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Proof step 25 from proof btree-refed-subseteq-alladrs (previous step)

# zbrar = BRANCH_SIZE, # asar = m + 1, # asar0 = m + 1, (( ( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, true, key-range(zbrar, n, m + 1, fi), iast, asar[n], asar0[n], ok) )⊗_ (0, n ) * ( ⌜zbrar[0 + n].branch?⌝ ∧ btree(r, zbrar[0 + n].child, zbrar[0 + n].adr, height0, true, key-range(zbrar, 0 + n, m + 1, fi), iast, asar[0 + n], asar0[0 + n], ok)) * (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, true, key-range(zbrar, n, m + 1, fi), iast, asar[n], asar0[n], ok) )⊗_ (0 + n + 1, m + 1 - n - 1 )) * r => mkznode(rp, zbrar, false, true, m + 1)) * P) )(zh0), n0 < BRANCH_SIZE, n < BRANCH_SIZE, m < BRANCH_SIZE, n < m + 1, disj(asar, 0, m + 1), disj(asar0, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), n0 ≤ m, n ≤ m, ∀ 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[n0] ⊆ union(asar0, 0, m + 1), asar[n] ⊆ union(asar0, 0, m + 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 26:
# asar0 = m + 1, # asar = m + 1, # zbrar = BRANCH_SIZE, (( ( ( ( btree(r, zbrar[n].child, zbrar[n].adr, height0, true, key-range(zbrar, n, m + 1, fi), iast, asar[n], asar0[n], ok) * (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, true, key-range(zbrar, n, m + 1, fi), iast, asar[n], asar0[n], ok) )⊗_ (0, n )) * (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, true, key-range(zbrar, n, m + 1, fi), iast, asar[n], asar0[n], ok) )⊗_ (1 + n, m - n )) * r => mkznode(rp, zbrar, false, true, m + 1)) * P) )(zh0), zbrar[n].branch?, m < BRANCH_SIZE, n < BRANCH_SIZE, n0 < BRANCH_SIZE, disj(asar0, 0, m + 1), disj(asar, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), n ≤ m, n0 ≤ m, ∀ 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), asar[n0] ⊆ union(asar0, 0, m + 1)

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜bv⌝ (zh) ↔ bv; (s)% ⊦ ((P ∧ P2))(zh) ↔ P(zh) ∧ P2(zh); (s)%sep-com: ⊦ P * P2 = P2 * P; from specification heap-sep[zheap]%sep-ass: ⊦ (P * P3) * P4 = P * P3 * P4; from specification heap-sep[zheap]%and-com: ⊦ (P ∧ P2) = (P2 ∧ P); from specification heap-sep[zheap]%and-ass: ⊦ ((P ∧ P3) ∧ P4) = (P ∧ P3 ∧ P4); from specification heap-sep[zheap]% ⊦ (⌜bv⌝ ∧ P) * P2 = (⌜bv⌝ ∧ P * P2); (s)% ⊦ n ≤ m → m + n0 - n = (m - n) + n0; (s)% ⊦ n + 0 = n; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)