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Proof step 34 from proof btree-refed-subseteq-alladrs (previous step)

# 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, fi), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * r0 => mkznode(rp, zbrar, false, false, m + 1)) * P) )(zh0), n < BRANCH_SIZE, m < 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), 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(asar0, 0, m + 1), adr ∈ union(asar, 0, m + 1), asar[n] ⊆ union(asar0, 0, m + 1) ++ adr, union(asar, 0, m + 1) ++ adr ⊆ union(asar0, 0, m + 1), asar[n] ⊆ union(asar0, 0, m + 1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 35:
⊦ m0 < m → nP0 ⊗_(n, m) = nP0 ⊗_(n, m0) * nP0(n + m0) * nP0 ⊗_(n + m0 + 1, m - m0 - 1)
Proof step 36:
# 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, fi), iast, asar[n], asar0[n], ok) )⊗_ (0, n ) * ( ⌜zbrar[0 + n].branch?⌝ ∧ btree(r0, zbrar[0 + n].child, zbrar[0 + n].adr, height0, false, key-range(zbrar, 0 + n, m + 1, fi), iast, asar[0 + n], asar0[0 + 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 + 1, m + 1 - n - 1 )) * r0 => mkznode(rp, zbrar, false, false, m + 1)) * P) )(zh0), m < BRANCH_SIZE, n < BRANCH_SIZE, n < m + 1, 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

Rule argument

spec: iter-n, inst: zheap name: Sep-unfold seq: ⊦ m0 < m → nP0 ⊗_(n, m) = nP0 ⊗_(n, m0) * nP0(n + m0) * nP0 ⊗_(n + m0 + 1, m - m0 - 1) subst: [m0, m, n, nP0] → [n, m + 1, 0, 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)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: n, m + 1, 0, 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)