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Proof step 44 from proof inv-gc (previous step)

# kx = # ax, replay-log(log0, fs0, fi ⊖ fo) = ri0 ⊖ ro, as ∩ range ri0[kx, ax] = ∅, ax.toSet ∩ log0.toSet = ∅, as ∩ log0.toSet = ∅, as ∩ ax.toSet = ∅, fo.inodes?, ro.inodes?, file-node-cons(fs0, ri0), log-cons(log0, fs0), log-cons(ax, fs0[ax, fs0[ri0[kx]]] \ as), log-cons(log0, fs0[ax, fs0[ri0[kx]]] \ as), new(ax, fs0), orphans-cons(fs0, ri0, ro), replay-safe(log0, fs0, fi ⊖ fo), ri-closed(ri0, fs0, ro), ri-cons(ri0, fs0), ri-cons(ri0[kx, ax], fs0[ax, fs0[ri0[kx]]] \ as), ri-inj(fi), ri-inj(ri0), ri-inj(ri0[kx, ax]), fo ⊆ dom fi, ro ⊆ dom ri0, kx.toSet ⊆ dom ri0 ⊦ dups(log0), dups(kx), dups(ax), orphans-cons(fs0[ax, fs0[ri0[kx]]] \ as, ri0[kx, ax], ro), inodekey(0) ∈ ri0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 45:
# kx = # ax, fs0[ri0[inodekey(ino)]].nlink = 0 ∧ ino ≠ ROOT_INO ↔ inodekey(ino) ∈ ro, replay-log(log0, fs0, fi ⊖ fo) = ri0 ⊖ ro, as ∩ range ri0[kx, ax] = ∅, ax.toSet ∩ log0.toSet = ∅, as ∩ log0.toSet = ∅, as ∩ ax.toSet = ∅, fo.inodes?, ro.inodes?, file-node-cons(fs0, ri0), log-cons(log0, fs0), log-cons(ax, fs0[ax, fs0[ri0[kx]]] \ as), log-cons(log0, fs0[ax, fs0[ri0[kx]]] \ as), new(ax, fs0), orphans-cons(fs0, ri0, ro), replay-safe(log0, fs0, fi ⊖ fo), ri-closed(ri0, fs0, ro), ri-cons(ri0, fs0), ri-cons(ri0[kx, ax], fs0[ax, fs0[ri0[kx]]] \ as), ri-inj(fi), ri-inj(ri0), ri-inj(ri0[kx, ax]), inodekey(ino) ∈ ri0[kx, ax], inodekey(ino) ∈ ri0, fo ⊆ dom fi, ro ⊆ dom ri0, kx.toSet ⊆ dom ri0 ⊦ (fs0[ax, fs0[ri0[kx]]] \ as)[ri0[kx, ax][inodekey(ino)]].nlink = 0 ∧ ino ≠ ROOT_INO ↔ inodekey(ino) ∈ ro, dups(log0), dups(kx), dups(ax), inodekey(0) ∈ ri0

Rule argument

Additional information

Used simplifier rules: ⊦ kx.toSet ⊆ dom ri ∧ ¬ key ∈ ri → ¬ key ∈ kx; (s)% ⊦ # kx = # ax → (key ∈ ri[kx, ax] ↔ key ∈ ri ∨ key ∈ kx); (ws)%⊦ orphans-cons(fs, ri, ro) → ( orphans-cons(fs0, ri0, ro0) ↔ (∀ ino. ( inodekey(ino) ∈ ri → ( fs[ri[inodekey(ino)]].nlink = 0 ∧ ino ≠ ROOT_INO ↔ inodekey(ino) ∈ ro)) → inodekey(ino) ∈ ri0 → ( fs0[ri0[inodekey(ino)]].nlink = 0 ∧ ino ≠ ROOT_INO ↔ inodekey(ino) ∈ ro0))); (ss)