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Proof step 17 from proof in-playload-data (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 18:
m + nused ≤ # zbrar ⊦ key ∈ payload(zbrar, m, nused) ↔ (∃ n. n < nused ∧ zbrar[m + n].entry? ∧ zbrar[m + n].key = key)
Proof step 19:
zbrar[n] = mkzentry(key, adr0), n + m0 < # zbrar, n < # zbrar, sorted(zbrar, 0, n + m0 + 1), adr ∈ payload(zbrar, n + 1, m0)[key, adr0], (n + 1) + m0 ≤ # zbrar, ∀ 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)), ∃ n0. n0 < m0 ∧ zbrar[(n + 1) + n0].entry? ∧ zbrar[(n + 1) + n0].key = key ⊦ ∃ m. n ≤ m ∧ ¬ n + m0 < m ∧ zbrar[m].entry? ∧ zbrar[m].adr = adr

Rule argument

spec: <toplevel> name: in-payload seq: m + nused ≤ # zbrar ⊦ key ∈ payload(zbrar, m, nused) ↔ (∃ n. n < nused ∧ zbrar[m + n].entry? ∧ zbrar[m + n].key = key) subst: [nused, m, zbrar, key] → [m0, n + 1, zbrar, key] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: m0, n + 1, zbrar, key