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Proof step 6 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)), nused < BRANCH_SIZE, n < BRANCH_SIZE, adr ∈ iast, n ≤ nused ⊦ (key ∈ key-range(load(brar, 0, nused + 1), n, nused + 1, ri) ↔ key ∈ key-range(zbrar, n, nused + 1, ri)) ∧ ( key ∈ key-range(load(brar, 0, nused + 1), n, nused + 1, ri) → key-range(load(brar, 0, nused + 1), n, nused + 1, ri)[key] = key-range(zbrar, n, nused + 1, ri)[key])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 7:
⊦ 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))
Proof step 8:
# 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])

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, load(brar, 0, nused + 1)] flags: <all paths>

Additional information

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