[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 52 from proof replayorphans-prepost (previous step)

ri0[key0] = ri1[key0], ro1.inodes?, ks0.inodes?, key ∈ ks0, key ∈ ro1, key0 ∈ ri0, key0 ∈ ri1, ks0 ⊆ ro1, ∀ key. (key ∈ ri0 ⊖ ks0 ↔ key ∈ ri1 ⊖ ro1) ∧ (ri0 ⊖ ks0)[key] = (ri1 ⊖ ro1)[key] ⊦ # ks0 = 0, # ro1 = 0, ks0 = ∅, ro1 = ∅, key0.data?, key0 ∈ ks0, key0 ∈ ro1, (key0 ∈ ri1 ⊖ ro1 → key0 ∈ ((ri0 \ datakeys(key, ri0)) -- key) ⊖ (ks0 -- key)) ∧ (((ri0 \ datakeys(key, ri0)) -- key) ⊖ (ks0 -- key))[key0] = ri1[key0]

Heuristic nonsplitting cut applied rule cut formula

And obtained the following new goals

Proof step 53:
ri0[key0] = ri1[key0], ks0.inodes?, ro1.inodes?, key0 ∈ ri1, key0 ∈ ri0, key ∈ ro1, key ∈ ks0, key0 ∈ ri1 ⊖ ro1, ks0 ⊆ ro1, ∀ key. (key ∈ ri0 ⊖ ks0 ↔ key ∈ ri1 ⊖ ro1) ∧ (ri0 ⊖ ks0)[key] = (ri1 ⊖ ro1)[key] ⊦ # ro1 = 0, # ks0 = 0, ks0 = ∅, ro1 = ∅, key0.data?, key0 ∈ ro1, key0 ∈ ks0, (key0 ∈ ri1 ⊖ ro1 → key0 ∈ ((ri0 \ datakeys(key, ri0)) -- key) ⊖ (ks0 -- key)) ∧ (((ri0 \ datakeys(key, ri0)) -- key) ⊖ (ks0 -- key))[key0] = ri1[key0]

Rule argument

fma: key0 ∈ ri1 ⊖ ro1

Additional information

Used simplifier rules: ⊦ key ∈ ri ⊖ ks ↔ key ∈ ri ∧ ¬ key ∈ ks ∧ (key.data? → ¬ inodekey(key.ino) ∈ ks); (ws) Used formulas: key0 ∈ ri1 ⊖ ro1