Theory Design_OCL

(*****************************************************************************
 * Featherweight-OCL --- A Formal Semantics for UML-OCL Version OCL 2.5
 *                       for the OMG Standard.
 *                       http://www.brucker.ch/projects/hol-testgen/
 *
 * Design_OCL.thy --- OCL Contracts and an Example.
 * This file is part of HOL-TestGen.
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theory
  Design_OCL
imports
  Design_UML
begin
text ‹\label{ex:employee-design:ocl}›

section‹OCL Part: Invariant›
text‹These recursive predicates can be defined conservatively
by greatest fix-point
constructions---automatically. See~\<^cite>‹"brucker.ea:hol-ocl-book:2006" and "brucker:interactive:2007"›
for details. For the purpose of this example, we state them as axioms
here.

\begin{ocl}
context Person
  inv label : self .boss <> null implies (self .salary  ≤  ((self .boss) .salary))
\end{ocl}
›

definition Person_label⇩i⇩n⇩v :: "Person ⇒ Boolean" 
where     "Person_label⇩i⇩n⇩v (self) ≡  
                 (self .boss <> null implies (self .salary  ≤⇩i⇩n⇩t  ((self .boss) .salary)))"
                                       

definition Person_label⇩i⇩n⇩v⇩A⇩T⇩p⇩r⇩e :: "Person ⇒ Boolean" 
where     "Person_label⇩i⇩n⇩v⇩A⇩T⇩p⇩r⇩e (self) ≡  
                 (self .boss@pre <> null implies (self .salary@pre ≤⇩i⇩n⇩t ((self .boss@pre) .salary@pre)))"

definition Person_label⇩g⇩l⇩o⇩b⇩a⇩l⇩i⇩n⇩v :: "Boolean"
where     "Person_label⇩g⇩l⇩o⇩b⇩a⇩l⇩i⇩n⇩v ≡ (Person .allInstances()->forAll⇩S⇩e⇩t(x | Person_label⇩i⇩n⇩v (x)) and 
                                  (Person .allInstances@pre()->forAll⇩S⇩e⇩t(x | Person_label⇩i⇩n⇩v⇩A⇩T⇩p⇩r⇩e (x))))"
                                  
                                  
lemma "τ ⊨ δ (X .boss) ⟹ τ ⊨ Person .allInstances()->includes⇩S⇩e⇩t(X .boss) ∧
                            τ ⊨ Person .allInstances()->includes⇩S⇩e⇩t(X) "
oops  
(* To be generated generically ... hard, but crucial lemma that should hold. 
   It means that X and it successor are object representation that actually
   occur in the state. *)

lemma REC_pre : "τ ⊨ Person_label⇩g⇩l⇩o⇩b⇩a⇩l⇩i⇩n⇩v 
       ⟹ τ ⊨ Person .allInstances()->includes⇩S⇩e⇩t(X) ― ‹‹X› represented object in state›
       ⟹ ∃ REC.  τ ⊨ REC(X)  ≜ (Person_label⇩i⇩n⇩v (X) and (X .boss <> null implies REC(X .boss)))"
oops (* Attempt to allegiate the burden of he following axiomatizations: could be
        a witness for a constant specification ...*)       

text‹This allows to state a predicate:›
                                       
axiomatization inv⇩P⇩e⇩r⇩s⇩o⇩n⇩_⇩l⇩a⇩b⇩e⇩l :: "Person ⇒ Boolean"
where inv⇩P⇩e⇩r⇩s⇩o⇩n⇩_⇩l⇩a⇩b⇩e⇩l_def:
"(τ ⊨ Person .allInstances()->includes⇩S⇩e⇩t(self)) ⟹ 
 (τ ⊨ (inv⇩P⇩e⇩r⇩s⇩o⇩n⇩_⇩l⇩a⇩b⇩e⇩l(self) ≜  (self .boss <> null implies  
                                  (self .salary  ≤⇩i⇩n⇩t  ((self .boss) .salary)) and
                                   inv⇩P⇩e⇩r⇩s⇩o⇩n⇩_⇩l⇩a⇩b⇩e⇩l(self .boss))))"

axiomatization inv⇩P⇩e⇩r⇩s⇩o⇩n⇩_⇩l⇩a⇩b⇩e⇩l⇩A⇩T⇩p⇩r⇩e :: "Person ⇒ Boolean"
where inv⇩P⇩e⇩r⇩s⇩o⇩n⇩_⇩l⇩a⇩b⇩e⇩l⇩A⇩T⇩p⇩r⇩e_def: 
"(τ ⊨ Person .allInstances@pre()->includes⇩S⇩e⇩t(self)) ⟹
 (τ ⊨ (inv⇩P⇩e⇩r⇩s⇩o⇩n⇩_⇩l⇩a⇩b⇩e⇩l⇩A⇩T⇩p⇩r⇩e(self) ≜ (self .boss@pre <> null implies 
                                   (self .salary@pre  ≤⇩i⇩n⇩t  ((self .boss@pre) .salary@pre)) and
                                    inv⇩P⇩e⇩r⇩s⇩o⇩n⇩_⇩l⇩a⇩b⇩e⇩l⇩A⇩T⇩p⇩r⇩e(self .boss@pre))))"


lemma inv_1 : 
"(τ ⊨ Person .allInstances()->includes⇩S⇩e⇩t(self)) ⟹
    (τ ⊨ inv⇩P⇩e⇩r⇩s⇩o⇩n⇩_⇩l⇩a⇩b⇩e⇩l(self) = ((τ ⊨ (self .boss ≐ null)) ∨
                               ( τ ⊨ (self .boss <> null) ∧ 
                                 τ ⊨ ((self .salary)  ≤⇩i⇩n⇩t  (self .boss .salary))  ∧
                                 τ ⊨ (inv⇩P⇩e⇩r⇩s⇩o⇩n⇩_⇩l⇩a⇩b⇩e⇩l(self .boss))))) "
oops (* Let's hope that this holds ... *)


lemma inv_2 : 
"(τ ⊨ Person .allInstances@pre()->includes⇩S⇩e⇩t(self)) ⟹
    (τ ⊨ inv⇩P⇩e⇩r⇩s⇩o⇩n⇩_⇩l⇩a⇩b⇩e⇩l⇩A⇩T⇩p⇩r⇩e(self)) =  ((τ ⊨ (self .boss@pre ≐ null)) ∨
                                     (τ ⊨ (self .boss@pre <> null) ∧
                                     (τ ⊨ (self .boss@pre .salary@pre ≤⇩i⇩n⇩t self .salary@pre))  ∧
                                     (τ ⊨ (inv⇩P⇩e⇩r⇩s⇩o⇩n⇩_⇩l⇩a⇩b⇩e⇩l⇩A⇩T⇩p⇩r⇩e(self .boss@pre)))))"
oops (* Let's hope that this holds ... *)

text‹A very first attempt to characterize the axiomatization by an inductive
definition - this can not be the last word since too weak (should be equality!)›
coinductive inv :: "Person ⇒ (𝔄)st ⇒ bool" where
 "(τ ⊨ (δ self)) ⟹ ((τ ⊨ (self .boss ≐ null)) ∨
                      (τ ⊨ (self .boss <> null) ∧ (τ ⊨ (self .boss .salary ≤⇩i⇩n⇩t self .salary))  ∧
                     ( (inv(self .boss))τ )))
                     ⟹ ( inv self τ)"


section‹OCL Part: The Contract of a Recursive Query›
text‹This part is analogous to the Analysis Model and skipped here.›


end