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Proof step 16 from proof btree-mark-dirty (previous step)

(( r => mkznode(rp0, zbrar, false, false, height0) * bnode(r, zbrar, 0, height0, height0, height1 + 1, false, ri, iast, alladrs -- adr, alladrs -- adr) * P) )(zh), nsize(height0, nstat), unmodified(adr, mkznode(rp0, zbrar, false, false, height0), iast), adr ∈ iast, adr ∈ alladrs, alladrs ⊆ dom iast, ∀ zh, r, rp0, zbrar, boolvar, height0, rp, adr, ri, iast, alladrs, nstat, P. zh[r] = mkznode(rp0, zbrar, boolvar, false, height0) ∧ ((btree(rp, r, adr, height1, false, ri, iast, alladrs, alladrs, nstat) * P))(zh) ∧ r ∈ zh ∧ adr ∈ alladrs ∧ adr ∈ iast ∧ alladrs ⊆ dom iast ∧ (height1 = 0 → alladrs = ∅ ++ adr) → # alladrs = 0 ∨ alladrs = ∅ ∨ ((btree(rp, r, adr, height1, true, ri, iast, alladrs -- adr, alladrs, nstat) * P))(zh[r, mkznode(rp0, zbrar, boolvar, true, height0)]) ⊦ # alladrs = 0, alladrs = ∅, ((btree(rp0, r, adr, height1 + 1, true, ri, iast, alladrs -- adr, alladrs, nstat) * P))(zh[r, mkznode(rp0, zbrar, false, true, height0)])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 17:
⊦ ((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 18:
(( r => mkznode(rp0, zbrar, false, false, height0) * bnode(r, zbrar, 0, height0, height0, height1 + 1, false, ri, iast, alladrs -- adr, alladrs -- adr) * P) )(zh), nsize(height0, nstat), unmodified(adr, mkznode(rp0, zbrar, false, false, height0), iast), adr ∈ alladrs, adr ∈ iast, alladrs ⊆ dom iast, ∀ zh, r, rp0, zbrar, boolvar, height0, rp, adr, ri, iast, alladrs, nstat, P. zh[r] = mkznode(rp0, zbrar, boolvar, false, height0) ∧ ((btree(rp, r, adr, height1, false, ri, iast, alladrs, alladrs, nstat) * P))(zh) ∧ r ∈ zh ∧ adr ∈ alladrs ∧ adr ∈ iast ∧ alladrs ⊆ dom iast ∧ (height1 = 0 → alladrs = ∅ ++ adr) → # alladrs = 0 ∨ alladrs = ∅ ∨ ((btree(rp, r, adr, height1, true, ri, iast, alladrs -- adr, alladrs, nstat) * P))(zh[r, mkznode(rp0, zbrar, boolvar, true, height0)]) ⊦ # alladrs = 0, alladrs = ∅, ∃ r1, zh1, zbrar0, dirty, nused. (r = null ⊃ ¬ dirty ∧ P(zh[r, mkznode(rp0, zbrar, false, true, height0)]);r1 = r ∧ zh1 = zh[r, mkznode(rp0, zbrar, false, true, height0)]) ∧ (dirty → true) ∧ (adr ∈ alladrs -- adr ↔ ¬ dirty) ∧ adr ∈ alladrs ∧ nsize(nused, nstat) ∧ unmodified(adr, mkznode(rp0, zbrar0, height1 + 1 = 0, dirty, nused), iast) ∧ (( r1 => mkznode(rp0, zbrar0, height1 + 1 = 0, dirty, nused) * bnode(r1, zbrar0, 0, nused, nused, height1 + 1, dirty, ri, iast, alladrs -- adr -- adr, alladrs -- adr) * 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[r, mkznode(rp0, zbrar, false, true, height0)], P, nstat, alladrs, alladrs -- adr, iast, ri, true, height1 + 1, adr, r, rp0] flags: <all paths>

Additional information

Used terms: zh[r, mkznode(rp0, zbrar, false, true, height0)], P, nstat, alladrs, alladrs -- adr, iast, ri, true, height1 + 1, adr, r, rp0