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Proof step 48 from proof in-playload-data (previous step)

zbrar[n] = mkzentry(zbrar[(n + 1) + n0].key, adr1), zbrar[m] = mkzentry(key, adr0), zbrar[(n + 1) + n0].entry?, n0 < m0, m < # zbrar, n + m0 < # zbrar, n < # zbrar, sorted(zbrar, 0, n + m0 + 1), m ≤ n + m0, n ≤ m, ∀ zbrar, n, adr. sorted(zbrar, 0, n + m0) ∧ n + m0 ≤ # zbrar → (adr ∈ payload(zbrar, n, m0) ↔ (∃ m1. n ≤ m1 ∧ m1 < n + m0 ∧ zbrar[m1].entry? ∧ zbrar[m1].adr = adr)) ⊦ adr0 ∈ payload(zbrar, n + 1, m0)[zbrar[(n + 1) + n0].key, adr1]

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[(n + 1) + n0], key0, adr2]

Additional information

Used simplifier rules: ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ ¬ m + n < m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ 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)% ⊦ m = m + n ↔ n = 0; (s)% ⊦ m + 1 ≠ 0; (s)%⊦ zbr.entry? → (key = zbr.key ∧ adr = zbr.adr ↔ zbr = mkzentry(key, adr)); (elim) Used terms: zbrar[(n + 1) + n0], key0, adr2