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Proof step 5 from proof btree-write-node (previous step)

zh[r] = mkznode(r0, zbrar, boolvar, true, m0), ((btree(rp, r, adr0, 0, dirty, ri, iast, ∅, ∅ ++ adr0, nstat) * P))(zh), btree-sub-clean(r, m0, zh), r ∈ zh ⊦ adr1 = adr0, ((btree(rp, r, adr1, 0, false, ri, iast[adr1, save(mkznode(r0, zbrar, boolvar, true, m0))], ∅ ++ adr1, ∅ ++ adr1, nstat) * P) )(zh[r, mkznode(r0, zbrar, boolvar, false, m0)]), adr1 ∈ iast

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ ((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 7:
zh[r] = mkznode(r0, zbrar, boolvar, true, m0), btree-sub-clean(r, m0, zh), r ∈ zh, ∃ r1, zh1, zbrar, dirty1, nused. (r = null ⊃ ¬ dirty1 ∧ P(zh);r1 = r ∧ zh1 = zh) ∧ (dirty1 → dirty) ∧ (adr0 ∈ ∅ ↔ ¬ dirty1) ∧ adr0 ∈ ∅ ++ adr0 ∧ nsize(nused, nstat) ∧ unmodified(adr0, mkznode(rp, zbrar, 0 = 0, dirty1, nused), iast) ∧ ((r1 => mkznode(rp, zbrar, 0 = 0, dirty1, nused) * bnode(r1, zbrar, 0, nused, nused, 0, dirty1, ri, iast, ∅ -- adr0, (∅ ++ adr0) -- adr0) * P) )(zh1) ⊦ adr1 = adr0, ((btree(rp, r, adr1, 0, false, ri, iast[adr1, save(mkznode(r0, zbrar, boolvar, true, m0))], ∅ ++ adr1, ∅ ++ adr1, nstat) * P) )(zh[r, mkznode(r0, zbrar, boolvar, false, m0)]), adr1 ∈ iast

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, P, nstat, ∅ ++ adr0, ∅, iast, ri, dirty, 0, adr0, r, rp] flags: <all paths>

Additional information

Used terms: zh, P, nstat, ∅ ++ adr0, ∅, iast, ri, dirty, 0, adr0, r, rp