def lbits

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/f64bl.bend:22

import Base
import bend-collections-laws-math@1.0.0.0/spec/lib/common.bend as C
import bend-collections-laws-math@1.0.0.0/src/math/f64.bend as F
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/width.bend as WW
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64add.bend as WA

22def lbits(+k: Nat, +n: Nat) -> {F.low_bits(k, n) == C.low(k, n) : Nat}:
23  match k:
24    case 0n:
25      {==}
26    case 1n+ +p:
27      +e1 = Equal.cong(Nat, Nat, z => Nat.add(z, Nat.double(F.low_bits(p, Nat.div(n, 2n)))), Nat.mod(n, 2n), C.bit(n), Equal.sym(Nat, C.bit(n), Nat.mod(n, 2n), WW.bit_mod(n)))
28      +e2 = Equal.cong(Nat, Nat, z => Nat.add(C.bit(n), Nat.double(F.low_bits(p, z))), Nat.div(n, 2n), C.half(n), Equal.sym(Nat, C.half(n), Nat.div(n, 2n), WA.half_div(n)))
29      +e3 = Equal.cong(Nat, Nat, z => Nat.add(C.bit(n), Nat.double(z)), F.low_bits(p, C.half(n)), C.low(p, C.half(n)), lbits(p, C.half(n)))
30      Equal.trans(Nat, Nat.add(Nat.mod(n, 2n), Nat.double(F.low_bits(p, Nat.div(n, 2n)))), Nat.add(C.bit(n), Nat.double(F.low_bits(p, Nat.div(n, 2n)))), Nat.add(C.bit(n), Nat.double(C.low(p, C.half(n)))), e1, Equal.trans(Nat, Nat.add(C.bit(n), Nat.double(F.low_bits(p, Nat.div(n, 2n)))), Nat.add(C.bit(n), Nat.double(F.low_bits(p, C.half(n)))), Nat.add(C.bit(n), Nat.double(C.low(p, C.half(n)))), e2, e3))