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Proof step 69 from proof completeness-rec (previous step)

x = x0, x0 ≠ [], i ≤ j, max(x0) ≤ j ⊦ 〈build_rec#(i; x, to)〉 (to = fail → (∀ t, x1. x0 ≠ depths(t, i) + x1))

User applied rule induction

And obtained the following new goals

Proof step 68:
∀ j0, i0, x, x0, to. i→n(j0 - i0) < i→n(j - i) → (x = x0 ∧ x0 ≠ [] ∧ i0 ≤ j0 ∧ max(x0) ≤ j0 → 〈build_rec#(i0; x, to)〉 (to = fail → (∀ t, x1. x0 ≠ depths(t, i0) + x1))), x = x0, x0 ≠ [], i ≤ j, max(x0) ≤ j ⊦ 〈build_rec#(i; x, to)〉 (to = fail → (∀ t, x1. x0 ≠ depths(t, i) + x1))

Rule argument

[INDHYPARG : [STANDARDINDUCTIONTYPE : ], true, false, i→n(j - i), [] → [], <]

Additional information

[[CONCRETESIMPRULES : ([CSIMPLESSPRED : <])], [EXTRATERMS : (i→n(j - i))]]