def rem_v

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/f64sqr.bend:312

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/spec/math/w64.bend as SW
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/src/math/natural.bend as M
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N

312def rem_v(+one: Nat, +h1: {one == 1n : Nat}, +nh: WU.U64, +h62: {C.fits(62n, SW.value(nh)) == True{} : Bool}, +h60: {C.fits(60n, SW.value(nh)) == False{} : Bool}, +e: Nat, +x: Nat, +hx: {Nat.add(x, 2180n) == e : Nat}, +q: U32, +u: U32, +hc: {Nat.add(Nat.mul(v(q), Nat.add(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh)))), v(u)) == C.shift(32n, Nat.sub(SW.value(nh), Nat.mul(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh))))) : Nat}, +hU: {Nat.is_le(v(u), Nat.add(M.isqrt(SW.value(nh)), M.isqrt(SW.value(nh)))) == True{} : Bool}, +hQ: {Nat.is_lt(v(q), C.shift(32n, one)) == True{} : Bool}, +g: Bool, +hg: {X.le(X.mul32(q, q), WU.U64{0, u}) == g : Bool}) -> {F.sq_rem(e, WU.U64{q, X.lo(X.isqrt(nh))}, WU.U64{0, u}, X.mul32(q, q), g) == SF.round(False{}, SW.jam(M.isqrt(C.shift(64n, SW.value(nh))), SF.b2n(Bool.not(Nat.is_eq(Nat.mul(M.isqrt(C.shift(64n, SW.value(nh))), M.isqrt(C.shift(64n, SW.value(nh)))), C.shift(64n, SW.value(nh)))))), x) : F.F64}:
313  match g:
314    case True{}:
315      brA_w(one, h1, nh, h62, h60, e, x, hx, q, u, hc, hU, hQ, hg)
316    case False{}:
317      +h0 = Equal.trans(Bool, Nat.is_le(Nat.mul(v(q), v(q)), C.shift(32n, v(u))), X.le(X.mul32(q, q), WU.U64{0, u}), False{}, Equal.sym(Bool, X.le(X.mul32(q, q), WU.U64{0, u}), Nat.is_le(Nat.mul(v(q), v(q)), C.shift(32n, v(u))), le_ba(q, u)), hg)
318      d1_v(one, h1, nh, h62, h60, e, x, hx, q, u, hc, hU, hQ, N.not_le_lt(Nat.mul(v(q), v(q)), C.shift(32n, v(u)), h0), X.le(X.sub(X.mul32(q, q), WU.U64{0, u}), X.sub(X.add(WU.U64{q, X.lo(X.isqrt(nh))}, WU.U64{q, X.lo(X.isqrt(nh))}), WU.U64{1, 0})), {==})