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-- This needs Lean v4.24.0 and https://github.com/verse-lab/velvet/ added as a dependency in the lakefile.toml
-- https://github.com/verse-lab/velvet/blob/master/Velvet/Examples/Examples.lean
import Velvet.Std

set_option loom.semantics.termination "total"
set_option loom.semantics.choice "demonic"

attribute [grind] Array.multiset_swap

method insertionSort_total (mut A : Array ℤ) return (u : Unit)
  require 1 ≤ A.size
  ensures ∀ i j, 0 ≤ i ∧ i ≤ j ∧ j < A.size → A[i]! ≤ A[j]!
  ensures AOld.toMultiset = A.toMultiset
  do
    let A' := A
    let N := A.size
    let mut i := 1
    while i ≠ A.size
    invariant A.size = N
    invariant 1 ≤ i ∧ i ≤ A.size
    invariant ∀ i' j', 0 ≤ i' ∧ i' < j' ∧ j' ≤ i - 1 → A[i']! ≤ A[j']!
    invariant A.toMultiset = A'.toMultiset
    decreasing A.size - i
    do
      let mut j := i
      while j ≠ 0
      invariant A.size = N
      invariant j ≤ i
      invariant ∀ i' j', 0 ≤ i' ∧ i' < j' ∧ j' ≤ i ∧ j' ≠ j → A[i']! ≤ A[j']!
      invariant A.toMultiset = A'.toMultiset
      decreasing j
      do
        if A[j]! < A[j - 1]! then
          swap! A[j - 1]! A[j]!
        j := j - 1
      i := i + 1
    return
set_option maxHeartbeats 1000000 in
prove_correct insertionSort_total by
  loom_solve