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Proof step 45 from proof completeness-rec (previous step)

〈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))), x = d ' + y, to2 = fail, d ' + y = depths(t0, i + 1) + x2, d ∈ depths(t0, i + 1), max(d ' + y) ≤ i1 + i, i < d, 0 ≤ i1 ⊦ 〈build_rec#(i + 1; x2, to2)〉〈if to2 = fail then to0 := fail else to0 := some(Node(t0, to2 .t))〉 (to0 = fail → (∀ t, x. d ' + y ≠ depths(t, i) + x))

User applied rule apply induction

And obtained the following new goals

Proof step 30:
〈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))), x = d ' + y, to2 = fail, d ' + y = depths(t0, i + 1) + x2, d ∈ depths(t0, i + 1), max(d ' + y) ≤ i1 + i, i < d, 0 ≤ i1, x2 ≠ x2 ∨ x2 = [] ∨ ¬ i + 1 ≤ i1 + i ∨ ¬ max(x2) ≤ i1 + i ⊦ 〈build_rec#(i + 1; x2, to2)〉〈if to2 = fail then to0 := fail else to0 := some(Node(t0, to2 .t))〉 (to0 = fail → (∀ t, x. d ' + y ≠ depths(t, i) + x))
Proof step 44:
〈x1 := x2〉〈build_rec#(i + 1; x2, to2)〉 (to2 = fail → (∀ t, x. x1 ≠ depths(t, i + 1) + x)), 〈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))), x = d ' + y, to2 = fail, d ' + y = depths(t0, i + 1) + x2, d ∈ depths(t0, i + 1), max(d ' + y) ≤ i1 + i, i < d, 0 ≤ i1 ⊦ 〈build_rec#(i + 1; x2, to2)〉〈if to2 = fail then to0 := fail else to0 := some(Node(t0, to2 .t))〉 (to0 = fail → (∀ t, x. d ' + y ≠ depths(t, i) + x))

Rule argument

[substlist-arg: [j0, i0, x, x0, to] → [i1 + i, i + 1, x2, x2, to2]]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ i + j - i + 1 = j - 1], [CSIMPSEQ : ⊦ i + j + i0 = (i + j) + i0], [CSIMPSEQ : ⊦ i + j = j + i], [CSIMPSEQ : ⊦ i→n(i - 1) < i→n(i) ↔ 0 < i], [CSIMPSEQ : ⊦ ¬ i < j ↔ j ≤ i], [CSIMPSEQ : ⊦ i ≤ j → (j ≤ i ↔ i = j)], [CSIMPSEQ : ⊦ (x + y) + z = x + y + z], [CSIMPSEQ : ⊦ i + 0 = i], [CSIMPFORWARD : ⊦ i ≤ j ∧ j < j0 → i < j0], [CSIMPSEQ : ⊦ ¬ max(d ' + x) < d])], [EXTRATERMS : (i1 + i, i + 1, x2, x2, to2)]]