2.4.13: Derivations with Identity predicate
Derivations with identity predicate require additional initial sequents and inference rules.
If \(t\) is a closed term, then \({} \Sequent \eq[t][t]\) is an initial sequent.
The rules for \(\eq[][]\) are ( \(t_1\) and \(t_2\) are closed terms):
If \(s\) and \(t\) are closed terms, then \(\eq[s][t], A(s) \Proves A(t)\) :
This may be familiar as the principle of substitutability of identicals, or Leibniz’ Law.
\(\Log{LK}\) proves that \(\eq[][]\) is symmetric and transitive:
In the proof on the left, the formula \(\eq[x][t_1]\) is our \(A(x)\) . On the right, we take \(A(x)\) to be \(\eq[t_1][x]\) .
Give derivations of the following sequents:
- \(\Sequent \lforall{x}{\lforall{y}{((x = y \land A(x)) \lif A(y))}}\)
- \(\lexists{x}{A(x)} \land \lforall{y}{\lforall{z}{((A(y) \land A(z)) \lif y = z)}} \Sequent \lexists{x}{(A(x) \land \lforall{y}{(A(y) \lif y = x)})}\)