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Proof step 8 from proof unmodified-key-range-node (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
# zbrar = BRANCH_SIZE, iast[adr] = indexnode(brar, bv1, nused + 1), indexnode(brar, bv1, nused + 1) = save(mkznode(rp, zbrar, bv1, false, nused + 1)), n < BRANCH_SIZE, nused < BRANCH_SIZE, adr ∈ iast, n ≤ nused ⊦ ( key ∈ (n = 0 ⊃ (nused = 0 ⊃ ri;ri |≤ load(brar[0]).key) ;(n = nused ⊃ ri |> load(brar[nused - 1]).key;(ri |> load(brar[n - 1]).key) |≤ load(brar[n]).key) ) ↔ key ∈ key-range(zbrar, n, nused + 1, ri)) ∧ ( key ∈ (n = 0 ⊃ (nused = 0 ⊃ ri;ri |≤ load(brar[0]).key) ;(n = nused ⊃ ri |> load(brar[nused - 1]).key;(ri |> load(brar[n - 1]).key) |≤ load(brar[n]).key) ) → (n = 0 ⊃ (nused = 0 ⊃ ri;ri |≤ load(brar[0]).key) ;(n = nused ⊃ ri |> load(brar[nused - 1]).key;(ri |> load(brar[n - 1]).key) |≤ load(brar[n]).key) ) [key ] = key-range(zbrar, n, nused + 1, ri)[key])

Rule argument

Additional information

Used simplifier rules: ⊦ key0 ∈ ri |> key1 ↔ key0 ∈ ri ∧ key1 < key0; (s)% ⊦ key0 ∈ ri |≤ key1 ↔ key0 ∈ ri ∧ ¬ key1 < key0; (s)% ⊦ m < n → ¬ n < m; (s)% ⊦ n ≠ n0 → (n < n0 ↔ ¬ n0 < n); (ss)% ⊦ m ≠ n → (m < n + 1 ↔ m < n); (s)% ⊦ ¬ m + n < m; (s)% ⊦ m < n → n ≠ 0; (s)%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)% ⊦ m < m + n ↔ n ≠ 0; (s)%n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s)% ⊦ m < 1 ↔ m = 0; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ m + n = m + n0 ↔ n = n0; (s)%n < nused, nused ≤ BRANCH_SIZE ⊦ load(brar, 0, nused)[n] = load(brar[n]); (s)% ⊦ m + 1 ≠ 0; (s)% ⊦ 0 < m ↔ m ≠ 0; (s)% ⊦ 1 = m + 1 ↔ m = 0; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ # load(brar, m, nused) = BRANCH_SIZE; (s)