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Proof step 13 from proof at-payload (previous step)

zbrar[m + n1 + 1] = mkzentry(zbrar[(m + 1) + n].key, adr), zbrar[(m + 1) + n].entry?, n < nused0, n < n1, m + nused0 < # zbrar, m < # zbrar, n1 < nused0, n + 1 < nused0, ∀ n0. ¬ n1 < n0 ∧ zbrar[m + n0].entry? → zbrar[m + n0].key ≠ zbrar[(m + 1) + n].key ⊦ (zbrar[m].entry? ⊃ payload(zbrar, m + 1, nused0)[zbrar[m].key, zbrar[m].adr];payload(zbrar, m + 1, nused0))[zbrar[(m + 1) + n].key] = adr

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 14:
zbrar[m + n1 + 1] = mkzentry(key, adr), zbrar[(m + 1) + n] = mkzentry(key, adr0), zbrar[(m + 1) + n].entry?, n + 1 < nused0, n1 < nused0, m < # zbrar, m + nused0 < # zbrar, n < n1, n < nused0, ∀ n0. ¬ n1 < n0 ∧ zbrar[m + n0].entry? → zbrar[m + n0].key ≠ key ⊦ (zbrar[m].entry? ⊃ payload(zbrar, m + 1, nused0)[zbrar[m].key, zbrar[m].adr];payload(zbrar, m + 1, nused0))[key] = adr

Rule argument

spec: zbranch name: elim-03 seq: ⊦ zbr.entry? → (key = zbr.key ∧ adr = zbr.adr ↔ zbr = mkzentry(key, adr)) subst: [zbr, key, adr] → [zbrar[(m + 1) + n], key, adr0]

Additional information

Used simplifier rules: a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%⊦ zbr.entry? → (key = zbr.key ∧ adr = zbr.adr ↔ zbr = mkzentry(key, adr)); (elim) Used terms: zbrar[(m + 1) + n], key, adr0