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Proof step 20 from proof btree-ri-unique (previous step)

adr ∈ as0 ↔ ¬ dirty0, (( rp0 => mkznode(rp, zbrar, false, dirty0, nused) * bnode(rp0, zbrar, 0, nused, nused, height0 + 1, dirty0, ri0, iast, as0 -- adr, alladrs0 -- adr) * P0) )(zh), ((btree(rp, rp0, adr, height0 + 1, dirty, ri1, iast, as1, alladrs1, ok) * P1))(zh), nused < BRANCH_SIZE, unmodified(adr, mkznode(rp, zbrar, false, dirty0, nused), iast), adr ∈ alladrs0, as0 ⊆ alladrs0, as0 ⊆ dom iast, as1 ⊆ alladrs1, as1 ⊆ dom iast, dirty0 → dirty, ∀ rp, rp0, adr, dirty, ri0, iast, as0, alladrs0, P0, zh, ri1, as1, alladrs1, P1. ((btree(rp, rp0, adr, height0, dirty, ri0, iast, as0, alladrs0, ok) * P0))(zh) ∧ ((btree(rp, rp0, adr, height0, dirty, ri1, iast, as1, alladrs1, ok) * P1))(zh) ∧ as0 ⊆ alladrs0 ∧ as0 ⊆ dom iast ∧ as1 ⊆ alladrs1 ∧ as1 ⊆ dom iast → ri0 = ri1 ∨ rp0 = null ⊦ # alladrs0 = 0, alladrs0 = ∅, ri0 = ri1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 21:
⊦ ((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 22:
adr ∈ as0 ↔ ¬ dirty0, (( rp0 => mkznode(rp, zbrar, false, dirty0, nused) * bnode(rp0, zbrar, 0, nused, nused, height0 + 1, dirty0, ri0, iast, as0 -- adr, alladrs0 -- adr) * P0) )(zh), nused < BRANCH_SIZE, unmodified(adr, mkznode(rp, zbrar, false, dirty0, nused), iast), adr ∈ alladrs0, as1 ⊆ dom iast, as1 ⊆ alladrs1, as0 ⊆ dom iast, as0 ⊆ alladrs0, dirty0 → dirty, ∀ rp, rp0, adr, dirty, ri0, iast, as0, alladrs0, P0, zh, ri1, as1, alladrs1, P1. ((btree(rp, rp0, adr, height0, dirty, ri0, iast, as0, alladrs0, ok) * P0))(zh) ∧ ((btree(rp, rp0, adr, height0, dirty, ri1, iast, as1, alladrs1, ok) * P1))(zh) ∧ as0 ⊆ alladrs0 ∧ as0 ⊆ dom iast ∧ as1 ⊆ alladrs1 ∧ as1 ⊆ dom iast → ri0 = ri1 ∨ rp0 = null, ∃ r, zh1, zbrar, dirty1, nused. (rp0 = null ⊃ ¬ dirty1 ∧ P1(zh);r = rp0 ∧ zh1 = zh) ∧ (dirty1 → dirty) ∧ (adr ∈ as1 ↔ ¬ dirty1) ∧ adr ∈ alladrs1 ∧ nsize(nused, ok) ∧ unmodified(adr, mkznode(rp, zbrar, height0 + 1 = 0, dirty1, nused), iast) ∧ (( r => mkznode(rp, zbrar, height0 + 1 = 0, dirty1, nused) * bnode(r, zbrar, 0, nused, nused, height0 + 1, dirty1, ri1, iast, as1 -- adr, alladrs1 -- adr) * P1) )(zh1) ⊦ # alladrs0 = 0, alladrs0 = ∅, ri0 = ri1

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, P1, ok, alladrs1, as1, iast, ri1, dirty, height0 + 1, adr, rp0, rp] flags: <all paths>

Additional information

Used terms: zh, P1, ok, alladrs1, as1, iast, ri1, dirty, height0 + 1, adr, rp0, rp