Update HOME_EXAM.md

HOME_EXAM.md

@@ -261,7 +261,7 @@ assignment:
 ```
 
 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
+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
 
 ```math
 \frac{\lang p, \sigma \rang}{\lang p(\overrightarrow{v}), \sigma \rang \ \Darr \ \lang n, \sigma \rang}
@@ -355,18 +355,24 @@ Greater than or equals (>=):
 
 Equals (==):
 ```math
-\frac{\Gamma \ \vdash \ e1 \ : \ bool \quad \Gamma \ \vdash \ e2 \ : \ bool}{\Gamma \ \vdash (e1 \ == \ e2) \ : \ bool} and
+\frac{\Gamma \ \vdash \ e1 \ : \ bool \quad \Gamma \ \vdash \ e2 \ : \ bool}{\Gamma \ \vdash (e1 \ == \ e2) \ : \ bool}
+```
+or
+```math
 \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 \ : \ bool \quad \Gamma \ \vdash \ e2 \ : \ bool}{\Gamma \ \vdash (e1 \ != \ e2) \ : \ bool}
+```
+or
+```math
 \frac{\Gamma \ \vdash \ e1 \ : \ i32 \quad \Gamma \ \vdash \ e2 \ : \ i32}{\Gamma \ \vdash (e1 \ != \ e2) \ : \ bool}
 ```
 
 ### Assignment
-Given the type $\tau$, variable $x$ and value $n$
+Given the type $`\tau`$, variable $`x`$ and value $`n`$
 
 ```math
 \frac{\Gamma \ \vdash \ x \ : \ \tau \quad \Gamma \ \vdash \ n \ : \ \tau}{\lang x := n, \sigma \rang \ \Darr \ \Gamma \ \vdash x \ : \ \tau}
@@ -378,28 +384,28 @@ Given the type $\tau$, variable $x$ and value $n$
 ```
 
 ### **while** statement
-The condition, $b$, of the while statement  
+The condition, $`b`$, of the while statement  
 
 ```math
 \frac{}{\Gamma \ \vdash b \ : \ bool}
 ```
 
 ### **if/elseif** statement
-The conditions, $b_i$, of the if- and elseif-statements
+The conditions, $`b_i`$, of the if- and elseif-statements
 
 ```math
 \frac{}{\Gamma \ \vdash b_i \ : \ bool}
 ```
 
 ### Functions
-Given the function $p$ with the type of its return type
+Given the function $`p`$ with the type of its return type
 
 ```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
+Given a function with parameters $`p1, \dotsb, p_i`$ and arguments $`a1, \dotsb, a_i`$ then for every parameter and argument
 
 ```math
 \frac{\Gamma \ \vdash p_i \ : \ \tau}{\Gamma \ \vdash a_i \ : \ \tau}
@@ -550,9 +556,9 @@ Error: Mismatched type for operation '=='
                 Note: expected i32, found bool
 Could not compile 'input.rs'
 ```
-By this it can be seen that the type checker assumes that the type of ```a``` is correct when checking the assignment of ```b``` which indeed would be correctly typed if the former variable would have been correctly assigned a i32 value. This is done by concluding that `5 * true` would evaluate to a i32 value if it was correctly typed and then the entire expression `5 + 5 * true` will be correctly typed.
+By this it can be seen that the type checker assumes that the type of `a` is correct when checking the assignment of `b` which indeed would be correctly typed if the former variable would have been correctly assigned a i32 value. This is done by concluding that `5 * true` would evaluate to a i32 value if it was correctly typed and then the entire expression `5 + 5 * true` will be correctly typed.
 
-Thus by following this pattern the type checker is able to continue type checking and reporting new errors, which can be seen when it reports that the assignment of ```c``` is incorrect (boolean equals requires the same type on each side).
+Thus by following this pattern the type checker is able to continue type checking and reporting new errors, which can be seen when it reports that the assignment of `c` is incorrect (boolean equals requires the same type on each side).
 
 The type checker also makes sure that functions with a specified return type always returns. This is done by going through each function and making sure that a tailing return statement or a return inside an else-statement exists. If this is not the case the error is reported along with all other type errors found.
 
@@ -567,7 +573,8 @@ It also needs to keep track of the kind of borrow that have been made to a resou
 
 # Your LLVM backend
 - Let your backend produces LLVM-IR for your example program.
-```LLVM IR
+
+```llvm
 ModuleID = 'program'
 source_filename = "program"
 target datalayout = "e-m:e-i64:64-f80:128-n8:16:32:64-S128"