Follow-up output: $m'$
Input relation: $\mathbf{X} \Rightarrow \mathbf{X}'$ where $\mathbf{X}'=\mathbf{X}\cup \{b\}$ and $b<m$
Output relation: $m'= m$
Pattern:

#### MR3------

Description:
Property: $\mathbf{X}'=\mathbf{X} - \{m\}$ (deleting all elements that are equal to m) $\Rightarrow$ $m'<m$
Source input: $\mathbf{X}$
Source output: $m$
Follow-up input: $\mathbf{X}'$
Follow-up output: $m'$
Input relation: $\mathbf{X} \Rightarrow \mathbf{X}'$ where $\mathbf{X}'=\mathbf{X} - \{m\}$(constructed by deleting all elements that are equal to m)
Output relation: $m'< m$
Pattern:

#### MR4------

Description:
Property: $\mathbf{X}'=\{0.0-x_1,0.0-x_2,\cdots ,0.0-x_n\} \Rightarrow m'=c$ where $c$ is the minimal value in $\mathbf{X}$
Source input: $\mathbf{X}$
Source output: $m$
Follow-up input: $\mathbf{X}'$
Follow-up output: $m'$
Input relation: $\mathbf{X} \Rightarrow \mathbf{X}'$ where $\mathbf{X}'=\{0.0-x_1,0.0-x_2,\cdots ,0.0-x_n\}$
Output relation: $m'= c$ where $c$ is the minimal value in $\mathbf{X}$
Pattern:

#### MR5------

Description:
Property: $\mathbf{X}'$ (changing the order of the data in $\mathbf{X}$) $\Rightarrow m'=m$
Source input: $\mathbf{X}$
Source output: $m$
Follow-up input: $\mathbf{X}'$
Follow-up output: $m'$
Input relation: $\mathbf{X} \Rightarrow \mathbf{X}'$ where $\mathbf{X}'$ is constructed by changing the order of the data in $\mathbf{X}$
Output relation: $m'= m$
Pattern:

#### MR6------

Description:
Property: $\mathbf{X}'=\{d*x_1,d*x_2,\cdots ,d*x_n\}$ where $d > 0 \Rightarrow m'=d*m$
Source input: $\mathbf{X}$
Source output: $m$
Follow-up input: $\mathbf{X}'$
Follow-up output: $m'$
Input relation: $\mathbf{X} \Rightarrow \mathbf{X}'$ where $\mathbf{X}'=\{d*x_1,d*x_2,\cdots ,d*x_n\}$ and $d > 0$
Output relation: $m'= d*m$
Pattern:

#### MR7------

Description:
Property: $\mathbf{X}'=\mathbf{X} \cup \{e\}$ where $e = m \Rightarrow m'= m$
Source input: $\mathbf{X}$
Source output: $m$
Follow-up input: $\mathbf{X}'$
Follow-up output: $m'$
Input relation: $\mathbf{X} \Rightarrow \mathbf{X}'$ where $\mathbf{X}'=\mathbf{X} \cup \{e\}$ and $e = m$
Output relation: $m'= m$
Pattern:

#### MR8------

Description:
Property: $\mathbf{X}'= \{f+x_1,f+x_2,\cdots ,f+x_n\} \Rightarrow m'= f+m$
Source input: $\mathbf{X}$
Source output: $m$
Follow-up input: $\mathbf{X}'$
Follow-up output: $m'$
Input relation: $\mathbf{X} \Rightarrow \mathbf{X}'$ where $\mathbf{X}'= \{f+x_1,f+x_2,\cdots ,f+x_n\}$
Output relation: $m'= f+m$
Pattern:
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