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Proof step 17 from proof completeness-rec (previous step)

〈build_rec#(i + 1; x, to2)〉 (x = [] ∧ to2 = some(t0)), ∀ 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 → (∀ t, x1. x0 ≠ depths(t, i0) + x1))), d ' + y = depths(t, i) + x0, to2 = fail, x = depths(t, i) + x0, depths(t0, i + 1) = depths(t, i) + x0, 0 ≠ i1, d ∈ depths(t, i), 0 < i1, i < d, max(depths(t, i) + x0) ≤ i1 + i ⊦

User applied rule constructor cut

And obtained the following new goals

Proof step 16:
〈build_rec#(i + 1; x, to2)〉 (x = [] ∧ to2 = some(t0)), ∀ 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 → (∀ t, x1. x0 ≠ depths(t, i0) + x1))), d ' + y = depths(t, i) + x0, to2 = fail, x = depths(t, i) + x0, depths(t0, i + 1) = depths(t, i) + x0, 0 ≠ i1, d ∈ depths(t, i), 0 < i1, i < d, max(depths(t, i) + x0) ≤ i1 + i, t = Node(t1, t2) ⊦

Rule argument

[var-arg: t]

Additional information

[[CONCRETESIMPRULES : ([CSIMPGEN : tree generated by leaf, node;], [CSIMPSEQ : ⊦ depths(Leaf, i) = i '], [CSIMPSEQ : ⊦ d ' + x = d0 ' + y ↔ d = d0 ∧ x = y], [CSIMPSEQ : ⊦ (x + y) + z = x + y + z], [CSIMPSEQ : ⊦ d ∈ d0 ' ↔ d = d0], [CSIMPFORWARD : ⊦ d ' + x = y → d ∈ y], [CSIMPSEQ : ⊦ d ∈ x + y ↔ ¬ ¬ (d ∈ x ∨ d ∈ y)], [CSIMPSEQ : ⊦ ¬ i < i])]]