enrich natlist, olist with predicates cutseq : natlist × list × bool; cutseq : natlist × list; in-range : natlist × nat × nat; procedures MAXCUT list : : natlist ; variables m : nat; n : nat; x : list; cut : natlist; ns : natlist; inc : bool; sz : nat; axioms cutseq-empty : ⊦ ¬ cutseq([], x); used for: ls; cutseq-one : ⊦ cutseq(n ', x) ↔ n = 0; used for: ls; cutseq-rec : ⊦ cutseq(cut + n ' + n0 ', x) ↔ ( ordered<(sublist(n, n0 - n, x)) ∧ (n0 < # x → ¬ x[n0 -1] < x[n0]) ∨ ordered≤(rev(sublist(n, n0 - n, x))) ∧ (n0 < # x → x[n0 -1] < x[n0])) ∧ n < n0 ∧ n0 ≤ # x ∧ cutseq(cut + n ', x); used for: ls; cutseq-empty-h : ⊦ ¬ cutseq([], x, inc); used for: ls; cutseq-one-h : ⊦ cutseq(n ', x, inc) ↔ n = 0; used for: ls; cutseq-rec-h : ⊦ cutseq(cut + n ' + n0 ', x, inc) ↔ (inc ⊃ ordered<(sublist(n, n0 - n, x));ordered≤(rev(sublist(n, n0 - n, x)))) ∧ n < n0 ∧ n0 ≤ # x ∧ cutseq(cut + n ', x); used for: ls; in-range-base : ⊦ in-range([], n, m); used for: ls, s; in-range-rec : ⊦ in-range(n0 ' + ns, n, m) ↔ n ≤ n0 ∧ n0 ≤ m ∧ in-range(ns, n, m); used for: ls, s; theorems MAXCUT-thm : ⊦ ⟨MAXCUT(x; ; cut)⟩ (cut.head = 0 ∧ cut.last = # x ∧ in-range(cut, 0, # x) ∧ cutseq(cut, x)); MAXCUT-non-empty : ⊦ ⟨/* _:use contract MAXCUT-thm and skip call; */; MAXCUT(x; ; cut)⟩ (# cut > 0); MAXCUT-begin-to-end : ⊦ ⟨/* _:use contract MAXCUT-thm and skip call; */; MAXCUT(x; ; cut)⟩ (cut.head = 0 ∧ cut.last = # x); MAXCUT-within-bounds : ⊦ ⟨/* _:use contract MAXCUT-thm and skip call; */; MAXCUT(x; ; cut)⟩ in-range(cut, 0, # x); MAXCUT-monotonic-maximal : ⊦ ⟨/* _:use contract MAXCUT-thm and skip call; */; MAXCUT(x; ; cut)⟩ cutseq(cut, x); declarations MAXCUT : MAXCUT(x; ; cut) {cut := 0 + []; let m = 0, n = 1 in {while n < # x do {/* _: invariant cutseq(cut, x) ∧ n = m + 1 ∧ m ≤ # x ∧ cut.last = m ∧ cut.head = 0 ∧ in-range(cut, 0, # x) wfbound # x - m; */; let inc = x[m] < x[n] in while n < # x ∧ (x[n - 1] < x[n] ↔ inc) do {/* _:invariant cutseq(cut + n ', x, inc) ∧ m < n ∧ n ≤ # x wfbound # x - n; */; n := n + 1}; cut := cut + n; m := n; n := m + 1}; if m < # x then cut := cut + # x} } ; end enrich