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Proof step 39 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]), inodekey(ino) ∈ ri0[kx, ax], fo ⊆ dom fi, ro ⊆ dom ri0, kx.toSet ⊆ dom ri0, inodekey(ino) ∈ ri0 ∧ ¬ fs0[ri0[inodekey(ino)]].directory → fs0[ri0[inodekey(ino)]].nsubdirs = 0 ⊦ (fs0[ax, fs0[ri0[kx]]] \ as)[ri0[kx, ax][inodekey(ino)]].nsubdirs = 0, (fs0[ax, fs0[ri0[kx]]] \ as)[ri0[kx, ax][inodekey(ino)]].directory, dups(log0), dups(kx), dups(ax), inodekey(0) ∈ ri0, ri0[kx, ax][inodekey(ino)] ∈ as

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 40:
# kx = # ax, replay-log(log0, fs0, fi ⊖ fo) = ri0 ⊖ ro, as ∩ ax.toSet = ∅, as ∩ log0.toSet = ∅, ax.toSet ∩ log0.toSet = ∅, as ∩ range ri0[kx, ax] = ∅, ro.inodes?, fo.inodes?, file-node-cons(fs0, ri0), log-cons(log0, fs0[ax, fs0[ri0[kx]]] \ as), log-cons(ax, fs0[ax, fs0[ri0[kx]]] \ as), log-cons(log0, fs0), new(ax, fs0), orphans-cons(fs0, ri0, ro), replay-safe(log0, fs0, fi ⊖ fo), ri-closed(ri0, fs0, ro), ri-cons(ri0[kx, ax], fs0[ax, fs0[ri0[kx]]] \ as), ri-cons(ri0, fs0), ri-inj(ri0[kx, ax]), ri-inj(ri0), ri-inj(fi), inodekey(ino) ∈ ri0[kx, ax], kx.toSet ⊆ dom ri0, ro ⊆ dom ri0, fo ⊆ dom fi, inodekey(ino) ∈ ri0 ∧ ¬ fs0[ri0[inodekey(ino)]].directory → fs0[ri0[inodekey(ino)]].nsubdirs = 0 ⊦ fs0[ax, fs0[ri0[kx]]][ri0[kx, ax][inodekey(ino)]].nsubdirs = 0, (fs0[ax, fs0[ri0[kx]]] \ as)[ri0[kx, ax][inodekey(ino)]].directory, dups(ax), dups(kx), dups(log0), inodekey(0) ∈ ri0

Rule argument

Additional information

Used simplifier rules: ⊦ key ∈ ri ∧ as ∩ range ri = ∅ → ¬ ri[key] ∈ as; (s)%int-c: ⊦ as ∩ as0 = as0 ∩ as; from specification set-union[node-store]%int-a: ⊦ (as ∩ as0) ∩ as1 = as ∩ as0 ∩ as1; from specification set-union[node-store]% ⊦ ¬ adr ∈ as → (fs \ as)[adr] = fs[adr]; (s)