Strictly Monotone Brouwer Trees for Well Founded Recursion over Multiple Arguments

Abstract

Ordinals can be used o prove he erminaion o depen-denly yped programs. Brouwer rees are a paricular ordinal noaion ha make i very easy o assign sizes o higher order daa srucures. Tey exend unary naural numbers wih a limi consrucor, so a uncion's size can be he leas upper bound o he sizes o values rom is image. Tese can hen be used o dene well-ounded recursion: any recursive calls are allowed so long as hey are on values whose sizes are sricly smaller han he curren size.Unorunaely, Brouwer rees are no algebraically wellbehaved. Tey can be characerized equaionally as a joinsemilatice, where he join akes he maximum o wo rees. However, his join does no inerac well wih he successor consrucor, so i does no inerac properly wih he sric ordering used in well-ounded recursion.We presen Sricly Monoone Brouwer rees (SMB-rees), a renemen o Brouwer rees ha are algebraically wellbehaved. SMB-rees are buil using uncions wih he same signaures as Brouwer ree consrucors, and hey saisy all Brouwer ree inequaliies. However, heir join operaor disribues over he successor, making hem suied or well-ounded recursion or equaional reasoning.Tis paper eaches how, using dependen pairs and care-ul deniions, an ill-behaved deniion can be urned ino a well-behaved one. Our approach is axiomaically lighweigh: i does no rely on Axiom K, univalence, quoien ypes, or Higher Inducive ypes. We implemen a recursively-dened maximum operaor or Brouwer rees ha maches on successors and handles hem specically. Ten, we dene SMB-rees as he subse o Brouwer rees or which he recursive maximum compues a leas upper bound. Finally, we show ha every Brouwer ree can be ransormed ino a corresponding SMB-ree by inniely joining i wih isel. All deniions and heorems are implemened in Agda. CCS Concepts• Teory o computation → ype theory; • Soware and its engineering → Soware verication;