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Proof step 24 from proof btree-write-node (previous step)

((r => mkznode(r0, zbrar, false, true, m0) * bnode(r, zbrar, 0, m0, m0, height0 + 1, true, ri, iast, as, alladrs -- adr0) * P))(zh), btree-sub-clean(r, m0, zh), nsize(m0, nstat), adr0 ∈ alladrs, as ⊆ alladrs, as ⊆ dom iast ⊦ # alladrs = 0, alladrs = ∅, adr0 ∈ as, adr1 ∈ alladrs, adr1 ∈ iast, m0 ≤ # zbrar ∧ # zbrar = BRANCH_SIZE ∧ m0 ≠ 0 ∧ sorted(zbrar, 0, m0 - 1) ∧ (∃ asar, asar0. # asar = # asar0 ∧ disj(asar, 0, # asar0) ∧ union(asar, 0, # asar0) = as -- adr1 ∧ # asar0 = m0 ∧ disj(asar0, 0, # asar0) ∧ union(asar0, 0, # asar0) = alladrs -- adr0 -- adr1 ∧ (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree (r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, # asar0, ri), iast[adr1, save(mkznode(r0, zbrar, false, true, # asar0))], asar[n], asar0[n], ok) )⊗_ (0, # asar0 ) * r =>_) * P) )(zh))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 25:
⊦ bnode(rp, zbrar, m, len, nused, height + 1, dirty, ri, iast, as, alladrs) = ( ⌜m + len ≤ nused ∧ nused ≤ # zbrar ∧ # zbrar = BRANCH_SIZE⌝ ∧ ⌜nused ≠ 0 ∧ sorted(zbrar, 0, nused - 1)⌝ ∧ _∃ (λ asar. _∃ (λ asar0. ⌜ # asar = nused ∧ disj(asar, 0, nused) ∧ union(asar, m, len) = as ∧ # asar0 = nused ∧ disj(asar0, 0, nused) ∧ union(asar0, m, len) = alladrs ⌝ ∧ (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(rp, zbrar[n].child, zbrar[n].adr, height, dirty, key-range(zbrar, n, nused, ri), iast, asar[n], asar0[n], ok) )⊗_ (m, len ))))
Proof step 26:
(( r => mkznode(r0, zbrar, false, true, m0) * ( ⌜0 + m0 ≤ m0 ∧ m0 ≤ # zbrar ∧ # zbrar = BRANCH_SIZE⌝ ∧ ⌜m0 ≠ 0 ∧ sorted(zbrar, 0, m0 - 1)⌝ ∧ _∃ (λ asar. _∃ (λ asar0. ⌜ # asar = m0 ∧ disj(asar, 0, m0) ∧ union(asar, 0, m0) = as ∧ # asar0 = m0 ∧ disj(asar0, 0, m0) ∧ union(asar0, 0, m0) = alladrs -- adr0 ⌝ ∧ (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, true, key-range(zbrar, n, m0, ri), iast, asar[n], asar0[n], ok) )⊗_ (0, m0 )))) * P) )(zh), btree-sub-clean(r, m0, zh), nsize(m0, nstat), adr0 ∈ alladrs, as ⊆ dom iast, as ⊆ alladrs ⊦ # alladrs = 0, alladrs = ∅, adr1 ∈ iast, adr1 ∈ alladrs, adr0 ∈ as, m0 ≤ # zbrar ∧ # zbrar = BRANCH_SIZE ∧ m0 ≠ 0 ∧ sorted(zbrar, 0, m0 - 1) ∧ (∃ asar, asar0. # asar = # asar0 ∧ disj(asar, 0, # asar0) ∧ union(asar, 0, # asar0) = as -- adr1 ∧ # asar0 = m0 ∧ disj(asar0, 0, # asar0) ∧ union(asar0, 0, # asar0) = alladrs -- adr0 -- adr1 ∧ (( ( (λ n. ⌜zbrar[n].branch?⌝ ∧ btree (r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, # asar0, ri), iast[adr1, save(mkznode(r0, zbrar, false, true, # asar0))], asar[n], asar0[n], ok) )⊗_ (0, # asar0 ) * r =>_) * P) )(zh))

Rule argument

spec: <toplevel> name: bnode seq: ⊦ bnode (rp, zbrar, m, len, nused, height + 1, dirty, ri, iast, as, alladrs) = ( ⌜ m + len ≤ nused ∧ nused ≤ # zbrar ∧ # zbrar = BRANCH_SIZE ⌝ ∧ ⌜nused ≠ 0 ∧ sorted(zbrar, 0, nused - 1)⌝ ∧ _∃ (λ asar. _∃ (λ asar0. ⌜ # asar = nused ∧ disj(asar, 0, nused) ∧ union(asar, m, len) = as ∧ # asar0 = nused ∧ disj(asar0, 0, nused) ∧ union(asar0, m, len) = alladrs ⌝ ∧ (λ n. ⌜zbrar[n].branch?⌝ ∧ btree (rp, zbrar[n].child, zbrar[n].adr, height, dirty, key-range(zbrar, n, nused, ri), iast, asar[n], asar0[n], ok) )⊗_ (m, len )))) subst: [alladrs, as, iast, ri, dirty, height, nused, len, m, zbrar, rp] → [alladrs -- adr0, as, iast, ri, true, height0, m0, m0, 0, zbrar, r] flags: <all paths>

Additional information

Used terms: alladrs -- adr0, as, iast, ri, true, height0, m0, m0, 0, zbrar, r