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Proof step 12 from proof unmodified-key-range-node (previous step)

# zbrar = BRANCH_SIZE, iast[adr] = indexnode(save(zbrar, 0, nused + 1), bv1, nused + 1), nused < BRANCH_SIZE, n < BRANCH_SIZE, adr ∈ iast, n ≤ nused ⊦ ( key ∈ (n = 0 ⊃ (nused = 0 ⊃ ri;ri |≤ zbrar[0].key);(n = nused ⊃ ri |> zbrar[nused - 1].key;(ri |> zbrar[n - 1].key) |≤ zbrar[n].key)) ↔ key ∈ key-range(zbrar, n, nused + 1, ri)) ∧ ( key ∈ (n = 0 ⊃ (nused = 0 ⊃ ri;ri |≤ zbrar[0].key);(n = nused ⊃ ri |> zbrar[nused - 1].key;(ri |> zbrar[n - 1].key) |≤ zbrar[n].key)) → (n = 0 ⊃ (nused = 0 ⊃ ri;ri |≤ zbrar[0].key);(n = nused ⊃ ri |> zbrar[nused - 1].key;(ri |> zbrar[n - 1].key) |≤ zbrar[n].key))[key] = key-range(zbrar, n, nused + 1, ri)[key])

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: range seq: ⊦ n < nused ∧ nused ≤ # zbrar → key-range(zbrar, n, nused, ri) = (n = 0 ⊃ (nused = 1 ⊃ ri;ri |≤ zbrar[0].key) ;(n + 1 = nused ⊃ ri |> zbrar[n - 1].key ;(ri |> zbrar[n - 1].key) |≤ zbrar[n].key ) ) subst: [ri, nused, n, zbrar] → [ri, nused + 1, n, zbrar] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m + 1 ≤ n ↔ m < n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ 1 = m + 1 ↔ m = 0; (s)% ⊦ m + n = m + n0 ↔ n = n0; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat% ⊦ ¬ n < n; (s) Used terms: ri, nused + 1, n, zbrar