Impl.fV: 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 45-49 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=45&end_line=49

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/lib/lemmas/spec/numeric.bend as S
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64add.bend as WA

# adding the carry
def fV(+a: WU.U64, +cb: Bool) -> {SW.value(X.add(a, WU.U64{X.b32(cb), 0})) == C.low(64n, Nat.add(SW.value(a), S.bit_value(cb))) : Nat}:
  Equal.trans(Nat, SW.value(X.add(a, WU.U64{X.b32(cb), 0})), C.low(64n, Nat.add(SW.value(a), SW.value(WU.U64{X.b32(cb), 0}))), C.low(64n, Nat.add(SW.value(a), S.bit_value(cb))),
    WA.add_value(a, WU.U64{X.b32(cb), 0}),
    Equal.cong(Nat, Nat, z => C.low(64n, Nat.add(SW.value(a), z)), SW.value(WU.U64{X.b32(cb), 0}), S.bit_value(cb), carry_value(cb)))

Uses (13):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.cong
  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=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/spec/numeric.bend&name=bit_value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/pcg/impl.bend&name=carry_value
  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/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=b32