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Proof step 55 from proof walk-term (previous step)

dirs[ino0] = mkdir(m, md, # links(ino0, dirs), m1, entries), err = ESUCCESS, ino = entries[str], afs-cons(dirs, files), px(user, md), ino0 ∈ dirs, str ∈ entries, links(ino0, dirs) ≠ ∅ ∨ ino0 = ROOT_INO, ∀ p0, err, dirs, files, ino, user, asynced. # p0 < #(str + p1) → err = ESUCCESS ∧ afs-cons(dirs, files) ∧ (ino ∈ dirs ∨ ino ∈ files) ∧ (links(ino, dirs) ≠ ∅ ∨ ino = ROOT_INO) → ⟪while p0 ≠ ε ∧ err = ESUCCESS do {choose inode with true in vfs_may_lookup#(ino, user, dirs, files, asynced; inode, err); if err = ESUCCESS then let dent = negdentry(p0.head) in {afs_lookup#(ino, dirs, files, asynced; dent, err); if err = ESUCCESS then {ino := dent.ino; p0 := p0.tail}}}⟫ true ⊦ ⟪while p1 ≠ ε ∧ err = ESUCCESS do {choose inode with true in vfs_may_lookup#(ino, user, dirs, files, asynced; inode, err); if err = ESUCCESS then let dent = negdentry(p1.head) in {afs_lookup#(ino, dirs, files, asynced; dent, err); if err = ESUCCESS then {ino := dent.ino; p1 := p1.tail}}}⟫ true

User applied rule apply induction

And obtained the following new goals

Proof step 56:
⟪p0 := p1; while p0 ≠ ε ∧ err = ESUCCESS do {choose inode with true in vfs_may_lookup#(ino, user, dirs, files, asynced; inode, err); if err = ESUCCESS then let dent = negdentry(p0.head) in {afs_lookup#(ino, dirs, files, asynced; dent, err); if err = ESUCCESS then {ino := dent.ino; p0 := p0.tail}}}⟫ true, dirs[ino0] = mkdir(m, md, # links(ino0, dirs), m1, entries), err = ESUCCESS, ino = entries[str], afs-cons(dirs, files), px(user, md), str ∈ entries, ino0 ∈ dirs, links(ino0, dirs) ≠ ∅ ∨ ino0 = ROOT_INO, ∀ p0, err, dirs, files, ino, user, asynced. # p0 < #(str + p1) → err = ESUCCESS ∧ afs-cons(dirs, files) ∧ (ino ∈ dirs ∨ ino ∈ files) ∧ (links(ino, dirs) ≠ ∅ ∨ ino = ROOT_INO) → ⟪while p0 ≠ ε ∧ err = ESUCCESS do {choose inode with true in vfs_may_lookup#(ino, user, dirs, files, asynced; inode, err); if err = ESUCCESS then let dent = negdentry(p0.head) in {afs_lookup#(ino, dirs, files, asynced; dent, err); if err = ESUCCESS then {ino := dent.ino; p0 := p0.tail}}}⟫ true ⊦ ⟪while p1 ≠ ε ∧ err = ESUCCESS do {choose inode with true in vfs_may_lookup#(ino, user, dirs, files, asynced; inode, err); if err = ESUCCESS then let dent = negdentry(p1.head) in {afs_lookup#(ino, dirs, files, asynced; dent, err); if err = ESUCCESS then {ino := dent.ino; p1 := p1.tail}}}⟫ true

Rule argument

subst: [p0, err, dirs, files, ino, user, asynced] → [p1, err, dirs, files, ino, user, asynced]

Additional information

Used simplifier rules: ⊦ p_ino ∈ dirs ∧ dirs[p_ino] = mkdir(m, md, n, n0, entries) ∧ str ∈ entries → links(entries[str], dirs) ≠ ∅; (s)%⊦ ino ∈ dirs ∧ dirs[ino] = mkdir(n0, md, n, n1, entries) ∧ str ∈ entries ∧ afs-cons(dirs, files) → (entries[str] ∈ files ↔ ¬ entries[str] ∈ dirs); (ss)% ⊦ str + p = str ' + p; (s)% ⊦ #(str ' + p) = (# p)+1; (s)% ⊦ m +1 = m + 1; (s)%a: ⊦ (p + p0) + p1 = p + p0 + p1; from specification list[path]% ⊦ 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 Used terms: p1, err, dirs, files, ino, user, asynced