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Proof step 8 from proof at-payload (previous step)

zbrar[m + n] = mkzentry(key, adr), n = n1 +1, m < # zbrar, m + nused0 < # zbrar, n ≤ nused0, ∀ n0. n0 < n ∧ zbrar[m + n0].entry? → zbrar[m + n0].key ≠ key, ∀ zbrar, m, n, key, adr. zbrar[m + n] = mkzentry(key, adr) ∧ n < nused0 ∧ m + nused0 ≤ # zbrar ∧ (∀ n0. n0 < n ∧ zbrar[m + n0].entry? → zbrar[m + n0].key ≠ key) → payload(zbrar, m, nused0)[key] = adr ⊦ (zbrar[m].entry? ⊃ payload(zbrar, m + 1, nused0)[zbrar[m].key, zbrar[m].adr];payload(zbrar, m + 1, nused0))[key] = adr

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
zbrar[m + n1 + 1] = mkzentry(key, adr), n1 < nused0, m < # zbrar, m + nused0 < # zbrar, ∀ n. ¬ n1 < n ∧ zbrar[m + n].entry? → zbrar[m + n].key ≠ key, ∀ zbrar, m, n, key, adr. zbrar[m + n] = mkzentry(key, adr) ∧ n < nused0 ∧ m + nused0 ≤ # zbrar ∧ (∀ n0. n0 < n ∧ zbrar[m + n0].entry? → zbrar[m + n0].key ≠ key) → payload(zbrar, m, nused0)[key] = adr ⊦ (zbrar[m].entry? ⊃ payload(zbrar, m + 1, nused0)[zbrar[m].key, zbrar[m].adr];payload(zbrar, m + 1, nused0))[key] = adr

Rule argument

Additional information

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