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Proof step 44 from proof btree-clean-alladrs (previous step)

# zbrar = BRANCH_SIZE, # asar = m + 1, # asar0 = m + 1, m = n0 + n, (( ( ( ( btree(r0, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok) * (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, n )) * (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) )⊗_ (1 + n, n0 )) * r0 => mkznode(rp, zbrar, false, false, m + 1)) * P) )(zh0), zbrar[n].branch?, n < BRANCH_SIZE, 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, adr0 ∈ asar0[n], n ≤ m, asar[n] ⊆ asar0[n], asar[n] ⊆ dom 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 ⊦ # union(asar0, 0, m + 1) = 0, # asar0[n] = 0, asar0[n] = ∅, union(asar0, 0, m + 1) = ∅, adr0 ∈ asar[n], adr0 ∈ union(asar, 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 45:
# zbrar = BRANCH_SIZE, # asar = (n0 + n) + 1, # asar0 = # asar, (( ( ( ( btree(r0, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, # asar, ri), iast, asar[n], asar0[n], ok) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree(r0, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar, ri), iast, asar[n1], asar0[n1], ok) )⊗_ (0, n )) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree(r0, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar, ri), iast, asar[n1], asar0[n1], ok) )⊗_ (1 + n, n0 )) * r0 => mkznode(rp, zbrar, false, false, # asar)) * P) )(zh0), zbrar[n].branch?, n < BRANCH_SIZE, n0 + n < BRANCH_SIZE, disj(asar, 0, # asar), disj(asar0, 0, # asar), nsize(# asar, nstat), sorted(zbrar, 0, n0 + n), unmodified(adr, mkznode(rp, zbrar, false, false, # asar), iast), adr ∈ iast, adr0 ∈ asar0[n], asar[n] ⊆ asar0[n], asar[n] ⊆ dom 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, ∀ n1. ¬ n0 + n < n1 → asar[n1] ⊆ union(asar0, 0, # asar) ++ adr, ∀ n1. ¬ n0 + n < n1 → asar[n1] ⊆ dom iast ⊦ # union(asar0, 0, # asar) = 0, # asar0[n] = 0, asar0[n] = ∅, union(asar0, 0, # asar) = ∅, adr0 ∈ asar[n], adr0 ∈ union(asar, 0, # asar), adr ∈ union(asar0, 0, # asar), adr ∈ union(asar, 0, # asar)

Rule argument

Additional information

Used simplifier rules: size-zero-01: ⊦ as ≠ ∅ → # as ≠ 0; (forward) from specification set-basic[node-store]%size-zero: ⊦ # as ≠ 0 → as ≠ ∅; (forward) from specification set-basic[node-store]%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m ≤ m + n; (s)