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Proof step 16 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, d ∈ depths(t, i + 1), max(d ' + y) ≤ i1 + i, i < d, 0 < i1, i1 + i < max(x1), max(d ' + y) < max(x1), x0 = x1 ∧ d1 = i + 1, ¬ is_empty(x0), h0 = head(x0) ⊦ 〈if h0 < d1 then to := fail else if h0 = d1 then { x0 := pop(x0) ; to := some(Leaf) } else let to0 = fail, to1 = fail in { build_rec#(d1 + 1; x0, to0) ; if to0 = fail then to := fail else { build_rec#(d1 + 1; x0, 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) + x0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 15:
〈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)

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ (x + y) + z = x + y + z], [CSIMPSEQ : ⊦ i + j + i0 = (i + j) + i0], [CSIMPSEQ : ⊦ i < j + 1 ↔ i ≤ j], [CSIMPSEQ : ⊦ is_empty(x) ↔ x = []], [CSIMPSEQ : ⊦ i + j = j + i])]]