theory MathExam
  imports "../../ontologies/mathex_onto"
    Real
begin
   
open_monitor*[exam::MathExam] 

section*[idir::Author, affiliation="''CentraleSupelec''", 
         email="''idir.aitsadoune@centralesupelec.fr''"]
{*Idir AIT SADOUNE*}

subsection*[header::Header,examSubject= "[algebra]",  examTitle="''Exam number 1''", 
            date="''02-05-2018''", timeAllowed="90::int"]
{* 
\begin{itemize}
\item Use black ink or black ball-point pen. 
\item Draw diagrams in pencil.
\item Answer all questions in the spaces provided.
\end{itemize}
*}

(*
text*[fig1::figure, width = "Some(textwidth 80)", 
      "file"="@{file ''../../../figures/Dogfood-Intro.png''}"]
     {* Ouroboros I : This paper from inside ... *}  
*)

subsubsection*[exo1 :: Exercise, Exercise.content="[q1::Task,q2::Task]"]
{* 
Here are the first four lines of a number pattern.
\begin{itemize}
\item Line 1 : @{term "1*6 + 2*4 = 2*7"}
\item Line 2 : @{term "2*7 + 2*5 = 3*8"}
\item Line 3 : @{term "3*8 + 2*6 = 4*9"}
\item Line 4 : @{term "4*9 + 2*7 = 5*10"} 
\end{itemize}
*}

(* a bit brutal, as long as lemma* does not yet work *)
(*<*)
lemma check_polynome :
  fixes x::real 
  shows "(x^3) - 6 * x^2 + 5 * x + 12 = (x-4) * (x+1) * (x - 3)"
        
proof -
  have * : "(x-4) * (x+1) * (x - 3) = (x-4) * ((x+1) * (x-3))"
         by simp
  have ** : "... = (x-4) * (x^2 - 2*x - 3)"
    apply(auto simp: right_diff_distrib add.commute semiring_normalization_rules(1)[symmetric])
    by (simp add: semiring_normalization_rules(29))
  have *** : "... = x^3 - 6 * x^2 + 5 * x + 12"
    apply(auto simp: right_diff_distrib left_diff_distrib add.commute semiring_normalization_rules(1)[symmetric])
    by (simp add: numeral_3_eq_3 semiring_normalization_rules(29))
  show ?thesis
    by(simp only: * ** ***)
qed
(*>*)
  
text*[a1::Answer_Formal_Step]{* First Step: Fill in term and justification *}
text*[a2::Answer_Formal_Step]{* Next Step: Fill in term and justification *}
text*[a3::Answer_Formal_Step]{* Next Step: Fill in term and justification *}
text*[a4::Answer_Formal_Step]{* Next Step: Fill in term and justification *}
  
text*[q1::Task, level="oneStar", mark="1::int", type="formal"] 
{* Complete Line 10 :  @{term "10*x + 2*y =  11*16"} *}

text*[q2::Task, level="threeStars", mark="3::int", type="formal"] 
{*
Prove that  @{term "n*(n+5) + 2*(n+3) "} is always the product of two numbers with a difference of 5.
*}


close_monitor*[exam] 

end