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Proof step 7 from proof at-payload (previous step)

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

User applied rule constructor cut

And obtained the following new goals

Proof step 8:
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

Rule argument

var: n

Additional information

Used simplifier rules: 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% ⊦ 0 ≤ m; (s)% ⊦ ¬ m < 0; (s)% ⊦ mkzentry(key, adr).entry?; (s)% ⊦ mkzentry(key, adr).key = key; (s)% ⊦ mkzentry(key, adr).adr = adr; (s)% ⊦ ri[key, adr][key] = adr; (s)