PROOF.AP.from_nat: def of bend-osdev-array@0.1.0.1
Import: import bend-osdev-array@0.1.0.1/PROOF.bend as PROOF
Lines 987-994 of bend-osdev-array@0.1.0.1/PROOF.bend: https://jend.leeeet.dev/read.txt?path=bend-osdev-array@0.1.0.1/PROOF.bend&start_line=987&end_line=994

import Base
import bend-osdev-array@0.1.0.1/array.bend as A

# from_nat then to_nat is the identity below 2^d, d <= 32
def AP.from_nat(k: Nat, +d: Nat, +h: {Nat.is_lt(k, A.Arrays.slots(d)) == True{} : Bool}, +hd: {Nat.is_le(d, 32n) == True{} : Bool}) -> {U32.to_nat(U32.from_nat(k)) == k : Nat}:
  match k:
    case 0n:
      {==}
    case 1n+ +p:
      +ih = AP.from_nat(p, d, AP.lt_succ_l(p, A.Arrays.slots(d), h), hd)
      Equal.trans(Nat, U32.to_nat(U32.inc(U32.from_nat(p))), 1n+U32.to_nat(U32.from_nat(p)), 1n+p, AP.u32_inc(U32.from_nat(p), d, AP.lt_rw(U32.to_nat(U32.from_nat(p)), p, A.Arrays.slots(d), ih, h), hd), AP.succ_cong(U32.to_nat(U32.from_nat(p)), p, ih))

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.trans
  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_le
  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.from_nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32.inc
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32.to_nat
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-array@0.1.0.1/PROOF.bend&name=AP.lt_rw
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-array@0.1.0.1/PROOF.bend&name=AP.lt_succ_l
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-array@0.1.0.1/PROOF.bend&name=AP.succ_cong
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-array@0.1.0.1/PROOF.bend&name=AP.u32_inc
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-array@0.1.0.1/array.bend&name=Arrays.slots