def le_step

bend-collections-laws-math@1.0.0.0 · proofs/math/natural/bits.bend:89

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-math@1.0.0.0/spec/lib/common.bend as SC

89# the lower bound, as 2^L <= max(1, 2 n)
90def le_step(+np: Nat, +h: Nat, +x: Nat, +e: {1n+np == Nat.add(Nat.double(h), Nat.mod(1n+np, 2n)) : Nat}, +ih: {Nat.is_le(SC.pow2(x), Nat.max(1n, Nat.double(h))) == True{} : Bool}) -> {Nat.is_le(SC.pow2(x), 1n+np) == True{} : Bool}:
91  match h:
92    case 0n:
93      N.le_trans(SC.pow2(x), 1n, 1n+np, ih, N.lt_succ_le_succ(0n, 1n+np, {==}))
94    case 1n+ +hp:
95      N.le_trans(SC.pow2(x), Nat.double(1n+hp), 1n+np, ih, N.le_trans(Nat.double(1n+hp), Nat.add(Nat.double(1n+hp), Nat.mod(1n+np, 2n)), 1n+np, N.le_add_right(Nat.double(1n+hp), Nat.mod(1n+np, 2n)), N.eq_le(Nat.add(Nat.double(1n+hp), Nat.mod(1n+np, 2n)), 1n+np, Equal.sym(Nat, 1n+np, Nat.add(Nat.double(1n+hp), Nat.mod(1n+np, 2n)), e))))