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Proof step 26 from proof btree-adrs (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 27:
⊦ m0 < m → nP0 ⊗_(n, m) = nP0 ⊗_(n, m0) * nP0(n + m0) * nP0 ⊗_(n + m0 + 1, m - m0 - 1)
Proof step 28:
# zbrar = BRANCH_SIZE, # asar = m + 1, # asar0 = m + 1, union(asar0, 0, m + 1) = alladrs -- adr0, (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(rp0, 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(rp0, 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(rp0, 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 )) * P) )(zh1), n < BRANCH_SIZE, m < BRANCH_SIZE, n < m + 1, ((rp0 => mkznode(rp, zbrar, false, true, m + 1)))(zh2), disj(asar, 0, m + 1), disj(asar0, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), adr0 ∈ alladrs, n ≤ m, zh2 ⊥ zh1, ∀ 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 = ∅, rp0 = null, adr0 ∈ union(asar, 0, m + 1), asar[n] ⊆ dom iast, alladrs ⊆ dom iast

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 (rp0, zbrar[n].child, zbrar[n].adr, height0, true, 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 (rp0, zbrar[n].child, zbrar[n].adr, height0, true, key-range(zbrar, n, m + 1, fi), iast, asar[n], asar0[n], ok)