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Proof step 48 from proof abs-has-rootpath-h (previous step)

((ino0 => mkdir(n0, md, m0, n, entries0) * {dirs0} * abs(sdt0, sdi)))(dirs2), ((abs(tree, ino1) * abs(sdt0, sdi) * {dirs0} * ino0 => mkdir(n0, md, m0, n, entries0)))(dirs1 ∪ dirs2), abs(tree, ino1)(dirs1), ino ∈ dirs2, sdi[str, ino1] ⊆ entries0, dirs1 ⊥ dirs2, ∀ str0. str0 ∈ sdt0[str, tree] → (∀ ino, dirs, ino0, dirs0, files. ino ∈ dirs ∧ ((abs(sdt0[str, tree][str0], ino0) * {dirs0} ))(dirs) → ino ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino, dirs, files))), ∀ sdt1, ino0, n0, md, m0, n, entries0, entries, dirs0, dirs, ino, files. # sdt1 < # sdt0[str, tree] → (((ino0 => mkdir(n0, md, m0, n, entries0) * abs(sdt1, entries)) * {dirs0} ))(dirs) ∧ ino ∈ dirs ∧ entries ⊆ entries0 ∧ (∀ str. str ∈ sdt1 → (∀ ino, dirs, ino0, dirs0, files. ino ∈ dirs ∧ ((abs(sdt1[str], ino0) * {dirs0} ))(dirs) → ino ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino, dirs, files)))) → ino ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino, dirs, files)) ⊦ ino ∈ dirs0, str ∈ sdi, str ∈ sdt0, ∃ p. ispath(p, ino0, ino, dirs1 ∪ dirs2, files)

User applied rule apply induction

And obtained the following new goals

Proof step 49:
((abs(tree, ino1) * abs(sdt0, sdi) * {dirs0} * ino0 => mkdir(n0, md, m0, n, entries0)))(dirs1 ∪ dirs2), ((ino0 => mkdir(n0, md, m0, n, entries0) * {dirs0} * abs(sdt0, sdi)))(dirs2), abs(tree, ino1)(dirs1), ino ∈ dirs2, sdi[str, ino1] ⊆ entries0, dirs1 ⊥ dirs2, ∀ str0. str0 ∈ sdt0[str, tree] → (∀ ino, dirs, ino0, dirs0, files. ino ∈ dirs ∧ ((abs(sdt0[str, tree][str0], ino0) * {dirs0} ))(dirs) → ino ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino, dirs, files))), ∀ sdt1, ino0, n0, md, m0, n, entries0, entries, dirs0, dirs, ino, files. # sdt1 < # sdt0[str, tree] → (((ino0 => mkdir(n0, md, m0, n, entries0) * abs(sdt1, entries)) * {dirs0} ))(dirs) ∧ ino ∈ dirs ∧ entries ⊆ entries0 ∧ (∀ str. str ∈ sdt1 → (∀ ino, dirs, ino0, dirs0, files. ino ∈ dirs ∧ ((abs(sdt1[str], ino0) * {dirs0} ))(dirs) → ino ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino, dirs, files)))) → ino ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino, dirs, files)) ⊦ str ∈ sdt0, str ∈ sdi, ino ∈ dirs0, ∃ p. ispath(p, ino0, ino, dirs1 ∪ dirs2, files), (((ino0 => mkdir(n0, md, m0, n, entries0) * abs(sdt0, sdi)) * {dirs0} ))(dirs2) ∧ ino ∈ dirs2 ∧ sdi ⊆ entries0 ∧ (∀ str. str ∈ sdt0 → (∀ ino, dirs, ino0, dirs0, files. ino ∈ dirs ∧ ((abs(sdt0[str], ino0) * {dirs0} ))(dirs) → ino ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino, dirs, files))))
Proof step 65:
((abs(tree, ino1) * abs(sdt0, sdi) * {dirs0} * ino0 => mkdir(n0, md, m0, n, entries0)))(dirs1 ∪ dirs2), ((ino0 => mkdir(n0, md, m0, n, entries0) * {dirs0} * abs(sdt0, sdi)))(dirs2), abs(tree, ino1)(dirs1), ino ∈ dirs2, sdi[str, ino1] ⊆ entries0, dirs1 ⊥ dirs2, ∀ str0. str0 ∈ sdt0[str, tree] → (∀ ino, dirs, ino0, dirs0, files. ino ∈ dirs ∧ ((abs(sdt0[str, tree][str0], ino0) * {dirs0} ))(dirs) → ino ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino, dirs, files))), ino ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino, dirs2, files)), ∀ sdt1, ino0, n0, md, m0, n, entries0, entries, dirs0, dirs, ino, files. # sdt1 < # sdt0[str, tree] → (((ino0 => mkdir(n0, md, m0, n, entries0) * abs(sdt1, entries)) * {dirs0} ))(dirs) ∧ ino ∈ dirs ∧ entries ⊆ entries0 ∧ (∀ str. str ∈ sdt1 → (∀ ino, dirs, ino0, dirs0, files. ino ∈ dirs ∧ ((abs(sdt1[str], ino0) * {dirs0} ))(dirs) → ino ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino, dirs, files)))) → ino ∈ dirs0 ∨ (∃ p. ispath(p, ino0, ino, dirs, files)) ⊦ str ∈ sdt0, str ∈ sdi, ino ∈ dirs0, ∃ p. ispath(p, ino0, ino, dirs1 ∪ dirs2, files)

Rule argument

subst: [sdt1, ino0, n0, md, m0, n, entries0, entries, dirs0, dirs, ino, files] → [sdt0, ino0, n0, md, m0, n, entries0, sdi, dirs0, dirs2, ino, files]

Additional information

Used simplifier rules: ⊦ ¬ str ∈ sdt → # sdt[str, tree] = # sdt + 1; (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ ¬ n < n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%precise(D) ⊦ D(dirs) ∧ dirs ⊥ dirs0 → (((D * D0))(dirs ∪ dirs0) ↔ D0(dirs0)); (s)%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: sdt0, ino0, n0, md, m0, n, entries0, sdi, dirs0, dirs2, ino, files