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Proof step 13 from proof btree-write-node (previous step)

zh[r] = mkznode(r0, zbrar, boolvar, true, m0), ((btree(rp, r, adr0, height0 + 1, dirty, ri, iast, as, alladrs, nstat) * P))(zh), btree-sub-clean(r, m0, zh), r ∈ zh, as ⊆ alladrs, as ⊆ dom iast ⊦ (( btree (rp, r, adr1, height0 + 1, false, ri, iast[adr1, save(mkznode(r0, zbrar, boolvar, true, m0))], as ++ adr1, (alladrs -- adr0) ++ adr1, nstat) * P) )(zh[r, mkznode(r0, zbrar, boolvar, false, m0)]), adr0 ∈ as, adr1 ∈ alladrs, adr1 ∈ iast

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ ((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 15:
zh[r] = mkznode(r0, zbrar, boolvar, true, m0), btree-sub-clean(r, m0, zh), r ∈ zh, as ⊆ dom iast, as ⊆ alladrs, ∃ r1, zh1, zbrar, dirty1, nused. (r = null ⊃ ¬ dirty1 ∧ P(zh);r1 = r ∧ zh1 = zh) ∧ (dirty1 → dirty) ∧ (adr0 ∈ as ↔ ¬ dirty1) ∧ adr0 ∈ alladrs ∧ nsize(nused, nstat) ∧ unmodified(adr0, mkznode(rp, zbrar, height0 + 1 = 0, dirty1, nused), iast) ∧ (( r1 => mkznode(rp, zbrar, height0 + 1 = 0, dirty1, nused) * bnode(r1, zbrar, 0, nused, nused, height0 + 1, dirty1, ri, iast, as -- adr0, alladrs -- adr0) * P) )(zh1) ⊦ (( btree (rp, r, adr1, height0 + 1, false, ri, iast[adr1, save(mkznode(r0, zbrar, boolvar, true, m0))], as ++ adr1, (alladrs -- adr0) ++ adr1, nstat) * P) )(zh[r, mkznode(r0, zbrar, boolvar, false, m0)]), adr1 ∈ iast, adr1 ∈ alladrs, adr0 ∈ as

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, alladrs, as, iast, ri, dirty, height0 + 1, adr0, r, rp] flags: <all paths>

Additional information

Used terms: zh, P, nstat, alladrs, as, iast, ri, dirty, height0 + 1, adr0, r, rp