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Proof step 34 from proof completeness-rec (previous step)

〈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))), x = depths(t0, i + 1) + x2, to2 = fail, (depths(t1, i + 1) + depths(t2, i + 1)) + x1 = depths(t0, i + 1) + x2, d ∈ depths(t1, i + 1), max(depths(t0, i + 1) + x2) ≤ i1 + i, i < d, 0 ≤ i1, d ∈ depths(t0, i + 1), ∀ t, x. x2 ≠ depths(t, i + 1) + x ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 31:
⊦ depths(t0, i) + x = depths(t1, i) + y ↔ x = y ∧ t0 = t1
Proof step 33:
〈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))), x = depths(t0, i + 1) + x2, to2 = fail, depths(t2, i + 1) + x1 = x2 ∧ t1 = t0, d ∈ depths(t1, i + 1), max(depths(t0, i + 1) + x2) ≤ i1 + i, i < d, 0 ≤ i1, d ∈ depths(t0, i + 1), ∀ t, x. x2 ≠ depths(t, i + 1) + x ⊦

Rule argument

[x0lemmaarg: spec-name: build-base instance-name: lemma-name: unique-prefix antecedent: succedent: depths(t0, i) + x = depths(t1, i) + y ↔ x = y ∧ t0 = t1 substlist: [y, t1, x, i, t0] → [x2, t0, depths(t2, i + 1) + x1, i + 1, t1] add preconditions?: F is rewrite lemma?: T current spec?: F rotate lemma?: F replace all?: T paths: ((0, 0, 5)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ i + j = j + i], [CSIMPSEQ : ⊦ (x + y) + z = x + y + z])], [EXTRATERMS : (x2, t0, depths(t2, i + 1) + x1, i + 1, t1)]]