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Proof step 48 from proof soundness-rec (previous step)

∀ 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 → x0 = depths(to .t, i0) + x)), x2 = x0, to3 = fail, x0 ≠ [], max(x0) ≤ i1 + i, 0 ≤ i1, i < head(x0) ⊦ 〈build_rec#(i + 1; x2, to3)〉〈if to3 = fail then to0 := fail else { to := fail ; build_rec#(i + 1; x2, to) ; if to = fail then to0 := fail else to0 := some(Node(to3 .t, to .t)) }〉 (to0 ≠ fail → x0 = depths(to0 .t, i) + x2)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 47:
∀ 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 → x0 = depths(to .t, i0) + x)), x2 = x0, to3 = fail, x0 ≠ [], max(x0) ≤ i1 + i, 0 ≤ i1, i < d, x0 = d + y ⊦ 〈build_rec#(i + 1; x2, to3)〉〈if to3 = fail then to0 := fail else { to := fail ; build_rec#(i + 1; x2, to) ; if to = fail then to0 := fail else to0 := some(Node(to3 .t, to .t)) }〉 (to0 ≠ fail → x0 = depths(to0 .t, i) + x2)

Rule argument

[x0lemmaarg: spec-name: list-data instance-name: intlist lemma-name: elim-list-c antecedent: x ≠ [] succedent: d = head(x) ∧ y = pop(x) ↔ x = d + y substlist: [x, d, y] → [x0, d, y] add preconditions?: F is rewrite lemma?: F current spec?: F rotate lemma?: T replace all?: T paths: () dummy: () ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPELIM : x ≠ [] ⊦ d = head(x) ∧ y = pop(x) ↔ x = d + y])], [EXTRATERMS : (x0, d, y)]]