Impl.fT: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/random/pcg/impl.bend as Impl
Lines 33-39 of bend-collections-laws-math@1.0.0.0/proofs/math/random/pcg/impl.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/pcg/impl.bend&start_line=33&end_line=39

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/w64.bend as SW
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/w64.bend as X
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64add.bend as WA
import bend-collections-laws-math@1.0.0.0/proofs/math/random/pcg/step.bend as ST

# the cross products mod 2^64
def fT(+hi: WU.U64, +lo: WU.U64, +ml: WU.U64, +mh: WU.U64) -> {C.low(64n, SW.value(X.add(X.mul(hi, ml), X.mul(lo, mh)))) == C.low(64n, Nat.add(Nat.mul(SW.value(hi), SW.value(ml)), Nat.mul(SW.value(lo), SW.value(mh)))) : Nat}:
  +m1 = X.mul(hi, ml)
  +m2 = X.mul(lo, mh)
  Equal.trans(Nat, C.low(64n, SW.value(X.add(m1, m2))), C.low(64n, Nat.add(SW.value(m1), SW.value(m2))), C.low(64n, Nat.add(Nat.mul(SW.value(hi), SW.value(ml)), Nat.mul(SW.value(lo), SW.value(mh)))),
    eqm_of(SW.value(X.add(m1, m2)), Nat.add(SW.value(m1), SW.value(m2)), WA.add_value(m1, m2)),
    ST.low_add(64n, SW.value(m1), Nat.mul(SW.value(hi), SW.value(ml)), SW.value(m2), Nat.mul(SW.value(lo), SW.value(mh)), eqm_of(SW.value(m1), Nat.mul(SW.value(hi), SW.value(ml)), WA.mul_value(hi, ml)), eqm_of(SW.value(m2), Nat.mul(SW.value(lo), SW.value(mh)), WA.mul_value(lo, mh))))

Uses (13):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.trans
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.add
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/pcg/impl.bend&name=eqm_of
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/pcg/step.bend&name=low_add
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64add.bend&name=add_value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64add.bend&name=mul_value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/lib/common.bend&name=low
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/w64.bend&name=value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/u64.bend&name=U64
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/w64.bend&name=add
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/w64.bend&name=mul