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Proof step 58 from proof path-split-wand (previous step)

((abs(tree, ino) * abs(sdt0, entries0) * ino0 => mkdir(# entries0 + 1, md, m0, m1, entries0[str, ino])))(dirs2 ∪ dirs ∪ dirs1), ((ino0 => mkdir(# entries0 + 1, md, m0, m1, entries0[str, ino]) * abs(sdt0, entries0)))(dirs1), abs(tree, ino)(dirs2 ∪ dirs), abs(tree[p0], ino1)(dirs2), abs(tree[p0], ino1)(dirs0), abs(tree, ino)(dirs0 ∪ dirs), p0 ∈ tree, dirs1 ⊥ dirs2, dirs2 ⊥ dirs, dirs1 ⊥ dirs, dirs0 ⊥ dirs1, dirs0 ⊥ dirs ⊦ (((abs(tree, ino) * abs(sdt0, entries0)) * ino0 => mkdir(# entries0 + 1, md, m0, m1, entries0[str, ino])))(dirs0 ∪ dirs ∪ dirs1), str ∈ entries0, str ∈ sdt0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 59:
⊦ (D * D0) * D1 = D * D0 * D1
Proof step 60:
((ino0 => mkdir(# entries0 + 1, md, m0, m1, entries0[str, ino]) * abs(sdt0, entries0)))(dirs1), ((abs(tree, ino) * abs(sdt0, entries0) * ino0 => mkdir(# entries0 + 1, md, m0, m1, entries0[str, ino])))(dirs2 ∪ dirs ∪ dirs1), abs(tree, ino)(dirs0 ∪ dirs), abs(tree[p0], ino1)(dirs0), abs(tree[p0], ino1)(dirs2), abs(tree, ino)(dirs2 ∪ dirs), p0 ∈ tree, dirs0 ⊥ dirs, dirs0 ⊥ dirs1, dirs1 ⊥ dirs, dirs2 ⊥ dirs, dirs1 ⊥ dirs2 ⊦ ((abs(tree, ino) * abs(sdt0, entries0) * ino0 => mkdir(# entries0 + 1, md, m0, m1, entries0[str, ino])))(dirs0 ∪ dirs ∪ dirs1), str ∈ sdt0, str ∈ entries0

Rule argument

spec: heap-sep, inst: dir-heap name: sep-ass seq: ⊦ (D * D0) * D1 = D * D0 * D1 subst: [D1, D0, D] → [ ino0 => mkdir (# entries0 + 1, md, m0, m1, entries0[str, ino]), abs(sdt0, entries0), abs(tree, ino)] flags: <all paths>

Additional information

Used simplifier rules: sep-ass: ⊦ (D * D0) * D1 = D * D0 * D1; from specification heap-sep[dir-heap]%sep-com: ⊦ D * D0 = D0 * D; from specification heap-sep[dir-heap] Used terms: ino0 => mkdir(# entries0 + 1, md, m0, m1, entries0[str, ino]), abs(sdt0, entries0), abs(tree, ino)