Hash.bucket_lt: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/hash/hash.bend as Hash
Lines 58-60 of bend-collections-laws-math@1.0.0.0/proofs/math/hash/hash.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/hash/hash.bend&start_line=58&end_line=60

import Base
import bend-collections-laws-math@1.0.0.0/src/math/hash.bend as HS
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/lib/word.bend as WD
import bend-collections-laws-math@1.0.0.0/proofs/lib/u32div.bend as UD

# THEOREM: with mask = 2^k - 1 (k low bits) the bucket indexes a table of 2^k.
def bucket_lt(+w: U32, +k: Nat, +mask: U32, +hm: {mask == U32{WD.mask(32n, k)} : U32}) -> {Nat.is_lt(UD.v(HS.bucket(w, mask)), WD.sc(k, 1n)) == True{} : Bool}:
  L.subst(U32, z => {Nat.is_lt(UD.v(HS.bucket(w, z)), WD.sc(k, 1n)) == True{} : Bool}, U32{WD.mask(32n, k)}, mask, Equal.sym(U32, mask, U32{WD.mask(32n, k)}, hm), bucket_ltw(w, k))

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=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.is_lt
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend&name=subst
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/u32div.bend&name=v
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/word.bend&name=mask
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/word.bend&name=sc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/hash/hash.bend&name=bucket_ltw
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/hash.bend&name=bucket