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Proof step 7 from proof completeness-rec (previous step)

〈build_rec#(i + 1; x, to3)〉 (x = x2 ∧ to3 = 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))), x = d ' + y, d ' + y = (depths(t0, i + 1) + depths(t1, i + 1)) + x1, to3 = fail, d ∈ depths(t0, i + 1), max(d ' + y) ≤ i1 + i, i < d, 0 ≤ i1, ∀ t, x. d ' + y ≠ depths(t, i + 1) + x ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

[EXRARG : pos: left 10, [QUANTTERMLIST : (t0, depths(t1, i + 1) + x1)]]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ i + j = j + i], [CSIMPSEQ : ⊦ (x + y) + z = x + y + z])], [EXTRATERMS : (t0, depths(t1, i + 1) + x1)]]