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Proof step 40 from proof btree-adrs (previous step)

# zbrar = BRANCH_SIZE, # asar = m + 1, # asar0 = m + 1, union(asar0, 0, m + 1) = alladrs -- adr0, (( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, m + 1, fi), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * P) )(zh1), m < BRANCH_SIZE, ((r => mkznode(rp, zbrar, false, false, m + 1)))(zh2), disj(asar, 0, m + 1), disj(asar0, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), unmodified(adr0, mkznode(rp, zbrar, false, false, m + 1), iast), adr0 ∈ alladrs, adr0 ∈ dom iast, zh2 ⊥ zh1, (rp0 = null ⊃ P(zh);r = rp0 ∧ zh2 ∪ zh1 = zh), ∀ rp, rp0, adr0, dirty, fi, iast, as, alladrs, nstat, P, zh. ((btree(rp, rp0, adr0, height0, dirty, fi, iast, as, alladrs, nstat) * P))(zh) → as ⊆ dom iast, ∀ n. ¬ m < n → asar[n] ⊆ alladrs ⊦ # alladrs = 0, alladrs = ∅, adr0 ∈ union(asar, 0, m + 1), alladrs ⊆ dom iast, union(asar, 0, m + 1) ⊆ dom iast

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 41:
# asar0 = m + 1, # asar = m + 1, # zbrar = BRANCH_SIZE, union(asar0, 0, m + 1) = alladrs -- adr0, (( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, m + 1, fi), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * P) )(zh1), m < BRANCH_SIZE, n < BRANCH_SIZE, ((r => mkznode(rp, zbrar, false, false, m + 1)))(zh2), disj(asar0, 0, m + 1), disj(asar, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), unmodified(adr0, mkznode(rp, zbrar, false, false, m + 1), iast), adr0 ∈ alladrs, adr0 ∈ iast, n ≤ m, zh2 ⊥ zh1, (rp0 = null ⊃ P(zh);r = rp0 ∧ zh2 ∪ zh1 = zh), ∀ n. ¬ m < n → asar[n] ⊆ alladrs, ∀ rp, rp0, adr0, dirty, fi, iast, as, alladrs, nstat, P, zh. ((btree(rp, rp0, adr0, height0, dirty, fi, iast, as, alladrs, nstat) * P))(zh) → as ⊆ dom iast ⊦ # alladrs = 0, alladrs = ∅, adr0 ∈ union(asar, 0, m + 1), alladrs ⊆ dom iast, asar[n] ⊆ dom iast

Rule argument

Additional information

Used simplifier rules: lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%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)%⊦ union(asar, n, m) ⊆ as ↔ (∀ n0. n ≤ n0 ∧ n0 < n + m → asar[n0] ⊆ as); (s)% ⊦ adr ∈ dom iast ↔ adr ∈ iast; (s)