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Proof step 11 from proof unmodified-key-range-node (previous step)

# zbrar = BRANCH_SIZE, iast[adr] = indexnode(brar, bv1, nused + 1), indexnode(brar, bv1, nused + 1) = indexnode (save(mkznode(rp, zbrar, bv1, false, nused + 1).zbranches, 0, mkznode(rp, zbrar, bv1, false, nused + 1).usedsize), mkznode(rp, zbrar, bv1, false, nused + 1).leaf, mkznode(rp, zbrar, bv1, false, nused + 1).usedsize), nused < BRANCH_SIZE, n < 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])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 12:
# 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])

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)% ⊦ load(save(zbrar[n])).key = zbrar[n].key; (s)%n < nused, nused ≤ BRANCH_SIZE ⊦ save(zbrar, 0, nused)[n] = save(zbrar[n]); (s)% ⊦ m + 1 ≤ n ↔ m < n; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)% ⊦ 0 < m ↔ m ≠ 0; (s)%⊦ indexnode(brar, boolvar, m) = indexnode(brar0, boolvar0, m0) ↔ brar = brar0 ∧ (boolvar ↔ boolvar0) ∧ m = m0; (ws)% ⊦ mkznode(r, zbrar, boolvar, boolvar0, m).leaf ↔ boolvar; (s)% ⊦ mkznode(r, zbrar, boolvar, boolvar0, m).usedsize = m; (s)% ⊦ mkznode(r, zbrar, boolvar, boolvar0, m).zbranches = zbrar; (s)