2.4.14: Soundness with Identity predicate
\(\Log{LK}\) with initial sequents and rules for identity is sound.
Proof. Initial sequents of the form \({} \Sequent \eq[t][t]\) are valid, since for every structure \(\Struct M\) , \(\Sat{M}{\eq[t][t]}\) . (Note that we assume the term \(t\) to be closed, i.e., it contains no variables, so variable assignments are irrelevant).
Suppose the last inference in a derivation is \(=\) . Then the premise is \(\eq[t_1][t_2], \Gamma \Sequent \Delta, A(t_1)\) and the conclusion is \(\eq[t_1][t_2], \Gamma \Sequent \Delta, A(t_2)\) . Consider a structure \(\Struct M\) . We need to show that the conclusion is valid, i.e., if \(\Sat{M}{\eq[t_1][t_2]}\) and \(\Sat{M}{\Gamma}\) , then either \(\Sat{M}{C}\) for some \(C \in \Delta\) or \(\Sat{M}{A(t_2)}\) .
By induction hypothesis, the premise is valid. This means that if \(\Sat{M}{\eq[t_1][t_2]}\) and \(\Sat{M}{\Gamma}\) either (a) for some \(C \in \Delta\) , \(\Sat{M}{C}\) or (b) \(\Sat{M}{A(t_1)}\) . In case (a) we are done. Consider case (b). Let \(s\) be a variable assignment with \(s(x) = \Value{t_1}{M}\) . By Proposition 5.12.3 , \(\Sat[,s]{M}{A(t_1)}\) . Since \(\varAssign{s}{s}{x}\) , by Proposition 5.13.3 , \(\Sat[,s]{M}{A(x)}\) . Since \(\Sat{M}{\eq[t_1][t_2]}\) , we have \(\Value{t_1}{M} = \Value{t_2}{M}\) , and hence \(s(x) = \Value{t_2}{M}\) . By applying Proposition 5.13.3 again, we also have \(\Sat[,s]{M}{A(t_2)}\) . By Proposition 5.12.3 , \(\Sat{M}{A(t_2)}\) . ◻