WordMultiplication.u32_refines: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/word_multiplication.bend as WordMultiplication
Lines 59-64 of bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/word_multiplication.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/word_multiplication.bend&start_line=59&end_line=64

import Base
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/lib/lemmas/proofs/numeric.bend as Numeric

def u32_refines(a: U32, b: U32) -> {U32.mul(a, b) == U32{S.from_nat(32n, Nat.mul(U32.to_nat(a), U32.to_nat(b)))} : U32}:
  match a b:
    case U32{+x} U32{+y}:
      %Equal.sym(Nat, Word.to_nat(32n, x), S.unsigned(32n, x), Numeric.primitive_word_interpretation(32n, x)) : {U32.mul(U32{x}, U32{y}) == U32{S.from_nat(32n, Nat.mul(_, Word.to_nat(32n, y)))} : U32}
      %Equal.sym(Nat, Word.to_nat(32n, y), S.unsigned(32n, y), Numeric.primitive_word_interpretation(32n, y)) : {U32.mul(U32{x}, U32{y}) == U32{S.from_nat(32n, Nat.mul(S.unsigned(32n, x), _))} : U32}
      Equal.cong(Word(32n), U32, w => U32{w}, Word.mul(32n, x, y), S.from_nat(32n, Nat.mul(S.unsigned(32n, x), S.unsigned(32n, y))), refines(32n, x, y))

Uses (14):
  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.sym
  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=Base/base.bend&name=U32
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32.mul
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32.to_nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word.mul
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word.to_nat
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/numeric.bend&name=primitive_word_interpretation
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/word_multiplication.bend&name=refines
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/spec/numeric.bend&name=from_nat
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/spec/numeric.bend&name=unsigned