def p2

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/f64round.bend:591

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/proofs/math/typed/width.bend as WW

591# the rounded significand overflows 53 bits exactly when m + 2^9 overflows 63
592def p2(+one: Nat, +h1: {one == 1n : Nat}, +m: Nat, +hm: {C.fits(63n, m) == True{} : Bool}) -> {C.fits(53n, SF.rne(m, 10n)) == C.fits(63n, Nat.add(m, 512n)) : Bool}:
593  +e1 = Equal.cong(Nat, Bool, z => C.fits(53n, z), SF.rne(m, 10n), SF.pick(Nat, Nat.is_eq(C.low(10n, m), 512n), Nat.sub(Nat.add(C.high(10n, m), C.high(10n, Nat.add(C.low(10n, m), 512n))), Nat.mod(Nat.add(C.high(10n, m), C.high(10n, Nat.add(C.low(10n, m), 512n))), 2n)), Nat.add(C.high(10n, m), C.high(10n, Nat.add(C.low(10n, m), 512n)))), Equal.sym(Nat, SF.pick(Nat, Nat.is_eq(C.low(10n, m), 512n), Nat.sub(Nat.add(C.high(10n, m), C.high(10n, Nat.add(C.low(10n, m), 512n))), Nat.mod(Nat.add(C.high(10n, m), C.high(10n, Nat.add(C.low(10n, m), 512n))), 2n)), Nat.add(C.high(10n, m), C.high(10n, Nat.add(C.low(10n, m), 512n)))), SF.rne(m, 10n), rp_c(C.high(10n, m), C.low(10n, m), WW.low_fits(10n, m), Nat.cmp(C.low(10n, m), 512n), {==})))
594  +e2 = fpk(one, h1, Nat.add(C.high(10n, m), C.high(10n, Nat.add(C.low(10n, m), 512n))), R_le(one, h1, m, hm), Nat.is_eq(C.low(10n, m), 512n))
595  +e3 = Equal.cong(Nat, Bool, z => C.fits(53n, z), Nat.add(C.high(10n, m), C.high(10n, Nat.add(C.low(10n, m), 512n))), C.high(10n, Nat.add(m, 512n)), Equal.sym(Nat, C.high(10n, Nat.add(m, 512n)), Nat.add(C.high(10n, m), C.high(10n, Nat.add(C.low(10n, m), 512n))), hsp(m)))
596  +e4 = Equal.sym(Bool, C.fits(63n, Nat.add(m, 512n)), C.fits(53n, C.high(10n, Nat.add(m, 512n))), fits_hc(10n, 53n, Nat.add(m, 512n)))
597  Equal.trans(Bool, C.fits(53n, SF.rne(m, 10n)), C.fits(53n, SF.pick(Nat, Nat.is_eq(C.low(10n, m), 512n), Nat.sub(Nat.add(C.high(10n, m), C.high(10n, Nat.add(C.low(10n, m), 512n))), Nat.mod(Nat.add(C.high(10n, m), C.high(10n, Nat.add(C.low(10n, m), 512n))), 2n)), Nat.add(C.high(10n, m), C.high(10n, Nat.add(C.low(10n, m), 512n))))), C.fits(63n, Nat.add(m, 512n)), e1, Equal.trans(Bool, C.fits(53n, SF.pick(Nat, Nat.is_eq(C.low(10n, m), 512n), Nat.sub(Nat.add(C.high(10n, m), C.high(10n, Nat.add(C.low(10n, m), 512n))), Nat.mod(Nat.add(C.high(10n, m), C.high(10n, Nat.add(C.low(10n, m), 512n))), 2n)), Nat.add(C.high(10n, m), C.high(10n, Nat.add(C.low(10n, m), 512n))))), C.fits(53n, Nat.add(C.high(10n, m), C.high(10n, Nat.add(C.low(10n, m), 512n)))), C.fits(63n, Nat.add(m, 512n)), e2, Equal.trans(Bool, C.fits(53n, Nat.add(C.high(10n, m), C.high(10n, Nat.add(C.low(10n, m), 512n)))), C.fits(53n, C.high(10n, Nat.add(m, 512n))), C.fits(63n, Nat.add(m, 512n)), e3, e4)))