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Proof step 42 from proof in-playload-data (previous step)

zbrar[m] = mkzentry(key, adr0), zbrar[n] = mkzentry(key0, adr1), n < # zbrar, n + m0 < # zbrar, m < # zbrar, sorted(zbrar, 0, n + m0 + 1), n ≤ m, m ≤ n + 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)) ⊦ adr0 ∈ payload(zbrar, n + 1, m0)[key0, adr1]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 43:
⊦ ¬ key ∈ ri → (adr ∈ ri[key, adr0] ↔ adr ∈ ri ∨ adr = adr0)
Proof step 44:
zbrar[n] = mkzentry(key0, adr1), zbrar[m] = mkzentry(key, adr0), m < # zbrar, n + m0 < # zbrar, n < # zbrar, sorted(zbrar, 0, n + m0 + 1), key0 ∈ payload(zbrar, n + 1, m0), 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)[key0, adr1]
Proof step 49:
zbrar[n] = mkzentry(key0, adr1), zbrar[m] = mkzentry(key, adr0), 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 = adr1, adr0 ∈ payload(zbrar, n + 1, m0), key0 ∈ payload(zbrar, n + 1, m0)

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] → [adr1, key0, payload(zbrar, n + 1, m0), adr0] flags: <all paths>

Additional information

Used terms: adr1, key0, payload(zbrar, n + 1, m0), adr0