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Proof step 2 from proof in-payload (previous step)

m + nused0 +1 ≤ # zbrar, ∀ m, zbrar, key. m + nused0 ≤ # zbrar → (key ∈ payload(zbrar, m, nused0) ↔ (∃ n. n < nused0 ∧ zbrar[m + n].entry? ∧ zbrar[m + n].key = key)) ⊦ key ∈ payload(zbrar, m, nused0 +1) ↔ (∃ n. n < nused0 +1 ∧ zbrar[m + n].entry? ∧ zbrar[m + n].key = key)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
m + nused0 < # zbrar, ∀ m, zbrar, key. m + nused0 ≤ # zbrar → (key ∈ payload(zbrar, m, nused0) ↔ (∃ n. n < nused0 ∧ zbrar[m + n].entry? ∧ zbrar[m + n].key = key)) ⊦ key ∈ payload(zbrar, m, nused0 + 1) ↔ (∃ n. ¬ nused0 < n ∧ zbrar[m + n].entry? ∧ zbrar[m + n].key = key)

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m + 1 ≤ n ↔ m < n; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ m +1 = m + 1; (s)