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Proof step 49 from proof completeness-rec (previous step)

〈build_rec#(i + 1; x, to2)〉 (to2 ≠ fail → x0 = depths(to2 .t, i + 1) + x), 〈build_rec#(i + 1; x, to2)〉 (x = x2 ∧ to2 = to1), ∀ j0, i0, x, x0, to. i→n(j0 - i0) < i→n(i1) → (x = x0 ∧ x0 ≠ [] ∧ i0 ≤ j0 ∧ max(x0) ≤ j0 → 〈build_rec#(i0; x, to)〉 (to = fail → (∀ t, x1. x0 ≠ depths(t, i0) + x1))), x = d ' + y, to2 = fail, to1 ≠ fail, max(d ' + y) ≤ i1 + i, i < d, 0 ≤ i1, x0 = x ⊦ 〈build_rec#(i + 1; x2, to2)〉〈if to2 = fail then to0 := fail else to0 := some(Node(to1 .t, to2 .t))〉 (to0 = fail → (∀ t, x. d ' + y ≠ depths(t, i) + x))

Heuristic contract and execute applied rule contract call left

And obtained the following new goals

Proof step 48:
〈build_rec#(i + 1; x, to2)〉 (x = x2 ∧ to2 = to1), ∀ j0, i0, x, x0, to. i→n(j0 - i0) < i→n(i1) → (x = x0 ∧ x0 ≠ [] ∧ i0 ≤ j0 ∧ max(x0) ≤ j0 → 〈build_rec#(i0; x, to)〉 (to = fail → (∀ t, x1. x0 ≠ depths(t, i0) + x1))), x = d ' + y, to2 = fail, to1 ≠ fail, max(d ' + y) ≤ i1 + i, i < d, 0 ≤ i1, x0 = x, to1 ≠ fail → x0 = depths(to1 .t, i + 1) + x2 ⊦ 〈build_rec#(i + 1; x2, to2)〉〈if to2 = fail then to0 := fail else to0 := some(Node(to1 .t, to2 .t))〉 (to0 = fail → (∀ t, x. d ' + y ≠ depths(t, i) + x))

Rule argument

[fmaposlist-arg: (pos: left 1, pos: left 2)]

Additional information

[[CONCRETESIMPRULES : ()]]