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Proof step 41 from proof completeness (previous step)

〈build_rec#(0; x, to)〉 (x = x1 ∧ to = to1), x ≠ [], 0 ≤ max(x), to1 ≠ fail → x = depths(to1 .t, 0) + x1 ⊦ 〈if x1 ≠ [] then to1 := fail〉 (to1 = fail → (∀ t. x ≠ depths(t, 0)))

User applied rule insert lemma

And obtained the following new goals

Proof step 30:
x ≠ [], x = x0, i ≤ j, max(x) ≤ j ⊦ 〈build_rec#(i; x, to)〉 (to = fail → ¬ (∃ t, x1. x0 = depths(t, i) + x1))
Proof step 40:
〈x1 := x〉〈build_rec#(0; x, to)〉 (to = fail → ¬ (∃ t, x. x1 = depths(t, 0) + x)), 〈build_rec#(0; x, to)〉 (x = x1 ∧ to = to1), x ≠ [], 0 ≤ max(x), to1 ≠ fail → x = depths(to1 .t, 0) + x1 ⊦ 〈if x1 ≠ [] then to1 := fail〉 (to1 = fail → (∀ t. x ≠ depths(t, 0)))

Rule argument

[x0lemmaarg: spec-name: instance-name: lemma-name: completeness-rec antecedent: x ≠ [], x = x0, i ≤ j, max(x) ≤ j succedent: 〈build_rec#(i; x, to)〉 (to = fail → ¬ (∃ t, x1. x0 = depths(t, i) + x1)) substlist: [x, x0, i, j, to] → [x, x, 0, max(x), to] add preconditions?: F is rewrite lemma?: F current spec?: T rotate lemma?: F replace all?: F paths: () dummy: () ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ i ≤ i])], [EXTRATERMS : (x, x, 0, max(x), to)]]