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Proof step 5 from proof at-payload (previous step)

zbrar[m + n] = mkzentry(key, adr), 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 ⊦ payload(zbrar, m, nused0 + 1)[key] = adr

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ m < # zbrar → payload(zbrar, m, nused + 1) = (zbrar[m].entry? ⊃ payload(zbrar, m + 1, nused)[zbrar[m].key, zbrar[m].adr];payload(zbrar, m + 1, nused))
Proof step 7:
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

Rule argument

spec: <toplevel> name: payload-rec seq: ⊦ m < # zbrar → payload(zbrar, m, nused + 1) = (zbrar[m].entry? ⊃ payload(zbrar, m + 1, nused)[zbrar[m].key, zbrar[m].adr] ;payload(zbrar, m + 1, nused) ) subst: [nused, m, zbrar] → [nused0, m, zbrar] flags: <all paths>

Additional information

Used simplifier rules: ⊦ ¬ m < n → ¬ m + m0 < n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: nused0, m, zbrar