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Proof step 15 from proof soundness-rec (previous step)

〈build_rec#(i + 1; x2, to2)〉 (x2 = x1 ∧ to2 = some(t)), ∀ 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 = d ' + y, to2 = fail, d ' + y = depths(t, i + 1) + x1, 0 ≠ i1, x1 ≠ [], d ∈ depths(t, i + 1), max(d ' + y) ≤ i1 + i, i < d, 0 < i1, i1 + i < max(x1), max(d ' + y) < max(x1) ⊦ 〈if head(x1) ≤ i then to := fail else if head(x1) = i + 1 then { x1 := pop(x1) ; to := some(Leaf) } else let to0 = fail, to1 = fail in { build_rec#(2 + i; x1, to0) ; if to0 = fail then to := fail else { build_rec#(2 + i; x1, to1) ; if to1 = fail then to := fail else to := some(Node(to0 .t, to1 .t)) } }〉〈to2 := to ; if to2 = fail then to0 := fail else to0 := some(Node(t, to2 .t))〉 (to0 ≠ fail → d ' + y = depths(to0 .t, i) + x1)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 14:
〈build_rec#(i + 1; x2, to2)〉 (x2 = x1 ∧ to2 = some(t)), ∀ 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 = d ' + y, to2 = fail, d ' + y = depths(t, i + 1) + x1, 0 ≠ i1, x1 ≠ [], d ∈ depths(t, i + 1), max(d ' + y) ≤ i1 + i, i < d, 0 < i1, i1 + i < max(x1), max(d ' + y) < max(x1), x1 = d0 + y0 ⊦ 〈if d0 ≤ i then to := fail else if d0 = i + 1 then { x1 := pop(x1) ; to := some(Leaf) } else let to0 = fail, to1 = fail in { build_rec#(2 + i; x1, to0) ; if to0 = fail then to := fail else { build_rec#(2 + i; x1, to1) ; if to1 = fail then to := fail else to := some(Node(to0 .t, to1 .t)) } }〉〈to2 := to ; if to2 = fail then to0 := fail else to0 := some(Node(t, to2 .t))〉 (to0 ≠ fail → d ' + y = depths(to0 .t, i) + x1)

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] → [x1, d0, y0] 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 : (x1, d0, y0)]]