[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 21 from proof inv-gc (previous step)

# kx = # ax, as ∩ range ri0[kx, ax] = ∅, ax.toSet ∩ log0.toSet = ∅, as ∩ log0.toSet = ∅, as ∩ ax.toSet = ∅, ro.inodes?, fo.inodes?, log-cons(log0, fs0), new(ax, fs0), node-cons(fs0[adr].key, fs0[adr]), ri-cons(ri0[kx, ax], fs0[ax, fs0[ri0[kx]]]), ri-cons(ri0, fs0), ri-inj(ri0), ri-inj(fi), adr ∈ log0, adr ∈ fs0, kx.toSet ⊆ dom ri0 ⊦ dups(kx), dups(ax), adr ∈ fs0[ax, fs0[ri0[kx]]] \ as ∧ node-cons((fs0[ax, fs0[ri0[kx]]] \ as)[adr].key, (fs0[ax, fs0[ri0[kx]]] \ as)[adr])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 22:
⊦ adr ∈ fs \ as ↔ adr ∈ fs ∧ ¬ adr ∈ as
Proof step 23:
# kx = # ax, as ∩ ax.toSet = ∅, as ∩ log0.toSet = ∅, ax.toSet ∩ log0.toSet = ∅, as ∩ range ri0[kx, ax] = ∅, fo.inodes?, ro.inodes?, log-cons(log0, fs0), new(ax, fs0), node-cons(fs0[adr].key, fs0[adr]), ri-cons(ri0, fs0), ri-cons(ri0[kx, ax], fs0[ax, fs0[ri0[kx]]]), ri-inj(fi), ri-inj(ri0), adr ∈ fs0, adr ∈ log0, kx.toSet ⊆ dom ri0 ⊦ dups(ax), dups(kx), (adr ∈ fs0[ax, fs0[ri0[kx]]] ∧ ¬ adr ∈ as) ∧ node-cons((fs0[ax, fs0[ri0[kx]]] \ as)[adr].key, (fs0[ax, fs0[ri0[kx]]] \ as)[adr])

Rule argument

spec: store+set, inst: node-store name: Diff-ext seq: ⊦ adr ∈ fs \ as ↔ adr ∈ fs ∧ ¬ adr ∈ as subst: [as, fs, adr] → [as, fs0[ax, fs0[ri0[kx]]], adr] flags: <all paths>

Additional information

Used terms: as, fs0[ax, fs0[ri0[kx]]], adr