def mulv

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/f64round.bend:340

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/proofs/lib/logic.bend as L
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/f64bits.bend as FB
import bend-collections-laws-math@1.0.0.0/proofs/lib/u32.bend as U
import bend-collections-laws-math@1.0.0.0/proofs/lib/word.bend as WD
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/spec/numeric.bend as S

340# a product below 2^32 is exact
341def mulv(+one: Nat, +h1: {one == 1n : Nat}, +a: U32, +c: U32, +h: {Nat.is_lt(Nat.mul(v(a), v(c)), C.shift(32n, one)) == True{} : Bool}) -> {v(U32.mul(a, c)) == Nat.mul(v(a), v(c)) : Nat}:
342  match a c:
343    case U32{+x} U32{+y}:
344      +ex = U.to_nat_word(x)
345      +ey = U.to_nat_word(y)
346      +es = Equal.trans(Nat, Nat.mul(v(U32{x}), v(U32{y})), Nat.mul(S.unsigned(32n, x), v(U32{y})), Nat.mul(S.unsigned(32n, x), S.unsigned(32n, y)), Equal.cong(Nat, Nat, z => Nat.mul(z, v(U32{y})), v(U32{x}), S.unsigned(32n, x), ex), Equal.cong(Nat, Nat, z => Nat.mul(S.unsigned(32n, x), z), v(U32{y}), S.unsigned(32n, y), ey))
347      +h2 = L.subst(Nat, z => {Nat.is_lt(z, C.shift(32n, one)) == True{} : Bool}, Nat.mul(v(U32{x}), v(U32{y})), Nat.mul(S.unsigned(32n, x), S.unsigned(32n, y)), es, h)
348      +h3 = L.subst(Nat, z => {Nat.is_lt(Nat.mul(S.unsigned(32n, x), S.unsigned(32n, y)), z) == True{} : Bool}, C.shift(32n, one), S.scale_binary(32n, one), Equal.sym(Nat, S.scale_binary(32n, one), C.shift(32n, one), FB.sb(32n, one)), h2)
349      +m = Word.mul(32n, x, y)
350      Equal.trans(Nat, U32.to_nat(U32{m}), S.unsigned(32n, m), Nat.mul(v(U32{x}), v(U32{y})), U.to_nat_word(m), Equal.trans(Nat, S.unsigned(32n, m), Nat.mul(S.unsigned(32n, x), S.unsigned(32n, y)), Nat.mul(v(U32{x}), v(U32{y})), WD.mul_exact(32n, one, h1, x, y, h3), Equal.sym(Nat, Nat.mul(v(U32{x}), v(U32{y})), Nat.mul(S.unsigned(32n, x), S.unsigned(32n, y)), es)))