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Proof step 28 from proof btree-clean-alladrs (previous step)

# zbrar = BRANCH_SIZE, # asar = m + 1, # asar0 = m + 1, (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r0, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * r0 => mkznode(rp, zbrar, false, false, m + 1)) * P) )(zh0), m < BRANCH_SIZE, disj(asar, 0, m + 1), disj(asar0, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), unmodified(adr, mkznode(rp, zbrar, false, false, m + 1), iast), adr ∈ iast, (r = null ⊃ P(zh);r0 = r ∧ zh0 = zh), ∀ rp, r, adr, ri, iast, as, alladrs, nstat, P, zh. ((btree(rp, r, adr, height0, false, ri, iast, as, alladrs, nstat) * P))(zh) → as = alladrs, ∀ n. ¬ m < n → asar[n] ⊆ union(asar0, 0, m + 1) ++ adr, ∀ n. ¬ m < n → asar[n] ⊆ dom iast, ∃ n0. 0 ≤ n0 ∧ n0 < 0 + m + 1 ∧ adr0 ∈ asar[n0] ⊦ # union(asar, 0, m + 1) = 0, union(asar, 0, m + 1) = ∅, adr0 ∈ union(asar0, 0, m + 1), adr ∈ union(asar0, 0, m + 1), adr ∈ union(asar, 0, m + 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 29:
# asar0 = m + 1, # asar = m + 1, # zbrar = BRANCH_SIZE, (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r0, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * r0 => mkznode(rp, zbrar, false, false, m + 1)) * P) )(zh0), m < BRANCH_SIZE, n < BRANCH_SIZE, disj(asar0, 0, m + 1), disj(asar, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), unmodified(adr, mkznode(rp, zbrar, false, false, m + 1), iast), adr0 ∈ asar[n], adr ∈ iast, n ≤ m, (r = null ⊃ P(zh);r0 = r ∧ zh0 = zh), ∀ n. ¬ m < n → asar[n] ⊆ dom iast, ∀ n. ¬ m < n → asar[n] ⊆ union(asar0, 0, m + 1) ++ adr, ∀ rp, r, adr, ri, iast, as, alladrs, nstat, P, zh. ((btree(rp, r, adr, height0, false, ri, iast, as, alladrs, nstat) * P))(zh) → as = alladrs ⊦ # asar[n] = 0, # union(asar, 0, m + 1) = 0, asar[n] = ∅, union(asar, 0, m + 1) = ∅, adr ∈ union(asar, 0, m + 1), adr ∈ union(asar0, 0, m + 1), adr0 ∈ union(asar0, 0, m + 1)

Rule argument

Additional information

Used simplifier rules: size-zero-01: ⊦ as ≠ ∅ → # as ≠ 0; (forward) from specification set-basic[node-store]%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%ne-01: ⊦ adr ∈ as → as ≠ ∅; (forward) from specification set-basic[node-store]%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ 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% ⊦ 0 ≤ m; (s)