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Proof step 3 from proof btree-inv-remove-refsize0 (previous step)

zh[rp] = mkznode(rp0, zbrar, boolvar, dirty, nused), ((btree(null, rp, adr, height, dirty0, ri, nodes(istore), as, alladrs, ok) * true))(zh), ((btree(null, null, adr, height0, false, fi, nodes(istore0), alladrs0, alladrs0, ok) * true))(zh), inv(istore0), inv(istore), refsize-cons(istore, as), refsize-cons(istore0, alladrs0), adr ∈ istore, alladrs0 ⊆ dom nodes(istore0), alladrs ⊆ dom nodes(istore), alladrs0 ⊆ dom nodes(istore), as ⊆ dom nodes(istore), as ⊆ alladrs, istore0 ⊑ istore, ¬ dirty ∨ rp = null ⊦ (∃ height, dirty, as, alladrs. ((btree(null, rp, adr, height, dirty, ri, nodes(remove-refsize0(istore)), as, alladrs, ok) * true))(zh) ∧ refsize-cons(remove-refsize0(istore), as) ∧ adr ∈ remove-refsize0(istore) ∧ alladrs ⊆ dom nodes(remove-refsize0(istore))) ∧ (∃ height, as, alladrs. ((btree(null, null, adr, height, false, fi, nodes(remove-refsize0(istore)), as, alladrs, ok) * true))(zh) ∧ refsize-cons(remove-refsize0(istore), as) ∧ alladrs ⊆ dom nodes(remove-refsize0(istore)))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ r = null ∨ ¬ zh0[r].dirty → ( ((btree(rp, r, adr, height, dirty, ri, iast, as, alladrs, nstat) * P))(zh0) ↔ ((btree(rp, r, adr, height, false, ri, iast, as, alladrs, nstat) * P))(zh0))
Proof step 5:
zh[rp] = mkznode(rp0, zbrar, boolvar, dirty, nused), ((btree(null, null, adr, height0, false, fi, nodes(istore0), alladrs0, alladrs0, ok) * true))(zh), ((btree(null, rp, adr, height, false, ri, nodes(istore), as, alladrs, ok) * true))(zh), inv(istore), inv(istore0), refsize-cons(istore0, alladrs0), refsize-cons(istore, as), adr ∈ istore, as ⊆ alladrs, as ⊆ dom nodes(istore), alladrs0 ⊆ dom nodes(istore), alladrs ⊆ dom nodes(istore), alladrs0 ⊆ dom nodes(istore0), istore0 ⊑ istore, ¬ dirty ∨ rp = null, rp = null ∨ ¬ zh[rp].dirty ⊦ (∃ height, dirty, as, alladrs. ((btree(null, rp, adr, height, dirty, ri, nodes(remove-refsize0(istore)), as, alladrs, ok) * true))(zh) ∧ refsize-cons(remove-refsize0(istore), as) ∧ adr ∈ remove-refsize0(istore) ∧ alladrs ⊆ dom nodes(remove-refsize0(istore))) ∧ (∃ height, as, alladrs. ((btree(null, null, adr, height, false, fi, nodes(remove-refsize0(istore)), as, alladrs, ok) * true))(zh) ∧ refsize-cons(remove-refsize0(istore), as) ∧ alladrs ⊆ dom nodes(remove-refsize0(istore)))

Rule argument

spec: <toplevel> name: btree-null-or-clean-below seq: ⊦ r = null ∨ ¬ zh0[r].dirty → ( (( btree (rp, r, adr, height, dirty, ri, iast, as, alladrs, nstat) * P) )(zh0) ↔ (( btree (rp, r, adr, height, false, ri, iast, as, alladrs, nstat) * P) )(zh0)) subst: [zh0, P, nstat, alladrs, as, iast, ri, dirty, height, adr, r, rp] → [zh, true, ok, alladrs, as, nodes(istore), ri, dirty0, height, adr, rp, null] flags: <all paths>

Additional information

Used simplifier rules: btree-root-in-heap: ⊦ r ≠ null ∧ (( btree (rp, r, adr, height, dirty, fi, iast, as, alladrs, nstat) * P) )(zh) → r ∈ zh;%sep-ass: ⊦ (P * P3) * P4 = P * P3 * P4; from specification heap-sep[zheap]%sep-com: ⊦ P * P2 = P2 * P; from specification heap-sep[zheap]% ⊦ r ∈ zh → r ≠ null; (s)% ⊦ mkznode(r, zbrar, boolvar, boolvar0, m).dirty ↔ boolvar0; (s) Used terms: zh, true, ok, alladrs, as, nodes(istore), ri, dirty0, height, adr, rp, null