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Proof step 38 from proof btree-refed-subseteq-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, fi), 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, fi), 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, fi), iast, asar[n], asar0[n], ok) )⊗_ (1 + n, n0 )) * r0 => mkznode(rp, zbrar, false, false, m + 1)) * P) )(zh0), zbrar[n].branch?, m < BRANCH_SIZE, n < 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), n ≤ m, (r = null ⊃ P(zh);r0 = r ∧ zh0 = zh), ∀ rp, r, adr, dirty, fi, iast, as, alladrs, nstat, P, zh. ((btree(rp, r, adr, height0, dirty, fi, iast, as, alladrs, nstat) * P))(zh) → as ⊆ alladrs ⊦ adr ∈ union(asar, 0, m + 1), adr ∈ union(asar0, 0, m + 1), asar[n] ⊆ union(asar0, 0, m + 1), union(asar, 0, m + 1) ++ adr ⊆ union(asar0, 0, m + 1), asar[n] ⊆ union(asar0, 0, m + 1) ++ adr

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 39:
# zbrar = BRANCH_SIZE, # asar = (n0 + n) + 1, # asar0 = # asar, (( ( ( ( btree(r0, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, # asar, fi), 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, fi), 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, fi), iast, asar[n1], asar0[n1], ok) )⊗_ (1 + n, n0 )) * r0 => mkznode(rp, zbrar, false, false, # asar)) * P) )(zh0), zbrar[n].branch?, n0 + n < BRANCH_SIZE, 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), (r = null ⊃ P(zh);r0 = r ∧ zh0 = zh), ∀ rp, r, adr, dirty, fi, iast, as, alladrs, nstat, P, zh. ((btree(rp, r, adr, height0, dirty, fi, iast, as, alladrs, nstat) * P))(zh) → as ⊆ alladrs ⊦ adr ∈ union(asar, 0, # asar), adr ∈ union(asar0, 0, # asar), asar[n] ⊆ union(asar0, 0, # asar), union(asar, 0, # asar) ++ adr ⊆ union(asar0, 0, # asar), asar[n] ⊆ union(asar0, 0, # asar) ++ adr

Rule argument

Additional information

Used simplifier rules: 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)