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Proof step 93 from proof payload-shift-left (previous step)

zbrar[n0] = mkzentry(zbrar[n1].key, adr), zbrar[n1].entry?, n1 + 1 < nused, n1 + 1 < # zbrar, n0 + 1 < # zbrar, n0 < n, n0 < # zbrar, n < # zbrar, nused < # zbrar, n0 < nused, n1 < n0, n1 < n, n1 < # zbrar, n1 < nused, n1 + 1 < n, 2 + n1 < # zbrar, sorted(zbrar, 0, nused + 1), zbrar[n1].key ∈ payload(shl(zbrar, n, nused + 1), 0, nused), n ≤ nused, ∀ n2. n2 < n0 ∧ zbrar[n2].entry? → zbrar[n2].key ≠ zbrar[n1].key ⊦ payload(shl(zbrar, n, nused + 1), 0, nused)[zbrar[n1].key] = adr, zbrar[n].entry?

Heuristic elimination applied rule insert elim lemma

And closed the goal!

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[n1], key, adr0]

Additional information

Used simplifier rules: ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ m < n → ¬ n < m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m + 1 ≤ n ↔ m < n; (s)%m0 < m, m1 < m, m ≤ # zbrar ⊦ sorted(zbrar, 0, m) ∧ zbrar[m0] = mkzentry(key, adr0) → (zbrar[m1] = mkzentry(key, adr1) ↔ m0 = m1 ∧ adr0 = adr1); (ws)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%⊦ zbr.entry? → (key = zbr.key ∧ adr = zbr.adr ↔ zbr = mkzentry(key, adr)); (elim) Used terms: zbrar[n1], key, adr0