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Proof step 29 from proof inv-gc (previous step)

# kx = # ax, 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-cons-strong(fs0, ri0, ro, fi, fo, log0), 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), kx.toSet ⊆ dom ri0, ro ⊆ dom ri0, fo ⊆ dom fi ⊦ dups(ax), dups(kx), dups(log0), inodekey(0) ∈ ri0, ri-closed(ri0[kx, ax], fs0[ax, fs0[ri0[kx]]] \ as, ro) ∧ (replay-safe(log0 + ax, fs0[ax, fs0[ri0[kx]]] \ as, fi ⊖ fo) ∧ replay(log0 + ax, fs0[ax, fs0[ri0[kx]]] \ as, fi, fo) = ri0[kx, ax] ⊖ ro) ∧ file-node-cons(fs0[ax, fs0[ri0[kx]]] \ as, ri0[kx, ax]) ∧ orphans-cons(fs0[ax, fs0[ri0[kx]]] \ as, ri0[kx, ax], ro)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 30:
# kx = # ax, 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-cons-strong(fs0, ri0, ro, fi, fo, log0), 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), inodekey(0) ∈ ri0, ri-closed(ri0[kx, ax], fs0[ax, fs0[ri0[kx]]] \ as, ro) ∧ replay-safe(ax, fs0[ax, fs0[ri0[kx]]], replay-log(log0, fs0, fi ⊖ fo)) ∧ replay-log(ax, fs0[ax, fs0[ri0[kx]]], replay-log(log0, fs0, fi ⊖ fo)) = ri0[kx, ax] ⊖ ro ∧ file-node-cons(fs0[ax, fs0[ri0[kx]]] \ as, ri0[kx, ax]) ∧ orphans-cons(fs0[ax, fs0[ri0[kx]]] \ as, ri0[kx, ax], ro)

Rule argument

Additional information

Used simplifier rules: ⊦ replay(log, fs, fi, fo) = replay-log(log, fs, fi ⊖ fo); (s)%⊦ replay(ax0 + ax1, fs, fi, fo) = replay-log(ax1, fs, replay(ax0, fs, fi, fo)); (s)%a: ⊦ (ax + ax0) + ax1 = ax + ax0 + ax1; from specification list[address-list-set]%# ax = # ndx ⊦ ax.toSet ∩ log.toSet = ∅ → replay-log(log, fs[ax, ndx], ri) = replay-log(log, fs, ri); (s)%⊦ as ∩ log.toSet = ∅ → replay-log(log, fs \ as, ri) = replay-log(log, fs, ri); (s)%⊦ replay-cons-strong(fs, ri, ro, fi, fo, log) → replay-safe(log, fs, fi ⊖ fo); (ss)%# ax = # ndx ⊦ ax.toSet ∩ log.toSet = ∅ → (replay-safe(log, fs[ax, ndx], ri) ↔ replay-safe(log, fs, ri)); (s)% ⊦ # ri[kx] = # kx; (s)% ⊦ # fs[ax] = # ax; (s)%⊦ as ∩ log.toSet = ∅ → (replay-safe(log, fs \ as, ri) ↔ replay-safe(log, fs, ri)); (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]%⊦ replay-safe(log0 + log1, fs, ri) ↔ replay-safe(log0, fs, ri) ∧ replay-safe(log1, fs, replay-log(log0, fs, ri)); (s)