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Proof step 35 from proof completeness-rec (previous step)

〈build_rec#(i + 1; x2, to2)〉 (x2 = x3 ∧ to2 = fail), 〈build_rec#(i + 1; x, to2)〉 (x = x2 ∧ to2 = some(t0)), ∀ 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 = (depths(t1, i + 1) + depths(t2, i + 1)) + x1, to2 = fail, (depths(t1, i + 1) + depths(t2, i + 1)) + x1 = depths(t0, i + 1) + x2, d ∈ depths(t1, i + 1), max((depths(t1, i + 1) + depths(t2, i + 1)) + x1) ≤ i1 + i, i < d, 0 ≤ i1, d ∈ depths(t0, i + 1), ∀ t, x. x2 ≠ depths(t, i + 1) + x ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 34:
〈build_rec#(i + 1; x2, to2)〉 (x2 = x3 ∧ to2 = fail), 〈build_rec#(i + 1; x, to2)〉 (x = x2 ∧ to2 = some(t0)), ∀ 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 = depths(t0, i + 1) + x2, to2 = fail, (depths(t1, i + 1) + depths(t2, i + 1)) + x1 = depths(t0, i + 1) + x2, d ∈ depths(t1, i + 1), max(depths(t0, i + 1) + x2) ≤ i1 + i, i < d, 0 ≤ i1, d ∈ depths(t0, i + 1), ∀ t, x. x2 ≠ depths(t, i + 1) + x ⊦

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ()]]