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Proof step 51 from proof completeness-rec (previous step)

〈build_rec#(i + 1; x, to2)〉 (x = x2 ∧ to2 = to1), ∀ 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, to1 ≠ fail, 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(to1 .t, to2 .t))〉 (to0 = fail → (∀ t, x. d ' + y ≠ depths(t, i) + x))

User applied rule insert lemma

And obtained the following new goals

Proof step 12:
x ≠ [], x = x0, i ≤ j, max(x) ≤ j ⊦ 〈build_rec#(i; x, to)〉 (to ≠ fail → x0 = depths(to .t, i) + x)
Proof step 50:
〈x1 := x〉〈build_rec#(i + 1; x, to2)〉 (to2 ≠ fail → x1 = depths(to2 .t, i + 1) + x), 〈build_rec#(i + 1; x, to2)〉 (x = x2 ∧ to2 = to1), ∀ 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, to1 ≠ fail, 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(to1 .t, to2 .t))〉 (to0 = fail → (∀ t, x. d ' + y ≠ depths(t, i) + x))

Rule argument

[x0lemmaarg: spec-name: instance-name: lemma-name: soundness-rec antecedent: x ≠ [], x = x0, i ≤ j, max(x) ≤ j succedent: 〈build_rec#(i; x, to)〉 (to ≠ fail → x0 = depths(to .t, i) + x) substlist: [x, x0, i, j, to] → [x, x, i + 1, i1 + i, to2] add preconditions?: F is rewrite lemma?: F current spec?: T rotate lemma?: F replace all?: F paths: () dummy: () ]

Additional information

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