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Proof step 27 from proof btree-mark-dirty (previous step)

zh[r] = mkznode(rp0, zbrar, false, false, height0), ((bnode(r, zbrar, 0, height0, height0, height1 + 1, false, ri, iast, alladrs -- adr, alladrs -- adr) * P))(zh -- r), nsize(height0, nstat), unmodified(adr, mkznode(rp0, zbrar, false, false, height0), iast), r ∈ zh, adr ∈ iast, adr ∈ alladrs, alladrs ⊆ dom iast, ∀ zh, r, rp0, zbrar, boolvar, height0, rp, adr, ri, iast, alladrs, nstat, P. zh[r] = mkznode(rp0, zbrar, boolvar, false, height0) ∧ ((btree(rp, r, adr, height1, false, ri, iast, alladrs, alladrs, nstat) * P))(zh) ∧ r ∈ zh ∧ adr ∈ alladrs ∧ adr ∈ iast ∧ alladrs ⊆ dom iast ∧ (height1 = 0 → alladrs = ∅ ++ adr) → # alladrs = 0 ∨ alladrs = ∅ ∨ ((btree(rp, r, adr, height1, true, ri, iast, alladrs -- adr, alladrs, nstat) * P))(zh[r, mkznode(rp0, zbrar, boolvar, true, height0)]) ⊦ # alladrs = 0, alladrs = ∅, ((bnode(r, zbrar, 0, height0, height0, height1 + 1, true, ri, iast, alladrs -- adr, alladrs -- adr) * P))(zh -- r)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 28:
⊦ 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 29:
zh[r] = mkznode(rp0, zbrar, false, false, height0), (( ( ⌜0 + height0 ≤ height0 ∧ height0 ≤ # zbrar ∧ # zbrar = BRANCH_SIZE⌝ ∧ ⌜height0 ≠ 0 ∧ sorted(zbrar, 0, height0 - 1)⌝ ∧ _∃ (λ asar. _∃ (λ asar0. ⌜ # asar = height0 ∧ disj(asar, 0, height0) ∧ union(asar, 0, height0) = alladrs -- adr ∧ # asar0 = height0 ∧ disj(asar0, 0, height0) ∧ union(asar0, 0, height0) = alladrs -- adr ⌝ ∧ (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height1, false, key-range(zbrar, n, height0, ri), iast, asar[n], asar0[n], ok) )⊗_ (0, height0 )))) * P) )(zh -- r), nsize(height0, nstat), unmodified(adr, mkznode(rp0, zbrar, false, false, height0), iast), adr ∈ alladrs, adr ∈ iast, r ∈ zh, alladrs ⊆ dom iast, ∀ zh, r, rp0, zbrar, boolvar, height0, rp, adr, ri, iast, alladrs, nstat, P. zh[r] = mkznode(rp0, zbrar, boolvar, false, height0) ∧ ((btree(rp, r, adr, height1, false, ri, iast, alladrs, alladrs, nstat) * P))(zh) ∧ r ∈ zh ∧ adr ∈ alladrs ∧ adr ∈ iast ∧ alladrs ⊆ dom iast ∧ (height1 = 0 → alladrs = ∅ ++ adr) → # alladrs = 0 ∨ alladrs = ∅ ∨ ((btree(rp, r, adr, height1, true, ri, iast, alladrs -- adr, alladrs, nstat) * P))(zh[r, mkznode(rp0, zbrar, boolvar, true, height0)]) ⊦ # alladrs = 0, alladrs = ∅, ((bnode(r, zbrar, 0, height0, height0, height1 + 1, true, ri, iast, alladrs -- adr, alladrs -- adr) * P))(zh -- r)

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 -- adr, alladrs -- adr, iast, ri, false, height1, height0, height0, 0, zbrar, r] flags: <all paths>

Additional information

Used terms: alladrs -- adr, alladrs -- adr, iast, ri, false, height1, height0, height0, 0, zbrar, r