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Proof step 6 from proof btree-reset-root-frame (previous step)

(( r0 => mkznode(rp, zbrar, height = 0, false, nused) * bnode(r0, zbrar, 0, nused, nused, height, false, ri, iast, alladrs -- adr, alladrs -- adr) * P) )(zh0), nused < BRANCH_SIZE, unmodified(adr, mkznode(rp, zbrar, height = 0, false, nused), iast), adr ∈ alladrs, adr ∈ iast, alladrs ⊆ dom iast, (r = null ⊃ P(zh);r0 = r ∧ zh0 = zh) ⊦ # alladrs = 0, alladrs = ∅, ((btree(rp, null, adr, height, false, ri, iast, alladrs, alladrs, ok) * true * P))(zh)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 7:
⊦ ((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 8:
(( r0 => mkznode(rp, zbrar, height = 0, false, nused) * bnode(r0, zbrar, 0, nused, nused, height, false, ri, iast, alladrs -- adr, alladrs -- adr) * P) )(zh0), nused < BRANCH_SIZE, unmodified(adr, mkznode(rp, zbrar, height = 0, false, nused), iast), adr ∈ iast, adr ∈ alladrs, alladrs ⊆ dom iast, (r = null ⊃ P(zh);r0 = r ∧ zh0 = zh) ⊦ # alladrs = 0, alladrs = ∅, ∃ r, zh1, zbrar, dirty, nused. (null = null ⊃ ¬ dirty ∧ ((true * P))(zh);r = null ∧ zh1 = zh) ∧ (dirty → false) ∧ (adr ∈ alladrs ↔ ¬ dirty) ∧ adr ∈ alladrs ∧ nsize(nused, ok) ∧ 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, alladrs -- adr, alladrs -- adr) * true * 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, true * P, ok, alladrs, alladrs, iast, ri, false, height, adr, null, rp] flags: <all paths>

Additional information

Used terms: zh, true * P, ok, alladrs, alladrs, iast, ri, false, height, adr, null, rp