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Proof step 19 from proof btree-clean-fwd-strong (previous step)

(( r2 => mkznode(rp0, zbrar0, false, false, nused) * bnode(r2, zbrar0, 0, nused, nused, height + 1, false, ri0, iast0, alladrs0 -- adr, alladrs0 -- adr) * P0) )(zh2), (( r => mkznode(rp1, zbrar, false, false, nused) * bnode(r, zbrar, 0, nused, nused, height0 + 1, false, ri1, iast1, alladrs1 -- adr, alladrs1 -- adr) * P1) )(zh), nsize(nused, nstat), unmodified(adr, mkznode(rp1, zbrar, false, false, nused), iast1), unmodified(adr, mkznode(rp0, zbrar0, false, false, nused), iast0), adr ∈ iast1, adr ∈ alladrs1, adr ∈ iast0, adr ∈ alladrs0, alladrs1 ⊆ dom iast1, alladrs0 ⊆ dom iast0, iast0 ⊆ iast1, (r0 = null ⊃ P0(zh0);r2 = r0 ∧ zh2 = zh0), (r1 = null ⊃ P1(zh1);r = r1 ∧ zh = zh1), ∀ rp1, r1, adr, height1, ri1, iast1, alladrs1, nstat, P1, zh1, rp0, r0, ri0, iast0, alladrs0, P0, zh0. ((btree(rp1, r1, adr, height1, false, ri1, iast1, alladrs1, alladrs1, nstat) * P1))(zh1) ∧ ((btree(rp0, r0, adr, height, false, ri0, iast0, alladrs0, alladrs0, nstat) * P0))(zh0) ∧ alladrs1 ⊆ dom iast1 ∧ alladrs0 ⊆ dom iast0 ∧ iast0 ⊆ iast1 → height = height1 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1 ⊦ # alladrs1 = 0, # alladrs0 = 0, alladrs0 = ∅, alladrs1 = ∅, height = height0 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 20:
⊦ ((P * P2))(zh) ↔ (∃ zh0, zh1. zh0 ⊥ zh1 ∧ zh = zh0 ∪ zh1 ∧ P(zh0) ∧ P2(zh1))
Proof step 21:
(( r2 => mkznode(rp0, zbrar0, false, false, nused) * bnode(r2, zbrar0, 0, nused, nused, height + 1, false, ri0, iast0, alladrs0 -- adr, alladrs0 -- adr) * P0) )(zh2), nsize(nused, nstat), unmodified(adr, mkznode(rp0, zbrar0, false, false, nused), iast0), unmodified(adr, mkznode(rp1, zbrar, false, false, nused), iast1), adr ∈ alladrs0, adr ∈ iast0, adr ∈ alladrs1, adr ∈ iast1, iast0 ⊆ iast1, alladrs0 ⊆ dom iast0, alladrs1 ⊆ dom iast1, (r1 = null ⊃ P1(zh1);r = r1 ∧ zh = zh1), (r0 = null ⊃ P0(zh0);r2 = r0 ∧ zh2 = zh0), ∀ rp1, r1, adr, height1, ri1, iast1, alladrs1, nstat, P1, zh1, rp0, r0, ri0, iast0, alladrs0, P0, zh0. ((btree(rp1, r1, adr, height1, false, ri1, iast1, alladrs1, alladrs1, nstat) * P1))(zh1) ∧ ((btree(rp0, r0, adr, height, false, ri0, iast0, alladrs0, alladrs0, nstat) * P0))(zh0) ∧ alladrs1 ⊆ dom iast1 ∧ alladrs0 ⊆ dom iast0 ∧ iast0 ⊆ iast1 → height = height1 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1, ∃ zh0, zh1. zh0 ⊥ zh1 ∧ zh = zh0 ∪ zh1 ∧ ((r => mkznode(rp1, zbrar, false, false, nused)))(zh0) ∧ ((bnode(r, zbrar, 0, nused, nused, height0 + 1, false, ri1, iast1, alladrs1 -- adr, alladrs1 -- adr) * P1))(zh1) ⊦ # alladrs0 = 0, # alladrs1 = 0, alladrs0 = ∅, alladrs1 = ∅, height = height0 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1

Rule argument

spec: heap-sep, inst: zheap name: sep seq: ⊦ ((P * P2))(zh) ↔ (∃ zh0, zh1. zh0 ⊥ zh1 ∧ zh = zh0 ∪ zh1 ∧ P(zh0) ∧ P2(zh1)) subst: [zh, P2, P] → [zh, bnode (r, zbrar, 0, nused, nused, height0 + 1, false, ri1, iast1, alladrs1 -- adr, alladrs1 -- adr) * P1, r => mkznode(rp1, zbrar, false, false, nused)] flags: <all paths>

Additional information

Used terms: zh, bnode (r, zbrar, 0, nused, nused, height0 + 1, false, ri1, iast1, alladrs1 -- adr, alladrs1 -- adr) * P1, r => mkznode(rp1, zbrar, false, false, nused)