Merge branch 'master' into 'master'

[Add] : Add 4.4 a)

See merge request etienne.levecque.etu/foundations-of-databases-exercise-corrections!2

Chapter 4/Exercise-4.4.md

+# Exercise 4.4
+
+(a) Let $`\varphi`$ and $`\psi`$ be equivalent conjunctive calculus formulas, and suppose that $`\Psi^{\prime}`$ is the result of replacing an occurrence of $`\varphi`$ by $`\psi`$ in conjunctive calculus formula $`\Psi`$. Prove that $`\Psi`$ and $`\Psi^{\prime}`$ are equivalent.
+
+(b) Prove that the application of the rewrite rules rename and merge-exists to a conjunctive calculus formula yields an equivalent formula.
+
+(c) Prove that these rules can be used to transform any conjunctive calculus formula into
+an equivalent formula in normal form.
+
+* * *
+### Rony
+
+a)
+
+Let's recall that : Conjunctive calculus formulas ϕ and ψ over **R** are equivalent if **they have the same
+free variables** and, for each **I** over **R** and valuation $`ν`$ over free(ϕ) = free(ψ), 
+
+**I |= ϕ[ν] iff I |= ψ[ν].**
+
+
+Let's say that : $`\Psi = \Phi \ \varphi \ \Phi^{\prime}`$ . In that case $`\Psi^{\prime} = \Phi \ \psi \ \Phi^{\prime}`$.
+
+$`\blacksquare`$ Clearly, $`free(\Psi)=free(\varphi) \ \text{and} \ free(\Psi^{\prime})=free(\psi)`$
+By hypothesis, $`\varphi`$ and $`\psi`$ are equivalent and so :
+
+$`free(\Psi)=free(\varphi)=free(\psi)=free(\Psi^{\prime})`$
+
+$`\blacksquare`$  Let's **I** an instance over **R** and $`v`$ a valuation over $`free(\Psi)=free(\Psi^{\prime})`$.
+
+Clearly  I|=$`\Psi[v]`$
+
+iff (I|=$`\Phi[v]`$ and I|=$`\varphi[v]`$ and  I|=$`\Phi^{\prime}[v]`$)    
+
+ iff (I|=$`\Phi[v]`$ and I|=$`\psi[v]`$ and  I|=$`\Phi^{\prime}[v]`$)   because  $`\varphi`$ and $`\psi`$ are equivalent.
+
+ iff I|=$`\Psi^{\prime}[v]`$
+ 
+ ### So Finally, $`\Psi`$ and $`\Psi^{\prime}`$ are equivalent.
+
+
+
+