def mul32

bend-collections-laws-crypto@1.0.0.0 · proofs/math/u64/u64div.bend:42

import Base
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend as L
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/word.bend as WD
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/u32div.bend as UD

42def mul32(+one: Nat, +h1: {one == 1n : Nat}, +x: U32, +y: U32, +h: {Nat.is_lt(Nat.mul(UD.v(x), UD.v(y)), WD.sc(32n, one)) == True{} : Bool}) -> {UD.v(U32.mul(x, y)) == Nat.mul(UD.v(x), UD.v(y)) : Nat}:
43  match x y:
44    case U32{+a} U32{+b}:
45      +cv = Equal.trans(Nat, Nat.mul(UD.v(U32{a}), UD.v(U32{b})), Nat.mul(WD.uw(32n, a), UD.v(U32{b})), Nat.mul(WD.uw(32n, a), WD.uw(32n, b)), Equal.cong(Nat, Nat, z => Nat.mul(z, UD.v(U32{b})), UD.v(U32{a}), WD.uw(32n, a), UD.vw(a)), Equal.cong(Nat, Nat, z => Nat.mul(WD.uw(32n, a), z), UD.v(U32{b}), WD.uw(32n, b), UD.vw(b)))
46      +hw = L.subst(Nat, z => {Nat.is_lt(z, WD.sc(32n, one)) == True{} : Bool}, Nat.mul(UD.v(U32{a}), UD.v(U32{b})), Nat.mul(WD.uw(32n, a), WD.uw(32n, b)), cv, h)
47      Equal.trans(Nat, UD.v(U32{Word.mul(32n, a, b)}), WD.uw(32n, Word.mul(32n, a, b)), Nat.mul(UD.v(U32{a}), UD.v(U32{b})), UD.vw(Word.mul(32n, a, b)), Equal.trans(Nat, WD.uw(32n, Word.mul(32n, a, b)), Nat.mul(WD.uw(32n, a), WD.uw(32n, b)), Nat.mul(UD.v(U32{a}), UD.v(U32{b})), WD.mul_exact(32n, one, h1, a, b, hw), Equal.sym(Nat, Nat.mul(UD.v(U32{a}), UD.v(U32{b})), Nat.mul(WD.uw(32n, a), WD.uw(32n, b)), cv)))