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Proof step 8 from proof in-playload-data (previous step)

n + m0 < # zbrar, sorted(zbrar, 0, n + m0 + 1), ∀ zbrar, n, adr. sorted(zbrar, 0, n + m0) ∧ n + m0 ≤ # zbrar → (adr ∈ payload(zbrar, n, m0) ↔ (∃ m1. n ≤ m1 ∧ m1 < n + m0 ∧ zbrar[m1].entry? ∧ zbrar[m1].adr = adr)) ⊦ adr ∈ payload(zbrar, n, m0 + 1) ↔ (∃ m. n ≤ m ∧ ¬ n + m0 < m ∧ zbrar[m].entry? ∧ zbrar[m].adr = adr)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ 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 10:
n + m0 < # zbrar, n < # zbrar, sorted(zbrar, 0, n + m0 + 1), ∀ zbrar, n, adr. sorted(zbrar, 0, n + m0) ∧ n + m0 ≤ # zbrar → (adr ∈ payload(zbrar, n, m0) ↔ (∃ m1. n ≤ m1 ∧ m1 < n + m0 ∧ zbrar[m1].entry? ∧ zbrar[m1].adr = adr)) ⊦ adr ∈ (zbrar[n].entry? ⊃ payload(zbrar, n + 1, m0)[zbrar[n].key, zbrar[n].adr];payload(zbrar, n + 1, m0)) ↔ (∃ m. n ≤ m ∧ ¬ n + m0 < m ∧ zbrar[m].entry? ∧ zbrar[m].adr = 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] → [m0, n, 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: m0, n, zbrar