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Proof step 51 from proof btree-write-node (previous step)

# asar0 = m + 1, # asar = m + 1, # zbrar = BRANCH_SIZE, zh[r] = mkznode(r0, zbrar, false, true, m + 1), zbrar[n] = mkzbranch(key, adr, r1), ((btree(r, r1, adr, height0, true, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok) * P0))(zh -- r), (( P * (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, true, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 )) )(zh -- r), n < BRANCH_SIZE, m < BRANCH_SIZE, btree-sub-clean(r, m + 1, zh), disj(asar0, 0, m + 1), disj(asar, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), r ∈ zh, n ≤ m, asar[n] ⊆ asar0[n], asar[n] ⊆ dom iast, ∀ n. ¬ m < n → asar[n] ⊆ dom iast, ∀ n. ¬ m < n → asar[n] ⊆ union(asar0, 0, m + 1) ++ adr0 ⊦ adr1 = adr0, (( btree (r, r1, adr, height0, false, key-range(zbrar, n, m + 1, ri), iast[adr1, save(mkznode(r0, zbrar, false, true, m + 1))], asar[n], asar0[n], ok) * P0) )(zh -- r), (( P * (λ n. ⌜zbrar[n].branch?⌝ ∧ btree (r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, m + 1, ri), iast[adr1, save(mkznode(r0, zbrar, false, true, m + 1))], asar[n], asar0[n], ok) )⊗_ (0, m + 1 )) )(zh -- r), adr0 ∈ union(asar, 0, m + 1), adr0 ∈ union(asar0, 0, m + 1), adr1 ∈ iast, adr1 ∈ union(asar0, 0, m + 1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 52:
⊦ iast0 ⊆ iast1 ∧ ((btree(rp, r0, adr, height, dirty0, ri, iast0, as, alladrs, nstat) * P))(zh0) → ((btree(rp, r0, adr, height, dirty0, ri, iast1, as, alladrs, nstat) * P))(zh0)
Proof step 53:
# zbrar = BRANCH_SIZE, # asar = m + 1, # asar0 = m + 1, zbrar[n] = mkzbranch(key, adr, r1), zh[r] = mkznode(r0, zbrar, false, true, m + 1), (( P * (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, true, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 )) )(zh -- r), ((btree(r, r1, adr, height0, true, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok) * P0))(zh -- r), m < BRANCH_SIZE, n < BRANCH_SIZE, btree-sub-clean(r, m + 1, zh), disj(asar, 0, m + 1), disj(asar0, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), r ∈ zh, n ≤ m, asar[n] ⊆ dom iast, asar[n] ⊆ asar0[n], ∀ n. ¬ m < n → asar[n] ⊆ union(asar0, 0, m + 1) ++ adr0, ∀ n. ¬ m < n → asar[n] ⊆ dom iast ⊦ adr1 = adr0, (( P * (λ n. ⌜zbrar[n].branch?⌝ ∧ btree (r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, m + 1, ri), iast[adr1, save(mkznode(r0, zbrar, false, true, m + 1))], asar[n], asar0[n], ok) )⊗_ (0, m + 1 )) )(zh -- r), (( btree (r, r1, adr, height0, false, key-range(zbrar, n, m + 1, ri), iast[adr1, save(mkznode(r0, zbrar, false, true, m + 1))], asar[n], asar0[n], ok) * P0) )(zh -- r), adr1 ∈ union(asar0, 0, m + 1), adr1 ∈ iast, adr0 ∈ union(asar0, 0, m + 1), adr0 ∈ union(asar, 0, m + 1), iast ⊆ iast[adr1, save(mkznode(r0, zbrar, false, true, m + 1))] ∧ ((btree(r, r1, adr, height0, false, key-range(zbrar, n, m + 1, ri), iast, asar[n], asar0[n], ok) * P0))(zh -- r)

Rule argument

spec: <toplevel> name: btree-istore-reduce seq: ⊦ iast0 ⊆ iast1 ∧ (( btree (rp, r0, adr, height, dirty0, ri, iast0, as, alladrs, nstat) * P) )(zh0) → (( btree (rp, r0, adr, height, dirty0, ri, iast1, as, alladrs, nstat) * P) )(zh0) subst: [iast0, zh0, P, nstat, alladrs, as, iast1, ri, dirty0, height, adr, r0, rp] → [iast, zh -- r, P0, ok, asar0[n], asar[n], iast[adr1, save(mkznode(r0, zbrar, false, true, m + 1))], key-range(zbrar, n, m + 1, ri), false, height0, adr, r1, r] flags: <all paths>

Additional information

Used terms: iast, zh -- r, P0, ok, asar0[n], asar[n], iast[adr1, save(mkznode(r0, zbrar, false, true, m + 1))], key-range(zbrar, n, m + 1, ri), false, height0, adr, r1, r