Bitsrel.sb_hi: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/bitsrel.bend as Bitsrel
Lines 79-89 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/bitsrel.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/bitsrel.bend&start_line=79&end_line=89

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/lib/nat.bend as N
import bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/width.bend as WW

# above 16: the bits of the rest
def sb_hi(i: Nat, +x: Nat, +r: Nat, +f16: {C.fits(16n, x) == True{} : Bool}) -> {sb(i, 16n, Nat.add(x, C.shift(16n, r))) == sb(i, 0n, r) : List<&2, Nat>}:
  match i:
    case 0n:
      {==}
    case 1n+ +j:
      %Equal.sym(List<&2, Nat>, sb(j, 16n, Nat.add(x, C.shift(16n, r))), sb(j, 0n, r), sb_hi(j, x, r, f16)) : {C.bit(C.high(Nat.add(16n, j), Nat.add(x, C.shift(16n, r)))) <> _ == C.bit(C.high(j, r)) <> sb(j, 0n, r) : List<&2, Nat>}
      %Equal.sym(Nat, C.high(Nat.add(16n, j), Nat.add(x, C.shift(16n, r))), C.high(j, C.high(16n, Nat.add(x, C.shift(16n, r)))), WW.high_comp(j, 16n, Nat.add(x, C.shift(16n, r)))) : {C.bit(_) <> sb(j, 0n, r) == C.bit(C.high(j, r)) <> sb(j, 0n, r) : List<&2, Nat>}
      %Equal.sym(Nat, C.high(16n, Nat.add(x, C.shift(16n, r))), Nat.add(C.high(16n, x), r), WW.high_add_shift(16n, x, r)) : {C.bit(C.high(j, _)) <> sb(j, 0n, r) == C.bit(C.high(j, r)) <> sb(j, 0n, r) : List<&2, Nat>}
      %Equal.sym(Nat, C.high(16n, x), 0n, N.eq_from_is_eq(C.high(16n, x), 0n, f16)) : {C.bit(C.high(j, Nat.add(_, r))) <> sb(j, 0n, r) == C.bit(C.high(j, r)) <> sb(j, 0n, r) : List<&2, Nat>}
      {==}

Uses (13):
  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=List
  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=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/bitsrel.bend&name=sb
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=eq_from_is_eq
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/width.bend&name=high_add_shift
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/width.bend&name=high_comp
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend&name=bit
  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=high
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend&name=shift