Abstract

Many proof tools use `tactic languages' as programs to direct their proofs. We present a simplified idealised tactic language, and describe its denotational semantics. The language has many applications outside theorem-proveo activ5QbG) The semantics is parametrised by a monad (plus additional structure). By instantiating this inv arious ways, the core semantics of a number of di#erent tactic languages is obtained. 1 Int roduct45 The notiB of a tactic as a program usedi the constructifi of a (machic[ assi46fi8 formal proof has become quie wie[S---#fifi[ Tacti# orifi---#z[ i the work of Gordon et al [GMW79] onEdi burgh LCF. The extent to whi h other`tacti4 based' systems istems[ t essentien[ the same style of programmifi faci---#---[I vari4 consi[I8#Bfi . InEdi burgh LCF, atacti does notit[8B construct a proof. Rather,i ti s usedi backwardreasoni[ to construct a vali#fiz[I functi[ whi h mayi46z8 prove thedesi6B property. Theoremhood i guarded by use of a `safe datatype', and only sound vali484[I functi[I may construct elements ofthi type. In other work, the type of theoremsi protected by havi8 the class oftacti--- icti protected, so thati i ia ossiSB tobui# unsound proofs. The account here tends towards the secondvion though the treatment oftacti6 i s actually so abstract that thi may not be an i[ edi---# t to i[ appli#[IS# i eipli sense. Whie. tacti[ arewiS---fi6[IS--- tacti programmi--- remai4 adiBfiBS task. Inthi paper, weconsi#[ abstractdescri[S#fi--- oftactifi[ wit the hope that modern algori------ desii techniSzS# such as thosedescri ed byBiS and de Moor [BdM97], can be brought to bear on thedi8S---fi[IS ontacti programmi#4 Earlia di#------S[ISS oftacti6 i n the abstract (wiract operati6z[ bii to any parti[ISS proof tool)i)[SS--- those by SchmiB [Sch84] and Mi4#...

Citations