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Proof step 3 from proof at-payload (previous step)

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

User applied rule structural induction

And obtained the following new goals

Proof step 4:
zbrar[m + n] = mkzentry(key, adr), n < nused0 +1, m + nused0 +1 ≤ # zbrar, ∀ 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 ⊦ payload(zbrar, m, nused0 +1)[key] = adr

Rule argument

indvar: nused

Additional information

Used simplifier rules: payload-base: ⊦ payload(zbrar, m, 0) = ∅;%nat freely generated by 0, +1;;% ⊦ n + 0 = n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ ¬ m < 0; (s)