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Proof step 32 from proof completeness-rec (previous step)

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

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

[EXRARG : pos: left 11, [QUANTTERMLIST : (t2, x1)]]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ i + j = j + i])], [EXTRATERMS : (t2, x1)]]