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Proof step 47 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 < 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)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 46:
∀ 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, to3 = fail, max(d ' + y) ≤ i1 + i, i < d, 0 ≤ i1 ⊦ 〈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 → d ' + y = depths(to0 .t, i) + x2)

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ d ∈ d0 ' ↔ d = d0], [CSIMPSEQ : ⊦ d ∈ x + y ↔ ¬ ¬ (d ∈ x ∨ d ∈ y)], [CSIMPFORWARD : ⊦ d ' + x = y → d ∈ y], [CSIMPSEQ : ⊦ (x + y) + z = x + y + z], [CSIMPSEQ : ⊦ d ' + x ≠ []], [CSIMPSEQ : ⊦ d + x = d ' + x])]]