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Proof step 16 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 = ∅, (( btree(r0, r, adr1, height0 + 1, false, ri, iast[adr1, save(mkznode(r0, zbrar, false, true, m0))], as ++ adr1, (alladrs -- adr0) ++ adr1, nstat) * P) )(zh[r, mkznode(r0, zbrar, false, false, m0)]), adr0 ∈ as, adr1 ∈ alladrs, adr1 ∈ iast

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 17:
⊦ ((btree(rp, r0, adr, height, dirty0, ri, iast, as, alladrs, nstat) * P))(zh0) ↔ (∃ r, zh, zbrar, dirty, nused. (r0 = null ⊃ ¬ dirty ∧ P(zh0);r = r0 ∧ zh = zh0) ∧ (dirty → dirty0) ∧ (adr ∈ as ↔ ¬ dirty) ∧ adr ∈ alladrs ∧ nsize(nused, nstat) ∧ unmodified(adr, mkznode(rp, zbrar, height = 0, dirty, nused), iast) ∧ (( r => mkznode(rp, zbrar, height = 0, dirty, nused) * bnode(r, zbrar, 0, nused, nused, height, dirty, ri, iast, as -- adr, alladrs -- adr) * P) )(zh))
Proof step 18:
((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 ⊆ dom iast, as ⊆ alladrs ⊦ # alladrs = 0, alladrs = ∅, adr1 ∈ iast, adr1 ∈ alladrs, adr0 ∈ as, ∃ r1, zh1, zbrar0, dirty, nused. (r = null ⊃ ¬ dirty ∧ P(zh[r, mkznode(r0, zbrar, false, false, m0)]);r1 = r ∧ zh1 = zh[r, mkznode(r0, zbrar, false, false, m0)]) ∧ (dirty → false) ∧ (adr1 ∈ as ++ adr1 ↔ ¬ dirty) ∧ adr1 ∈ (alladrs -- adr0) ++ adr1 ∧ nsize(nused, nstat) ∧ unmodified(adr1, mkznode(r0, zbrar0, height0 + 1 = 0, dirty, nused), iast[adr1, save(mkznode(r0, zbrar, false, true, m0))]) ∧ (( r1 => mkznode(r0, zbrar0, height0 + 1 = 0, dirty, nused) * bnode (r1, zbrar0, 0, nused, nused, height0 + 1, dirty, ri, iast[adr1, save(mkznode(r0, zbrar, false, true, m0))], (as ++ adr1) -- adr1, ((alladrs -- adr0) ++ adr1) -- adr1) * P) )(zh1)

Rule argument

spec: <toplevel> name: btree seq: ⊦ (( btree (rp, r0, adr, height, dirty0, ri, iast, as, alladrs, nstat) * P) )(zh0) ↔ (∃ r, zh, zbrar, dirty, nused. (r0 = null ⊃ ¬ dirty ∧ P(zh0);r = r0 ∧ zh = zh0) ∧ (dirty → dirty0) ∧ (adr ∈ as ↔ ¬ dirty) ∧ adr ∈ alladrs ∧ nsize(nused, nstat) ∧ unmodified (adr, mkznode(rp, zbrar, height = 0, dirty, nused), iast) ∧ (( r => mkznode(rp, zbrar, height = 0, dirty, nused) * bnode (r, zbrar, 0, nused, nused, height, dirty, ri, iast, as -- adr, alladrs -- adr) * P) )(zh)) subst: [zh0, P, nstat, alladrs, as, iast, ri, dirty0, height, adr, r0, rp] → [zh[r, mkznode(r0, zbrar, false, false, m0)], P, nstat, (alladrs -- adr0) ++ adr1, as ++ adr1, iast[adr1, save(mkznode(r0, zbrar, false, true, m0))], ri, false, height0 + 1, adr1, r, r0] flags: <all paths>

Additional information

Used terms: zh[r, mkznode(r0, zbrar, false, false, m0)], P, nstat, (alladrs -- adr0) ++ adr1, as ++ adr1, iast[adr1, save(mkznode(r0, zbrar, false, true, m0))], ri, false, height0 + 1, adr1, r, r0