ProofPcg.mulg: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/random/proof_pcg.bend as ProofPcg
Lines 66-72 of bend-collections-laws-math@1.0.0.0/proofs/math/random/proof_pcg.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/proof_pcg.bend&start_line=66&end_line=72

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

# the value of a product mod 2^64 from the values of its factors; the
# factors stay variables here, so no product is ever normalized
def mulg(+x: WU.U64, +o: WU.U64, +xv: Nat, +hx: {SW.value(x) == xv : Nat}, +ov: Nat, +ho: {SW.value(o) == ov : Nat}) -> {SW.value(X.mul(x, o)) == C.low(64n, Nat.mul(xv, ov)) : Nat}:
  Equal.trans(Nat, SW.value(X.mul(x, o)), C.low(64n, Nat.mul(SW.value(x), SW.value(o))), C.low(64n, Nat.mul(xv, ov)), WA.mul_value(x, o),
    Equal.trans(Nat, C.low(64n, Nat.mul(SW.value(x), SW.value(o))), C.low(64n, Nat.mul(xv, SW.value(o))), C.low(64n, Nat.mul(xv, ov)),
      Equal.cong(Nat, Nat, z => C.low(64n, Nat.mul(z, SW.value(o))), SW.value(x), xv, hx),
      Equal.cong(Nat, Nat, z => C.low(64n, Nat.mul(xv, z)), SW.value(o), ov, ho)))

Uses (9):
  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.mul
  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=mul