Arith.fits_mul: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/poly1305/arith.bend as Arith
Lines 106-108 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/poly1305/arith.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/poly1305/arith.bend&start_line=106&end_line=108

import Base
import bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend as C
import bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/width.bend as WW

# x fits a bits, y fits b bits: x y fits a + b bits
def fits_mul(+a: Nat, +b: Nat, +x: Nat, +y: Nat, +hx: {C.fits(a, x) == True{} : Bool}, +hy: {C.fits(b, y) == True{} : Bool}) -> {C.fits(Nat.add(a, b), Nat.mul(x, y)) == True{} : Bool}:
  WW.fits_of_lt(Nat.add(a, b), Nat.mul(x, y), lt_subst(Nat.mul(x, y), Nat.mul(C.pow2(a), C.pow2(b)), C.pow2(Nat.add(a, b)), pow2_add(a, b), lt_mul(x, y, C.pow2(a), C.pow2(b), WW.lt_of_fits(a, x, hx), WW.lt_of_fits(b, y, hy))))

Uses (11):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  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=Base/base.bend&name=Nat.mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/poly1305/arith.bend&name=lt_mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/poly1305/arith.bend&name=lt_subst
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/poly1305/arith.bend&name=pow2_add
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/width.bend&name=fits_of_lt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/width.bend&name=lt_of_fits
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend&name=fits
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend&name=pow2