Ordinals can be used o prove he erminaion o depen-denly yped programs. Brouwer rees are a paricular ordinal noaion ha make i very easy o assign sizes o higher order daa srucures. Tey exend unary naural numbers wih a limi consrucor, so a uncion's size can be he leas upper bound o he sizes o values rom is image. Tese can hen be used o dene well-ounded recursion: any recursive calls are allowed so long as hey are on values whose sizes are sricly smaller han he curren size.Unorunaely, Brouwer rees are no algebraically wellbehaved. Tey can be characerized equaionally as a joinsemilatice, where he join akes he maximum o wo rees. However, his join does no inerac well wih he successor consrucor, so i does no inerac properly wih he sric ordering used in well-ounded recursion.We presen Sricly Monoone Brouwer rees (SMB-rees), a renemen o Brouwer rees ha are algebraically wellbehaved. SMB-rees are buil using uncions wih he same signaures as Brouwer ree consrucors, and hey saisy all Brouwer ree inequaliies. However, heir join operaor disribues over he successor, making hem suied or well-ounded recursion or equaional reasoning.Tis paper eaches how, using dependen pairs and care-ul deniions, an ill-behaved deniion can be urned ino a well-behaved one. Our approach is axiomaically lighweigh: i does no rely on Axiom K, univalence, quoien ypes, or Higher Inducive ypes. We implemen a recursively-dened maximum operaor or Brouwer rees ha maches on successors and handles hem specically. Ten, we dene SMB-rees as he subse o Brouwer rees or which he recursive maximum compues a leas upper bound. Finally, we show ha every Brouwer ree can be ransormed ino a corresponding SMB-ree by inniely joining i wih isel. All deniions and heorems are implemened in Agda.
CCS Concepts• Teory o computation → ype theory; • Soware and its engineering → Soware verication;
