def hlf_upper

bend-collections-laws-math@1.0.0.0 · proofs/lib/u32half.bend:90

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend as L
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N

90def hlf_upper(+a: Nat, +b: Nat, +h: {Nat.is_lt(a, b) == True{} : Bool}) -> {Nat.is_lt(hlf(Nat.add(a, b)), b) == True{} : Bool}:
91  match a b:
92    case 0n 0n:
93      Empty.absurd({Nat.is_lt(hlf(Nat.add(0n, 0n)), 0n) == True{} : Bool}, L.false_true(h))
94    case 0n 1n+ +q:
95      N.le_lt_trans(hlf(1n+q), q, 1n+q, hlf_succ_le(q), N.lt_succ(q))
96    case 1n+p 0n:
97      Empty.absurd({Nat.is_lt(hlf(Nat.add(1n+p, 0n)), 0n) == True{} : Bool}, L.false_true(h))
98    case 1n+ +p 1n+ +q:
99      %Equal.sym(Nat, Nat.add(p, 1n+q), 1n+Nat.add(p, q), N.add_succ(p, q)) : {Nat.is_lt(hlf(1n+_), 1n+q) == True{} : Bool}
100      hlf_upper(p, q, h)