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zsj7(^xT32@-W^04G+_djW3dZT#h7k%n!;;q5$d{Vdqf%4zclRq+yG!&El4rB7N zQ9%A3;^UQe9P}~>D=bV$iDH_VV^o3_Z5owh)J3FZnwcXi$&}K|+|H^lNNW~iL|+ce zY?f@4tqEWPFIlBaR)mv?sm2<`))5>oS=FKv9Tfx{A-6MH(D2qdSQDB#hPzL}RzvG^ zI(2=qJR)OMmu%x&WbGM1eW0ZZIO6EJruZPFO#>?H1hER>H+5R9_VJq2e37|lUtm(d0GNJ;lSGL*UGe(@4_@JZlmEK-pPwkPJ&L=BnVZK~H!Eut S9zFrV_mcx1ovf-H%KrjM_Y~v+ diff --git a/semester4/fmfp/formal-methods-functional-programming-summary.tex b/semester4/fmfp/formal-methods-functional-programming-summary.tex index 2c992a8..6311613 100644 --- a/semester4/fmfp/formal-methods-functional-programming-summary.tex +++ b/semester4/fmfp/formal-methods-functional-programming-summary.tex @@ -60,6 +60,7 @@ \input{parts/01_formal-reasoning/03_first-order-logic/00_syntax.tex} \input{parts/01_formal-reasoning/03_first-order-logic/01_semantics.tex} \input{parts/01_formal-reasoning/03_first-order-logic/02_quantifiers.tex} +\input{parts/01_formal-reasoning/04_equality.tex} % \input{parts/01_formal-reasoning/} diff --git a/semester4/fmfp/parts/01_formal-reasoning/04_equality.tex b/semester4/fmfp/parts/01_formal-reasoning/04_equality.tex index e69de29..638f503 100644 --- a/semester4/fmfp/parts/01_formal-reasoning/04_equality.tex +++ b/semester4/fmfp/parts/01_formal-reasoning/04_equality.tex @@ -0,0 +1,43 @@ +\subsection{Equality} +Since equality is such an important concept, it isn't just a predicate, but a separate First-Order Logic (FOL), +called \bi{FOL with equality}. + +The language is extended by $t_1 = t_2 \in \textit{Form}$ if $t_1, t_2 \in \textit{Term}$, +the semantic entailment $\models$ is also extended by ``$\cI \models t_1 = t_2$ if $\cI(t_1) = \cI(t_2)$''. +This definition is the exact intuition of equality of two terms, in that they are equal if their value under the interpretation is equal. + +Equality is an equivalence relation, so the following rules apply (ref = reflexivity, sym = symmetry, trans = transitivity): +\begin{align*} + \begin{prooftree} + \infer0[ref]{\Gamma \vdash t = t} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\Gamma \vdash t = s} + \infer1[sym]{\Gamma \vdash s = t} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\Gamma \vdash t = s} + \hypo{\Gamma \vdash s = r} + \infer2[sym]{\Gamma \vdash s = r} + \end{prooftree} +\end{align*} + +Equality is also a congruence on terms and (definable) relations +\begin{align*} + \begin{prooftree} + \hypo{\Gamma \vdash t_1 = s_1} + \hypo{\dots} + \hypo{\Gamma \vdash t_n = s_n} + \infer3[$cong_1$]{\Gamma f(t_1, \ldots, t_n) = f(s_1, \ldots, s_n)} + \end{prooftree} + \\ + \begin{prooftree} + \hypo{\Gamma \vdash t_1 = s_1} + \hypo{\dots} + \hypo{\Gamma \vdash t_n = s_n} + \hypo{\Gamma \vdash p(t_1, \ldots, t_n)} + \infer4[$cong_2$]{\Gamma p(s_1, \ldots, s_n)} + \end{prooftree} +\end{align*}