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Proof step 48 from proof inv-commit-rec (previous step)

istore2[m1] = indexblock(indx, lax, m4), zh[r0] = mkznode(r, zbrar, boolvar, true, m0), istore[adr1] = istore2[adr1], zh0[r0] = mkznode(r1, zbrar0, boolvar0, false, m5), remove-refsize0(gstore0) = gstore0, remove-refsize0(istore0) = istore0, elems nx ∩ ginval = ∅, dom gstore0 ∪ elems nx = dom gstore ∪ ginval, ((btree(null, null, adr1, height0, false, fi, nodes(istore0), alladrs0, alladrs0, ok) * true))(zh), ((btree(null, r0, adr1, height, dirty, ri, nodes(istore), as, alladrs, ok) * true))(zh), ((btree(null, r0, @(m1, m2, m3), height, false, ri, nodes(istore2) | alladrs1, alladrs1, alladrs1, ok) * true))(zh0), inv(indexblock(indx, lax, m4)), inv(istore2), inv(gstore \ ginval), inv(gstore0), inv(istore), inv(istore0), refsize-cons(istore2, alladrs1), refsize-cons(istore, as), refsize-cons(istore0, alladrs0), m1 ∈ istore2, @(m1, m2, m3) ∈ istore2, @(m1, m2, m3) ∈ alladrs1, adr1 ∈ istore2, r0 ∈ zh0, m ∈ istore, adr1 ∈ istore, r0 ∈ zh, m ∈ istore2, m6 ∈ istore2, alladrs1 ∩ dom nodes(istore) ⊆ alladrs, alladrs1 ⊆ dom nodes(istore2), alladrs1 ∩ dom nodes(istore) ⊆ as, as ⊆ alladrs, as ⊆ dom nodes(istore), alladrs0 ⊆ dom nodes(istore), alladrs0 ⊆ dom nodes(istore0), alladrs ⊆ dom nodes(istore), istore ⊑ istore2, istore0 ⊑ istore2, istore0 ⊑ istore, height = 0 → alladrs1 = ∅ ++ @(m1, m2, m3), height = 0 → as = ∅ ⊦ # alladrs1 = 0, alladrs1 = ∅, ((btree(null, r0, adr1, height, true, ri, nodes(istore2), alladrs1 -- @(m1, m2, m3), (alladrs1 -- @(m1, m2, m3)) ++ adr1, ok) * true) )(zh0[r0, mkznode(r1, zbrar0, boolvar0, true, m5)]), dups(nx), adr1 ∈ as, ¬ adr1 ∈ alladrs1 ∧ ((btree(null, r0, @(m1, m2, m3), height, true, ri, nodes(istore2), alladrs1 -- @(m1, m2, m3), alladrs1, ok) * true) )(zh0[r0, mkznode(r1, zbrar0, boolvar0, true, m5)])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 49:
⊦ r ∈ zh ∧ zh[r] = mkznode(rp0, zbrar, boolvar, false, height0) ∧ ((btree(rp, r, adr, height, false, ri, iast, as, alladrs, nstat) * P))(zh) → ((btree(rp, r, adr, height, true, ri, iast, as -- adr, alladrs, nstat) * P))(zh[r, mkznode(rp0, zbrar, boolvar, true, height0)])
Proof step 50:
istore[adr1] = istore2[adr1], zh[r0] = mkznode(r, zbrar, boolvar, true, m0), istore2[m1] = indexblock(indx, lax, m4), zh0[r0] = mkznode(r1, zbrar0, boolvar0, false, m5), remove-refsize0(istore0) = istore0, remove-refsize0(gstore0) = gstore0, elems nx ∩ ginval = ∅, dom gstore0 ∪ elems nx = dom gstore ∪ ginval, ((btree(null, r0, @(m1, m2, m3), height, false, ri, nodes(istore2) | alladrs1, alladrs1, alladrs1, ok) * true))(zh0), ((btree(null, r0, adr1, height, dirty, ri, nodes(istore), as, alladrs, ok) * true))(zh), ((btree(null, null, adr1, height0, false, fi, nodes(istore0), alladrs0, alladrs0, ok) * true))(zh), ((btree(null, r0, @(m1, m2, m3), height, false, ri, nodes(istore2), alladrs1, alladrs1, ok) * true))(zh0), inv(istore0), inv(istore), inv(gstore0), inv(gstore \ ginval), inv(istore2), inv(indexblock(indx, lax, m4)), refsize-cons(istore0, alladrs0), refsize-cons(istore, as), refsize-cons(istore2, alladrs1), m6 ∈ istore2, m ∈ istore2, r0 ∈ zh, adr1 ∈ istore, m ∈ istore, adr1 ∈ istore2, @(m1, m2, m3) ∈ alladrs1, @(m1, m2, m3) ∈ istore2, m1 ∈ istore2, r0 ∈ zh0, alladrs ⊆ dom nodes(istore), alladrs0 ⊆ dom nodes(istore0), alladrs0 ⊆ dom nodes(istore), as ⊆ dom nodes(istore), as ⊆ alladrs, alladrs1 ∩ dom nodes(istore) ⊆ as, alladrs1 ⊆ dom nodes(istore2), alladrs1 ∩ dom nodes(istore) ⊆ alladrs, istore0 ⊑ istore, istore0 ⊑ istore2, istore ⊑ istore2, height = 0 → as = ∅, height = 0 → alladrs1 = ∅ ++ @(m1, m2, m3) ⊦ # alladrs1 = 0, alladrs1 = ∅, ((btree(null, r0, adr1, height, true, ri, nodes(istore2), alladrs1 -- @(m1, m2, m3), (alladrs1 -- @(m1, m2, m3)) ++ adr1, ok) * true) )(zh0[r0, mkznode(r1, zbrar0, boolvar0, true, m5)]), dups(nx), adr1 ∈ as, ¬ adr1 ∈ alladrs1 ∧ true

Rule argument

spec: btree-inv name: btree-mark-dirty seq: ⊦ r ∈ zh ∧ zh[r] = mkznode(rp0, zbrar, boolvar, false, height0) ∧ (( btree (rp, r, adr, height, false, ri, iast, as, alladrs, nstat) * P) )(zh) → (( btree (rp, r, adr, height, true, ri, iast, as -- adr, alladrs, nstat) * P) )(zh[r, mkznode(rp0, zbrar, boolvar, true, height0)]) subst: [height0, boolvar, zbrar, rp0, zh, P, nstat, alladrs, as, iast, ri, height, adr, r, rp] → [m5, boolvar0, zbrar0, r1, zh0, true, ok, alladrs1, alladrs1, nodes(istore2), ri, height, @(m1, m2, m3), r0, null] flags: <all paths>

Additional information

Used simplifier rules: sep-ass: ⊦ (P * P3) * P4 = P * P3 * P4; from specification heap-sep[zheap]%sep-com: ⊦ P * P2 = P2 * P; from specification heap-sep[zheap]%⊦ (( btree (rp, r0, adr, height, dirty0, ri, iast | as0, as, alladrs, nstat) * P) )(zh0) → (( btree (rp, r0, adr, height, dirty0, ri, iast, as, alladrs, nstat) * P) )(zh0); (ss) Used terms: m5, boolvar0, zbrar0, r1, zh0, true, ok, alladrs1, alladrs1, nodes(istore2), ri, height, @(m1, m2, m3), r0, null