def fz_g

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/f64bits.bend:220

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/spec/math/f64.bend as SF
import bend-collections-laws-math@1.0.0.0/src/math/f64.bend as F
import bend-collections-laws-math@1.0.0.0/src/math/w64.bend as X
import bend-collections-laws-math@1.0.0.0/src/math/u64.bend as WU
import bend-collections-laws-math@1.0.0.0/proofs/lib/word.bend as WD
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/u32laws.bend as LW

220# the fraction is zero
221def fz_g(+l: U32, +h: U32, +m20: U32, +hm20: {m20 == U32{WD.mask(32n, 20n)} : U32}) -> {X.is_zero(WU.U64{l, U32.and(h, m20)}) == Nat.is_eq(SF.frac(F.Bits{l, h}), 0n) : Bool}:
222  +f = C.low(20n, v(h))
223  +e1 = Equal.cong(Bool, Bool, z => Bool.and(z, U32.is_zero(U32.and(h, m20))), U32.is_zero(l), Nat.is_eq(v(l), 0n), LW.zero_nat(l))
224  +e2 = Equal.trans(Bool, U32.is_zero(U32.and(h, m20)), Nat.is_eq(v(U32.and(h, m20)), 0n), Nat.is_eq(f, 0n), LW.zero_nat(U32.and(h, m20)), Equal.cong(Nat, Bool, z => Nat.is_eq(z, 0n), v(U32.and(h, m20)), f, andm(h, 20n, m20, hm20)))
225  +e3 = Equal.cong(Bool, Bool, z => Bool.and(Nat.is_eq(v(l), 0n), z), U32.is_zero(U32.and(h, m20)), Nat.is_eq(f, 0n), e2)
226  Equal.trans(Bool, Bool.and(U32.is_zero(l), U32.is_zero(U32.and(h, m20))), Bool.and(Nat.is_eq(v(l), 0n), U32.is_zero(U32.and(h, m20))), Nat.is_eq(Nat.add(v(l), C.shift(32n, f)), 0n), e1, Equal.trans(Bool, Bool.and(Nat.is_eq(v(l), 0n), U32.is_zero(U32.and(h, m20))), Bool.and(Nat.is_eq(v(l), 0n), Nat.is_eq(f, 0n)), Nat.is_eq(Nat.add(v(l), C.shift(32n, f)), 0n), e3, Equal.sym(Bool, Nat.is_eq(Nat.add(v(l), C.shift(32n, f)), 0n), Bool.and(Nat.is_eq(v(l), 0n), Nat.is_eq(f, 0n)), z2(v(l), 32n, f))))