Equality and extensionality in automated higher order theorem proving

This thesis focuses on equality and extensionality in automated higher-order theorem proving based on Church's simply typed -calculus (classical type theory). First, a landscape of various semantical notions is presented that is motivated by the di erent roles equality adopts in them. Each of the semantical notions in this landscape | including Henkin semantics | is then linked with an abstract consistency principle that can be employed for analysing the connection between syntax and semantics of higher-order calculi. Apart from this proof theoretic tools, the main contributions of this thesis are the three new calculi ER (extensional higher-order resolution), EP (extensional higher-order paramodulation) and ERUE (extensional higher-order RUE-resolution) which improve the mechanisation of de ned and primitive equality in classical type theory. In contrast to the refutation approaches for classical type theory developed so far, these calculi reach Henkin completeness without requiring additional extensionality axioms. The key idea is to allow for recursive calls from higher-order uni cation to the overall refutation search. Calculus ER, which in contrast to EP and ERUE , considers equality only as a de ned notion, has been implemented in the theorem prover Leo and the suitability of this approach for proving simple theorems about sets has been demonstrated in a case study. Quo facto, quando orientur controversiae, non magis disputatione opus erit inter duos philosophos, quam inter duos Computistas. Su ciet enim calamos in manus sumere sedere ad abacos, et sibi mutuo (accito si placet amico) dicere: calculemus. Gottfried Wilhelm Leibniz (C.I. Gerhardt (ed.), Die philosophischen Schriften von Gottfried Wilhelm Leibniz, vol. 7, Berlin 1890, p. 200.) Chapter