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Proof step 10 from proof btree-clean-fwd (previous step)

((btree(rp1, r1, adr, height1, false, ri1, iast, alladrs1, alladrs1, nstat) * P1))(zh1), ((btree(rp0, r0, adr, height + 1, false, ri0, iast, alladrs0, alladrs0, nstat) * P0))(zh0), alladrs1 ⊆ dom iast, alladrs0 ⊆ dom iast, ∀ rp0, r0, adr, ri0, iast, as0, alladrs0, nstat, P0, zh0, rp1, r1, height1, ri1, as1, alladrs1, P1, zh1. ((btree(rp0, r0, adr, height, false, ri0, iast, as0, alladrs0, nstat) * P0))(zh0) ∧ ((btree(rp1, r1, adr, height1, false, ri1, iast, as1, alladrs1, nstat) * P1))(zh1) → height = height1 ∧ as0 = as1 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1 ⊦ height + 1 = height1 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
⊦ ((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 12:
((btree(rp0, r0, adr, height + 1, false, ri0, iast, alladrs0, alladrs0, nstat) * P0))(zh0), alladrs0 ⊆ dom iast, alladrs1 ⊆ dom iast, ∀ rp0, r0, adr, ri0, iast, as0, alladrs0, nstat, P0, zh0, rp1, r1, height1, ri1, as1, alladrs1, P1, zh1. ((btree(rp0, r0, adr, height, false, ri0, iast, as0, alladrs0, nstat) * P0))(zh0) ∧ ((btree(rp1, r1, adr, height1, false, ri1, iast, as1, alladrs1, nstat) * P1))(zh1) → height = height1 ∧ as0 = as1 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1, ∃ r, zh, zbrar, dirty, nused. (r1 = null ⊃ ¬ dirty ∧ P1(zh1);r = r1 ∧ zh = zh1) ∧ (dirty → false) ∧ (adr ∈ alladrs1 ↔ ¬ dirty) ∧ adr ∈ alladrs1 ∧ nsize(nused, nstat) ∧ unmodified(adr, mkznode(rp1, zbrar, height1 = 0, dirty, nused), iast) ∧ (( r => mkznode(rp1, zbrar, height1 = 0, dirty, nused) * bnode(r, zbrar, 0, nused, nused, height1, dirty, ri1, iast, alladrs1 -- adr, alladrs1 -- adr) * P1) )(zh) ⊦ height + 1 = height1 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1

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] → [zh1, P1, nstat, alladrs1, alladrs1, iast, ri1, false, height1, adr, r1, rp1] flags: <all paths>

Additional information

Used terms: zh1, P1, nstat, alladrs1, alladrs1, iast, ri1, false, height1, adr, r1, rp1