def nt.lt_trans

bend-osdev-u32@0.1.0.0 · PROOF_order.bend:410

import Base

410def nt.lt_trans(x: Nat, y: Nat, z: Nat, h1: {Nat.is_lt(x, y) == True{} : Bool}, h2: {Nat.is_lt(y, z) == True{} : Bool}) -> {Nat.is_lt(x, z) == True{} : Bool}:
411  match x y z:
412    case 0n 0n _:
413      Empty.absurd({Nat.is_lt(0n, z) == True{} : Bool}, ab.ft(h1))
414    case 0n 1n+yp 0n:
415      Empty.absurd({Nat.is_lt(0n, 0n) == True{} : Bool}, ab.ft(h2))
416    case 0n 1n+yp 1n+zp:
417      {==}
418    case 1n+xp 0n _:
419      Empty.absurd({Nat.is_lt(1n+xp, z) == True{} : Bool}, ab.ft(h1))
420    case 1n+xp 1n+yp 0n:
421      Empty.absurd({Nat.is_lt(1n+xp, 0n) == True{} : Bool}, ab.ft(h2))
422    case 1n+xp 1n+yp 1n+zp:
423      nt.lt_trans(xp, yp, zp, h1, h2)