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Proof step 47 from proof completeness-rec (previous step)

〈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))), d ' + y = depths(to1 .t, i + 1) + x2, to2 = fail, x = d ' + y, to1 ≠ fail, d ∈ depths(to1 .t, i + 1), 0 ≤ i1, i < d, max(d ' + y) ≤ i1 + i ⊦ 〈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 elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 46:
〈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))), d ' + y = depths(t0, i + 1) + x2, to2 = fail, x = d ' + y, to1 ≠ fail, d ∈ depths(t0, i + 1), 0 ≤ i1, i < d, max(d ' + y) ≤ i1 + i, to1 = some(t0) ⊦ 〈build_rec#(i + 1; x2, to2)〉〈if to2 = fail then to0 := fail else to0 := some(Node(t0, to2 .t))〉 (to0 = fail → (∀ t, x. d ' + y ≠ depths(t, i) + x))

Rule argument

[x0lemmaarg: spec-name: treeorfail instance-name: lemma-name: elim antecedent: succedent: to ≠ fail → (t = to .t ↔ to = some(t)) substlist: [to, t] → [to1, t0] add preconditions?: F is rewrite lemma?: F current spec?: F rotate lemma?: T replace all?: T paths: () dummy: () ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPELIM : ⊦ to ≠ fail → (t = to .t ↔ to = some(t))])], [EXTRATERMS : (to1, t0)]]