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Proof step 30 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, x1 = [] ⊦ 〈build_rec#(i + 1; x1, to2)〉〈if to2 = fail then to0 := fail else to0 := some(Node(t, to2 .t))〉 (to0 ≠ fail → d ' + y = depths(to0 .t, i) + x1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 29:
〈build_rec#(i + 1; x2, to2)〉 (x2 = [] ∧ 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)), x1 = [], x2 = depths(t, i + 1), to2 = fail, d ' + y = depths(t, i + 1), 0 ≠ i1, d ∈ depths(t, i + 1), max(depths(t, i + 1)) ≤ i1 + i, i < d, 0 < i1 ⊦ 〈build_rec#(i + 1; x1, to2)〉〈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 : ⊦ x + [] = x])]]