U64int.sal: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/typed/u64int.bend as U64int
Lines 357-379 of bend-collections-laws-math@1.0.0.0/proofs/math/typed/u64int.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/u64int.bend&start_line=357&end_line=379

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/spec/math/generic.bend as SG
import bend-collections-laws-math@1.0.0.0/spec/math/natural.bend as NS
import bend-collections-laws-math@1.0.0.0/src/math/generic.bend as G
import bend-collections-laws-math@1.0.0.0/src/math/num.bend as NM
import bend-collections-laws-math@1.0.0.0/src/math/instances.bend as I
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/u64laws.bend as LW
import bend-collections-laws-math@1.0.0.0/src/math/u64.bend as WU

def sal(+xs: List<&2, WU.U64>, +r: Maybe<&2, WU.U64>, +n: Nat, +c: Bool, +hr: {r == G.fits(WU.U64, Bool.not(c), of(n)) : Maybe<&2, WU.U64>}, +hc: {C.fits(64n, n) == c : Bool}) -> {G.ok(WU.U64, G.sum_go(~WU.U64, ~I.u64_op, ~I.u64_is, xs, r)) == SG.checked(~WU.U64, ~SG.u64_of, 64n, Nat.add(n, NS.lsum(vals(xs)))) : Result<&2, &2, NM.NumError, WU.U64>}:
  match xs c:
    case Nil{} True{}:
      %Equal.sym(Maybe<&2, WU.U64>, r, Some{of(n)}, hr) : {G.ok(WU.U64, _) == SG.checked(~WU.U64, ~SG.u64_of, 64n, Nat.add(n, 0n)) : Result<&2, &2, NM.NumError, WU.U64>}
      %Equal.sym(Nat, Nat.add(n, 0n), n, N.add_zero(n)) : {Done{of(n)} == SG.checked(~WU.U64, ~SG.u64_of, 64n, _) : Result<&2, &2, NM.NumError, WU.U64>}
      %Equal.sym(Bool, C.fits(64n, n), True{}, hc) : {Done{of(n)} == SG.checked_pick(~WU.U64, ~SG.u64_of, n, _) : Result<&2, &2, NM.NumError, WU.U64>}
      {==}
    case Nil{} False{}:
      %Equal.sym(Maybe<&2, WU.U64>, r, None{}, hr) : {G.ok(WU.U64, _) == SG.checked(~WU.U64, ~SG.u64_of, 64n, Nat.add(n, 0n)) : Result<&2, &2, NM.NumError, WU.U64>}
      %Equal.sym(Nat, Nat.add(n, 0n), n, N.add_zero(n)) : {Fail{NM.Overflow{}} == SG.checked(~WU.U64, ~SG.u64_of, 64n, _) : Result<&2, &2, NM.NumError, WU.U64>}
      %Equal.sym(Bool, C.fits(64n, n), False{}, hc) : {Fail{NM.Overflow{}} == SG.checked_pick(~WU.U64, ~SG.u64_of, n, _) : Result<&2, &2, NM.NumError, WU.U64>}
      {==}
    case Con{x, t} False{}:
      %Equal.sym(Maybe<&2, WU.U64>, r, None{}, hr) : {G.ok(WU.U64, G.sum_go(~WU.U64, ~I.u64_op, ~I.u64_is, Con{x, t}, _)) == SG.checked(~WU.U64, ~SG.u64_of, 64n, Nat.add(n, NS.lsum(vals(Con{x, t})))) : Result<&2, &2, NM.NumError, WU.U64>}
      +s = Nat.add(n, NS.lsum(vals(Con{x, t})))
      %Equal.sym(Bool, C.fits(64n, s), False{}, unfit_mono(64n, n, s, N.le_add_right(n, NS.lsum(vals(Con{x, t}))), hc)) : {Fail{NM.Overflow{}} == SG.checked_pick(~WU.U64, ~SG.u64_of, s, _) : Result<&2, &2, NM.NumError, WU.U64>}
      {==}
    case Con{+x, +t} True{}:
      +n2 = Nat.add(n, v(x))
      +hn = Equal.cong(Nat, Nat, z => Nat.add(z, v(x)), v(of(n)), n, LW.vo(n, hc))
      %Equal.sym(Maybe<&2, WU.U64>, r, Some{of(n)}, hr) : {G.ok(WU.U64, G.sum_go(~WU.U64, ~I.u64_op, ~I.u64_is, Con{x, t}, _)) == SG.checked(~WU.U64, ~SG.u64_of, 64n, Nat.add(n, Nat.add(v(x), NS.lsum(vals(t))))) : Result<&2, &2, NM.NumError, WU.U64>}
      %NA.add_assoc(n, v(x), NS.lsum(vals(t))) : {G.ok(WU.U64, G.sum_go(~WU.U64, ~I.u64_op, ~I.u64_is, t, G.cadd(~WU.U64, ~I.u64_op, ~I.u64_is, of(n), x))) == SG.checked(~WU.U64, ~SG.u64_of, 64n, _) : Result<&2, &2, NM.NumError, WU.U64>}
      sal(t, G.cadd(~WU.U64, ~I.u64_op, ~I.u64_is, of(n), x), n2, C.fits(64n, n2), cadd_rel(of(n), x, n2, hn, C.fits(64n, n2), {==}), {==})

Uses (31):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool.not
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.cong
  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=Maybe
  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=Result
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=add_assoc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=add_zero
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=le_add_right
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/u64int.bend&name=cadd_rel
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/u64int.bend&name=of
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/u64int.bend&name=unfit_mono
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/u64int.bend&name=v
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/u64int.bend&name=vals
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/u64laws.bend&name=vo
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/lib/common.bend&name=fits
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/generic.bend&name=checked
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/generic.bend&name=checked_pick
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/generic.bend&name=u64_of
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/natural.bend&name=lsum
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/generic.bend&name=cadd
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/generic.bend&name=fits
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/generic.bend&name=ok
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/generic.bend&name=sum_go
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/instances.bend&name=u64_is
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/instances.bend&name=u64_op
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/num.bend&name=NumError
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/u64.bend&name=U64