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Proof step 37 from proof completeness-rec (previous step)

〈build_rec#(i + 1; x, to2)〉 (x = x2 ∧ to2 = some(t0)), 〈build_rec#(i + 1; x2, to2)〉 (x2 = x3 ∧ to2 = fail), ∀ 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))), depths(t, i) + x1 = depths(t0, i + 1) + x2, to2 = fail, x = depths(t, i) + x1, d ' + y = depths(t, i) + x1, d ∈ depths(t, i), 0 ≤ i1, i < d, max(depths(t, i) + x1) ≤ i1 + i, d ∈ depths(t0, i + 1), ∀ t, x. x2 ≠ depths(t, i + 1) + x, t = Node(t1, t2) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 36:
〈build_rec#(i + 1; x2, to2)〉 (x2 = x3 ∧ to2 = fail), 〈build_rec#(i + 1; x, to2)〉 (x = x2 ∧ 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(t1, i + 1) + depths(t2, i + 1)) + x1, x = d ' + y, to2 = fail, d ' + y = depths(t0, i + 1) + x2, d ∈ depths(t1, i + 1), max(d ' + y) ≤ i1 + i, i < d, 0 ≤ i1, d ∈ depths(t0, i + 1), ∀ t, x. x2 ≠ depths(t, i + 1) + x ⊦

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ d ∈ x → d ∈ x + y], [CSIMPFORWARD : ⊦ d ' + x = y → d ∈ y], [CSIMPFORWARD : ⊦ depths(t, i) + x = d ' + y → d ∈ depths(t, i)], [CSIMPSEQ : ⊦ (x + y) + z = x + y + z], [CSIMPSEQ : ⊦ d ∈ x + y ↔ ¬ ¬ (d ∈ x ∨ d ∈ y)], [CSIMPSEQ : ⊦ depths(Node(t0, t1), i) = depths(t0, i + 1) + depths(t1, i + 1)], [CSIMPSEQ : ⊦ i + j = j + i])]]