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Proof step 42 from proof payload-shift-left (previous step)

zbrar[n] = mkzentry(key, adr), n < # zbrar, nused < # zbrar, sorted(zbrar, 0, nused + 1), n ≤ nused ⊦ (∃ n0. n0 < nused ∧ (n0 < n ⊃ zbrar[n0];zbrar[n0 + 1]).entry? ∧ (n0 < n ⊃ zbrar[n0];zbrar[n0 + 1]).key = key0) ↔ key0 ≠ key ∧ key0 ∈ payload(zbrar, 0, nused + 1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 43:
m + nused ≤ # zbrar ⊦ key ∈ payload(zbrar, m, nused) ↔ (∃ n. n < nused ∧ zbrar[m + n].entry? ∧ zbrar[m + n].key = key)
Proof step 44:
zbrar[n] = mkzentry(key, adr), nused < # zbrar, n < # zbrar, sorted(zbrar, 0, nused + 1), n ≤ nused, 0 + nused + 1 ≤ # zbrar ⊦ (∃ n0. n0 < nused ∧ (n0 < n ⊃ zbrar[n0];zbrar[n0 + 1]).entry? ∧ (n0 < n ⊃ zbrar[n0];zbrar[n0 + 1]).key = key0) ↔ key0 ≠ key ∧ (∃ n. n < nused + 1 ∧ zbrar[0 + n].entry? ∧ zbrar[0 + n].key = key0)

Rule argument

spec: <toplevel> name: in-payload seq: m + nused ≤ # zbrar ⊦ key ∈ payload(zbrar, m, nused) ↔ (∃ n. n < nused ∧ zbrar[m + n].entry? ∧ zbrar[m + n].key = key) subst: [nused, m, zbrar, key] → [nused + 1, 0, zbrar, key0] flags: <all paths>

Additional information

Used simplifier rules: a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s) Used terms: nused + 1, 0, zbrar, key0