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Proof step 14 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], ∀ 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 15:
⊦ ¬ key ∈ ri → (adr ∈ ri[key, adr0] ↔ adr ∈ ri ∨ adr = adr0)
Proof step 16:
zbrar[n] = mkzentry(key, adr0), n + m0 < # zbrar, n < # zbrar, sorted(zbrar, 0, n + m0 + 1), key ∈ payload(zbrar, n + 1, m0), adr ∈ payload(zbrar, n + 1, m0)[key, adr0], ∀ 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
Proof step 21:
zbrar[n] = mkzentry(key, adr0), n + m0 < # zbrar, n < # zbrar, sorted(zbrar, 0, n + m0 + 1), ∀ 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)), adr ∈ payload(zbrar, n + 1, m0) ∨ adr = adr0 ⊦ key ∈ payload(zbrar, n + 1, m0), ∃ m. n ≤ m ∧ ¬ n + m0 < m ∧ zbrar[m].entry? ∧ zbrar[m].adr = adr

Rule argument

spec: store+set, inst: node-index name: Data-in-other seq: ⊦ ¬ key ∈ ri → (adr ∈ ri[key, adr0] ↔ adr ∈ ri ∨ adr = adr0) subst: [adr0, key, ri, adr] → [adr0, key, payload(zbrar, n + 1, m0), adr] flags: <all paths>

Additional information

Used terms: adr0, key, payload(zbrar, n + 1, m0), adr