Update HOME_EXAM.md

HOME_EXAM.md

@@ -156,77 +156,112 @@ No error messages, except the already existing error messages from LALRPOP, have
 Given a expression $e$ in the state $\sigma$  that evaluates to a value $n$ or boolean $b$ the following can be said for the operators in the language
 
 #### Arithmetic
-Addition:\
+Addition:
 ```math
 \frac{\lang e1, \sigma \rang \ \Downarrow \ n1 \; \lang e2, \sigma \rang \ \Downarrow \ n2}{\lang e1 + e2, \sigma \rang \ \Downarrow \ n1 + n2}
 ```
 
-Subtraction:\
-$\frac{\lang e1, \sigma \rang \ \Downarrow \ n1 \; \lang e2, \sigma \rang \ \Downarrow \ n2}{\lang e1 - e2, \sigma \rang \ \Downarrow \ n1 - n2}$
+Subtraction:
+```math
+\frac{\lang e1, \sigma \rang \ \Downarrow \ n1 \; \lang e2, \sigma \rang \ \Downarrow \ n2}{\lang e1 - e2, \sigma \rang \ \Downarrow \ n1 - n2}
+```
 
-Multiplication:\
-$\frac{\lang e1, \sigma \rang \ \Downarrow \ n1 \; \lang e2, \sigma \rang \ \Downarrow \ n2}{\lang e1 * e2, \sigma \rang \ \Downarrow \ n1 * n2}$
+Multiplication:
+```math
+\frac{\lang e1, \sigma \rang \ \Downarrow \ n1 \; \lang e2, \sigma \rang \ \Downarrow \ n2}{\lang e1 * e2, \sigma \rang \ \Downarrow \ n1 * n2}
+```
 
-Division:\
-$\frac{\lang e1, \sigma \rang \ \Downarrow \ n1 \; \lang e2, \sigma \rang \ \Downarrow \ n2}{\lang e1 / e2, \sigma \rang \ \Downarrow \ n1 / n2}$
+Division:
+```math
+\frac{\lang e1, \sigma \rang \ \Downarrow \ n1 \; \lang e2, \sigma \rang \ \Downarrow \ n2}{\lang e1 / e2, \sigma \rang \ \Downarrow \ n1 / n2}
+```
 
-Unary:\
-$\frac{\lang e1, \sigma \rang \ \Downarrow \ n1}{\lang -e1, \sigma \rang \ \Downarrow \ -n1}$
+Unary:
+```math
+\frac{\lang e1, \sigma \rang \ \Downarrow \ n1}{\lang -e1, \sigma \rang \ \Downarrow \ -n1}
+```
 
 #### Boolean
-AND:\
-$\frac{\lang e1, \sigma \rang \Downarrow \ b1 \; \lang e2, \sigma \rang \Downarrow \ b1}{\lang e1 \ \&\& \ e2, \sigma \rang \ \Downarrow \ b1 \ \text{AND} \ b2}$
+AND:
+```math
+\frac{\lang e1, \sigma \rang \Downarrow \ b1 \; \lang e2, \sigma \rang \Downarrow \ b1}{\lang e1 \ \&\& \ e2, \sigma \rang \ \Downarrow \ b1 \ \text{AND} \ b2}
+```
 
-OR:\
-$\frac{\lang e1, \sigma \rang \Downarrow \ b1 \; \lang e2, \sigma \rang \Downarrow \ b2}{\lang e1 \ || \ e2, \sigma \rang \ \Downarrow \ b1 \ \text{OR} \ b2}$
+OR:
+```math
+\frac{\lang e1, \sigma \rang \Downarrow \ b1 \; \lang e2, \sigma \rang \Downarrow \ b2}{\lang e1 \ || \ e2, \sigma \rang \ \Downarrow \ b1 \ \text{OR} \ b2}
+```
 
-Less than (<):\
-$\frac{\lang e1, \sigma \rang \Downarrow \ n1 \; \lang e2, \sigma \rang \Downarrow \ n2}{\lang e1 \ < \ e2, \sigma \rang \ \Downarrow \ n1 \ < \ n2}$
+Less than (<):
+```math
+\frac{\lang e1, \sigma \rang \Downarrow \ n1 \; \lang e2, \sigma \rang \Downarrow \ n2}{\lang e1 \ < \ e2, \sigma \rang \ \Downarrow \ n1 \ < \ n2}
+```
 
-Greaten than (>):\
-$\frac{\lang e1, \sigma \rang \Downarrow \ n1 \; \lang e2, \sigma \rang \Downarrow \ n2}{\lang e1 \ > \ e2, \sigma \rang \ \Downarrow \ n1 \ > \ n2}$
+Greaten than (>):
+```math
+\frac{\lang e1, \sigma \rang \Downarrow \ n1 \; \lang e2, \sigma \rang \Downarrow \ n2}{\lang e1 \ > \ e2, \sigma \rang \ \Downarrow \ n1 \ > \ n2}
+```
 
-Less than or equals (<=):\
-$\frac{\lang e1, \sigma \rang \Downarrow \ n1 \; \lang e2, \sigma \rang \Downarrow \ n2}{\lang e1 \ <= \ e2, \sigma \rang \ \Downarrow \ n1 \ <= \ n2}$
+Less than or equals (<=):
+```math
+\frac{\lang e1, \sigma \rang \Downarrow \ n1 \; \lang e2, \sigma \rang \Downarrow \ n2}{\lang e1 \ <= \ e2, \sigma \rang \ \Downarrow \ n1 \ <= \ n2}
+```
 
-Greater than or equals (>=):\
-$\frac{\lang e1, \sigma \rang \Downarrow \ n1 \; \lang e2, \sigma \rang \Downarrow \ n2}{\lang e1 \ >= \ e2, \sigma \rang \ \Downarrow \ n1 \ >= \ n2}$
+Greater than or equals (>=):
+```math
+\frac{\lang e1, \sigma \rang \Downarrow \ n1 \; \lang e2, \sigma \rang \Downarrow \ n2}{\lang e1 \ >= \ e2, \sigma \rang \ \Downarrow \ n1 \ >= \ n2}
+```
 
-Equals (==):\
-$\frac{\lang e1, \sigma \rang \Downarrow \ n1 \; \lang e2, \sigma \rang \Downarrow \ n2}{\lang e1 \ == \ e2, \sigma \rang \ \Downarrow \ n1 \ == \ n2}$
+Equals (==):
+```math
+\frac{\lang e1, \sigma \rang \Downarrow \ n1 \; \lang e2, \sigma \rang \Downarrow \ n2}{\lang e1 \ == \ e2, \sigma \rang \ \Downarrow \ n1 \ == \ n2}
+```
 
-Not equals (!=):\
-$\frac{\lang e1, \sigma \rang \Downarrow \ n1 \; \lang e2, \sigma \rang \Downarrow \ n2}{\lang e1 \ != \ e2, \sigma \rang \ \Downarrow \ n1 \ != \ n2}$
+Not equals (!=):
+```math
+\frac{\lang e1, \sigma \rang \Downarrow \ n1 \; \lang e2, \sigma \rang \Downarrow \ n2}{\lang e1 \ != \ e2, \sigma \rang \ \Downarrow \ n1 \ != \ n2}
+```
 
 #### Commands
 
 **if** statement:
 
-$\frac{\lang b, \sigma \rang \ \Downarrow \ \text{true} \quad \lang c1, \sigma \rang \ \Downarrow \ \sigma'}{\lang \text{if } b \text{ then } c1 \text{ else } c2, \sigma \rang \ \Downarrow \ \sigma'}$
+```math
+\frac{\lang b, \sigma \rang \ \Downarrow \ \text{true} \quad \lang c1, \sigma \rang \ \Downarrow \ \sigma'}{\lang \text{if } b \text{ then } c1 \text{ else } c2, \sigma \rang \ \Downarrow \ \sigma'}
 
-$\frac{\lang b, \sigma \rang \ \Downarrow \ \text{false} \quad \lang c2, \sigma \rang \ \Downarrow \ \sigma''}{\lang \text{if } b \text{ then } c1 \text{ else } c2, \sigma \rang \ \Downarrow \ \sigma''}$
+\frac{\lang b, \sigma \rang \ \Downarrow \ \text{false} \quad \lang c2, \sigma \rang \ \Downarrow \ \sigma''}{\lang \text{if } b \text{ then } c1 \text{ else } c2, \sigma \rang \ \Downarrow \ \sigma''}
+```
 
 Not sure how to do this for elseif
 
 **while** statement:
+```math
+\frac{\lang b, \sigma \rang \ \Downarrow \ \text{false} \quad \lang c, \sigma \rang \Downarrow \sigma' }{\text{while } b \text{ do } c, \sigma \rang \ \Downarrow \ \sigma}
 
-$\frac{\lang b, \sigma \rang \ \Downarrow \ \text{false} \quad \lang c, \sigma \rang \Downarrow \sigma' }{\text{while } b \text{ do } c, \sigma \rang \ \Downarrow \ \sigma}$
-
-$\frac{\lang b, \sigma \rang \ \Downarrow \ \text{true} \quad \lang c, \sigma \rang \ \Downarrow \ \sigma' \quad \lang \text{while } b \text{ do } c,\sigma' \rang \ \Downarrow \ \sigma''}{\lang \text{while } b \text{ do } c, \sigma \rang \ \Downarrow \ \sigma''}$
+\frac{\lang b, \sigma \rang \ \Downarrow \ \text{true} \quad \lang c, \sigma \rang \ \Downarrow \ \sigma' \quad \lang \text{while } b \text{ do } c,\sigma' \rang \ \Downarrow \ \sigma''}{\lang \text{while } b \text{ do } c, \sigma \rang \ \Downarrow \ \sigma''}
+```
 
 **let** statement:\
-$\frac{}{\lang \text{let }x := n, \sigma \rang \ \Downarrow \ \sigma [x := n]}$
+```math
+\frac{}{\lang \text{let }x := n, \sigma \rang \ \Downarrow \ \sigma [x := n]}
+```
 
-**return** statement:\
-$\frac{\lang e, \sigma \rang \Downarrow n}{\lang \text{return } e, \sigma \rang \Downarrow n }$
+**return** statement:
+```math
+\frac{\lang e, \sigma \rang \Downarrow n}{\lang \text{return } e, \sigma \rang \Downarrow n }
+```
 
-assignment:\
-$\frac{}{\lang x := n, \sigma \rang \ \Downarrow \ \sigma [x := n]}$
+assignment:
+```math
+\frac{}{\lang x := n, \sigma \rang \ \Downarrow \ \sigma [x := n]}
+```
 
-functions: \
+functions:
 Given a list of arguments $\overrightarrow{v} = [v_1, \ldots, v_n]$ the call of function $p$ will evaulate to a value $n$ (the return value, if any) according to
 
-$\frac{\lang p, \sigma \rang}{\lang p(\overrightarrow{v}), \sigma \rang \ \Darr \ \lang n, \sigma \rang}$
+```math
+\frac{\lang p, \sigma \rang}{\lang p(\overrightarrow{v}), \sigma \rang \ \Darr \ \lang n, \sigma \rang}
+```
 
 - Explain (in text) what an interpretation of your example should produce, do that by dry running your given example step by step. Relate back to the SOS rules. You may skip repetions to avoid cluttering.
 
@@ -258,75 +293,113 @@ the interpreter will panic with the message:
 
 ### Expressions
 #### Arithmetic
-Addition:\
-$\frac{\Gamma \ \vdash \ e1 \ : \ i32 \quad \Gamma \ \vdash \ e2 \ : \ i32}{\Gamma \ \vdash (e1 + e2) \ : \ i32}$
+Addition:
+```math
+\frac{\Gamma \ \vdash \ e1 \ : \ i32 \quad \Gamma \ \vdash \ e2 \ : \ i32}{\Gamma \ \vdash (e1 + e2) \ : \ i32}
+```
 
-Subtraction:\
-$\frac{\Gamma \ \vdash \ e1 \ : \ i32 \quad \Gamma \ \vdash \ e2 \ : \ i32}{\Gamma \ \vdash (e1 - e2) \ : \ i32}$
+Subtraction:
+```math
+\frac{\Gamma \ \vdash \ e1 \ : \ i32 \quad \Gamma \ \vdash \ e2 \ : \ i32}{\Gamma \ \vdash (e1 - e2) \ : \ i32}
+```
 
-Division:\
-$\frac{\Gamma \ \vdash \ e1 \ : \ i32 \quad \Gamma \ \vdash \ e2 \ : \ i32}{\Gamma \ \vdash (e1 / e2) \ : \ i32}$
+Division:
+```math
+\frac{\Gamma \ \vdash \ e1 \ : \ i32 \quad \Gamma \ \vdash \ e2 \ : \ i32}{\Gamma \ \vdash (e1 / e2) \ : \ i32}
+```
 
-Multiplication:\
-$\frac{\Gamma \ \vdash \ e1 \ : \ i32 \quad \Gamma \ \vdash \ e2 \ : \ i32}{\Gamma \ \vdash (e1 * e2) \ : \ i32}$
+Multiplication:
+```math
+\frac{\Gamma \ \vdash \ e1 \ : \ i32 \quad \Gamma \ \vdash \ e2 \ : \ i32}{\Gamma \ \vdash (e1 * e2) \ : \ i32}
+```
 
-Unary:\
-$\frac{\Gamma \ \vdash \ e1 \ : \ i32}{\Gamma \ \vdash (-e1) \ : \ i32}$
+Unary:
+```math
+\frac{\Gamma \ \vdash \ e1 \ : \ i32}{\Gamma \ \vdash (-e1) \ : \ i32}
+```
 
 #### Boolean
-AND:\
-$\frac{\Gamma \ \vdash \ e1 \ : \ bool \quad \Gamma \ \vdash \ e2 \ : \ bool}{\Gamma \ \vdash (e1 \ \&\& \ e2) \ : \ bool}$
+AND:
+```math
+\frac{\Gamma \ \vdash \ e1 \ : \ bool \quad \Gamma \ \vdash \ e2 \ : \ bool}{\Gamma \ \vdash (e1 \ \&\& \ e2) \ : \ bool}
+```
 
-OR:\
-$\frac{\Gamma \ \vdash \ e1 \ : \ bool \quad \Gamma \ \vdash \ e2 \ : \ bool}{\Gamma \ \vdash (e1 \ || \ e2) \ : \ bool}$
+OR:
+```math
+\frac{\Gamma \ \vdash \ e1 \ : \ bool \quad \Gamma \ \vdash \ e2 \ : \ bool}{\Gamma \ \vdash (e1 \ || \ e2) \ : \ bool}
+```
 
-Less than (<):\
-$\frac{\Gamma \ \vdash \ e1 \ : \ i32 \quad \Gamma \ \vdash \ e2 \ : \ i32}{\Gamma \ \vdash (e1 \ < \ e2) \ : \ bool}$
+Less than (<):
+```math
+\frac{\Gamma \ \vdash \ e1 \ : \ i32 \quad \Gamma \ \vdash \ e2 \ : \ i32}{\Gamma \ \vdash (e1 \ < \ e2) \ : \ bool}
+```
 
-Greaten than (>):\
-$\frac{\Gamma \ \vdash \ e1 \ : \ i32 \quad \Gamma \ \vdash \ e2 \ : \ i32}{\Gamma \ \vdash (e1 \ > \ e2) \ : \ bool}$
+Greaten than (>):
+```math
+\frac{\Gamma \ \vdash \ e1 \ : \ i32 \quad \Gamma \ \vdash \ e2 \ : \ i32}{\Gamma \ \vdash (e1 \ > \ e2) \ : \ bool}
+```
 
-Less than or equals (<=):\
-$\frac{\Gamma \ \vdash \ e1 \ : \ i32 \quad \Gamma \ \vdash \ e2 \ : \ i32}{\Gamma \ \vdash (e1 \ <= \ e2) \ : \ bool}$
+Less than or equals (<=):
+```math
+\frac{\Gamma \ \vdash \ e1 \ : \ i32 \quad \Gamma \ \vdash \ e2 \ : \ i32}{\Gamma \ \vdash (e1 \ <= \ e2) \ : \ bool}
+```
 
-Greater than or equals (>=):\
-$\frac{\Gamma \ \vdash \ e1 \ : \ i32 \quad \Gamma \ \vdash \ e2 \ : \ i32}{\Gamma \ \vdash (e1 \ >= \ e2) \ : \ bool}$
+Greater than or equals (>=):
+```math
+\frac{\Gamma \ \vdash \ e1 \ : \ i32 \quad \Gamma \ \vdash \ e2 \ : \ i32}{\Gamma \ \vdash (e1 \ >= \ e2) \ : \ bool}
+```
 
-Equals (==):\
-$\frac{\Gamma \ \vdash \ e1 \ : \ bool \quad \Gamma \ \vdash \ e2 \ : \ bool}{\Gamma \ \vdash (e1 \ == \ e2) \ : \ bool}$ and
-$\frac{\Gamma \ \vdash \ e1 \ : \ i32 \quad \Gamma \ \vdash \ e2 \ : \ i32}{\Gamma \ \vdash (e1 \ == \ e2) \ : \ bool}$
+Equals (==):
+```math
+\frac{\Gamma \ \vdash \ e1 \ : \ bool \quad \Gamma \ \vdash \ e2 \ : \ bool}{\Gamma \ \vdash (e1 \ == \ e2) \ : \ bool} and
+\frac{\Gamma \ \vdash \ e1 \ : \ i32 \quad \Gamma \ \vdash \ e2 \ : \ i32}{\Gamma \ \vdash (e1 \ == \ e2) \ : \ bool}
+```
 
-Not equals (!=):\
-$\frac{\Gamma \ \vdash \ e1 \ : \ bool \quad \Gamma \ \vdash \ e2 \ : \ bool}{\Gamma \ \vdash (e1 \ != \ e2) \ : \ bool}$ and
-$\frac{\Gamma \ \vdash \ e1 \ : \ i32 \quad \Gamma \ \vdash \ e2 \ : \ i32}{\Gamma \ \vdash (e1 \ != \ e2) \ : \ bool}$
+Not equals (!=):
+```math
+\frac{\Gamma \ \vdash \ e1 \ : \ bool \quad \Gamma \ \vdash \ e2 \ : \ bool}{\Gamma \ \vdash (e1 \ != \ e2) \ : \ bool} and
+\frac{\Gamma \ \vdash \ e1 \ : \ i32 \quad \Gamma \ \vdash \ e2 \ : \ i32}{\Gamma \ \vdash (e1 \ != \ e2) \ : \ bool}
+```
 
 ### Assignment
 Given the type $\tau$, variable $x$ and value $n$
 
-$\frac{\Gamma \ \vdash \ x \ : \ \tau \quad \Gamma \ \vdash \ n \ : \ \tau}{\lang x := n, \sigma \rang \ \Darr \ \Gamma \ \vdash x \ : \ \tau}$
+```math
+\frac{\Gamma \ \vdash \ x \ : \ \tau \quad \Gamma \ \vdash \ n \ : \ \tau}{\lang x := n, \sigma \rang \ \Darr \ \Gamma \ \vdash x \ : \ \tau}
+```
 
 ### **let** assignment
-$\frac{\Gamma \ \vdash \ n \ : \ \tau}{\lang \text{let} \  x \ : \ \tau \ := \ n, \sigma \rang \ \Darr \ \Gamma \ \vdash x \ : \ \tau}$
+```math
+\frac{\Gamma \ \vdash \ n \ : \ \tau}{\lang \text{let} \  x \ : \ \tau \ := \ n, \sigma \rang \ \Darr \ \Gamma \ \vdash x \ : \ \tau}
+```
 
 ### **while** statement
 The condition, $b$, of the while statement  
 
-$\frac{}{\Gamma \ \vdash b \ : \ bool}$
+```math
+\frac{}{\Gamma \ \vdash b \ : \ bool}
+```
 
 ### **if/elseif** statement
 The conditions, $b_i$, of the if- and elseif-statements
 
-$\frac{}{\Gamma \ \vdash b_i \ : \ bool}$
+```math
+\frac{}{\Gamma \ \vdash b_i \ : \ bool}
+```
 
 ### Functions
 Given the function $p$ with the type of its return type
 
-$\frac{\Gamma \ \vdash p \ : \ \tau \quad \lang p, \sigma \rang \ \Darr \ n}{\Gamma \ \vdash n \ : \ \tau}$
+```math
+\frac{\Gamma \ \vdash p \ : \ \tau \quad \lang p, \sigma \rang \ \Darr \ n}{\Gamma \ \vdash n \ : \ \tau}
+```
 
 #### Parameters and arguments
 Given a function with parameters $p1, \dotsb, p_i$ and arguments $a1, \dotsb, a_i$ then for every parameter and argument
 
-$\frac{\Gamma \ \vdash p_i \ : \ \tau}{\Gamma \ \vdash a_i \ : \ \tau}$
+```math
+\frac{\Gamma \ \vdash p_i \ : \ \tau}{\Gamma \ \vdash a_i \ : \ \tau}
+```
 
 - Demonstrate each "type rule" by an example