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Proof step 8 from proof btree-repl-dirty-address (previous step)

((r => mkznode(rp, zbrar0, height = 0, true, nused) * bnode(r, zbrar0, 0, nused, nused, height, true, ri, iast, as, alladrs -- adr0) * P))(zh), nsize(nused, nstat), adr0 ∈ alladrs, as ⊆ dom iast, as ⊆ alladrs, height = 0 → as = ∅ ⊦ # alladrs = 0, alladrs = ∅, ((btree(rp, r, adr1, height, true, ri, iast, as, (alladrs -- adr0) ++ adr1, nstat) * P))(zh), adr1 ∈ alladrs, adr0 ∈ as

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ ((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 10:
((r => mkznode(rp, zbrar0, height = 0, true, nused) * bnode(r, zbrar0, 0, nused, nused, height, true, ri, iast, as, alladrs -- adr0) * P))(zh), nsize(nused, nstat), adr0 ∈ alladrs, as ⊆ alladrs, as ⊆ dom iast, height = 0 → as = ∅ ⊦ # alladrs = 0, alladrs = ∅, adr0 ∈ as, adr1 ∈ alladrs, ∃ r1, zh1, zbrar, dirty, nused. (r = null ⊃ ¬ dirty ∧ P(zh);r1 = r ∧ zh1 = zh) ∧ (dirty → true) ∧ (adr1 ∈ as ↔ ¬ dirty) ∧ adr1 ∈ (alladrs -- adr0) ++ adr1 ∧ nsize(nused, nstat) ∧ unmodified(adr1, mkznode(rp, zbrar, height = 0, dirty, nused), iast) ∧ (( r1 => mkznode(rp, zbrar, height = 0, dirty, nused) * bnode(r1, zbrar, 0, nused, nused, height, dirty, ri, iast, as -- 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, P, nstat, (alladrs -- adr0) ++ adr1, as, iast, ri, true, height, adr1, r, rp] flags: <all paths>

Additional information

Used terms: zh, P, nstat, (alladrs -- adr0) ++ adr1, as, iast, ri, true, height, adr1, r, rp