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Proof step 37 from proof replayorphans-prepost (previous step)

(key0 ∈ ri0 ∧ key0.data? ∧ inodekey(key0.ino) ∈ ks0 ⊃ (∅)[key0];(key0 ∈ ks0 ⊃ (∅)[key0];ri0[key0])) = (key0 ∈ ri1 ∧ key0.data? ∧ inodekey(key0.ino) ∈ ro1 ⊃ (∅)[key0];(key0 ∈ ro1 ⊃ (∅)[key0];ri1[key0])), key0 ∈ ri0 ∧ ¬ key0 ∈ ks0 ∧ (¬ key0.data? ∨ ¬ inodekey(key0.ino) ∈ ks0) ↔ key0 ∈ ri1 ∧ ¬ key0 ∈ ro1 ∧ (¬ key0.data? ∨ ¬ inodekey(key0.ino) ∈ ro1), ro1.inodes?, ks0.inodes?, key ∈ ks0, key ∈ ro1, ks0 ⊆ ro1, ∀ key1. (key1 ∈ ri0 ∧ ¬ key1 ∈ ks0 ∧ (¬ key1.data? ∨ ¬ inodekey(key1.ino) ∈ ks0) ↔ key1 ∈ ri1 ∧ ¬ key1 ∈ ro1 ∧ (¬ key1.data? ∨ ¬ inodekey(key1.ino) ∈ ro1)) ∧ (key1 ∈ ri0 ∧ key1.data? ∧ inodekey(key1.ino) ∈ ks0 ⊃ (∅)[key1];(key1 ∈ ks0 ⊃ (∅)[key1];ri0[key1])) = (key1 ∈ ri1 ∧ key1.data? ∧ inodekey(key1.ino) ∈ ro1 ⊃ (∅)[key1];(key1 ∈ ro1 ⊃ (∅)[key1];ri1[key1])) ⊦ # ks0 = 0, # ro1 = 0, ks0 = ∅, ro1 = ∅, key0 ∈ ri0 ∧ ¬ key0 ∈ ks0 ∧ ¬ key0.data?, ( key0 ∈ ri0 ∧ (¬ key0.data? ∨ ¬ key.inode? ∨ key0.ino ≠ key.ino) ∧ ¬ key0 ∈ ks0 ∧ ¬ key0 ∈ datakeys(ks0, (ri0 \ datakeys(key, ri0)) -- key) \ datakeys(key, (ri0 \ datakeys(key, ri0)) -- key) ↔ key0 ∈ ri1 ∧ ¬ key0 ∈ ro1 ∧ (¬ key0.data? ∨ ¬ inodekey(key0.ino) ∈ ro1)) ∧ (key = key0 ⊃ (∅)[key0] ;(key0 ∈ datakeys(ks0, (ri0 \ datakeys(key, ri0)) -- key) \ datakeys(key, (ri0 \ datakeys(key, ri0)) -- key) ⊃ (∅)[key0] ;(key0 ∈ ks0 ⊃ (∅)[key0];(key0 ∈ ri0 ∧ key0.data? ∧ key0.ino = key.ino ⊃ (∅)[key0];ri0[key0])) ) ) = (key0 ∈ ri1 ∧ key0.data? ∧ inodekey(key0.ino) ∈ ro1 ⊃ (∅)[key0];(key0 ∈ ro1 ⊃ (∅)[key0];ri1[key0]))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 38:
⊦ key ∈ ks \ ks0 ↔ key ∈ ks ∧ ¬ key ∈ ks0
Proof step 39:
(key0 ∈ ri0 ∧ key0.data? ∧ inodekey(key0.ino) ∈ ks0 ⊃ (∅)[key0];(key0 ∈ ks0 ⊃ (∅)[key0];ri0[key0])) = (key0 ∈ ri1 ∧ key0.data? ∧ inodekey(key0.ino) ∈ ro1 ⊃ (∅)[key0];(key0 ∈ ro1 ⊃ (∅)[key0];ri1[key0])), key0 ∈ ri0 ∧ ¬ key0 ∈ ks0 ∧ (¬ key0.data? ∨ ¬ inodekey(key0.ino) ∈ ks0) ↔ key0 ∈ ri1 ∧ ¬ key0 ∈ ro1 ∧ (¬ key0.data? ∨ ¬ inodekey(key0.ino) ∈ ro1), ks0.inodes?, ro1.inodes?, key ∈ ro1, key ∈ ks0, ks0 ⊆ ro1, ∀ key1. (key1 ∈ ri0 ∧ ¬ key1 ∈ ks0 ∧ (¬ key1.data? ∨ ¬ inodekey(key1.ino) ∈ ks0) ↔ key1 ∈ ri1 ∧ ¬ key1 ∈ ro1 ∧ (¬ key1.data? ∨ ¬ inodekey(key1.ino) ∈ ro1)) ∧ (key1 ∈ ri0 ∧ key1.data? ∧ inodekey(key1.ino) ∈ ks0 ⊃ (∅)[key1];(key1 ∈ ks0 ⊃ (∅)[key1];ri0[key1])) = (key1 ∈ ri1 ∧ key1.data? ∧ inodekey(key1.ino) ∈ ro1 ⊃ (∅)[key1];(key1 ∈ ro1 ⊃ (∅)[key1];ri1[key1])) ⊦ # ro1 = 0, # ks0 = 0, ks0 = ∅, ro1 = ∅, ( key0 ∈ ri0 ∧ (¬ key0.data? ∨ ¬ key.inode? ∨ key0.ino ≠ key.ino) ∧ ¬ key0 ∈ ks0 ∧ ¬(key0 ∈ datakeys(ks0, (ri0 \ datakeys(key, ri0)) -- key) ∧ ¬ key0 ∈ datakeys(key, (ri0 \ datakeys(key, ri0)) -- key)) ↔ key0 ∈ ri1 ∧ ¬ key0 ∈ ro1 ∧ (¬ key0.data? ∨ ¬ inodekey(key0.ino) ∈ ro1)) ∧ (key = key0 ⊃ (∅)[key0] ;(key0 ∈ datakeys(ks0, (ri0 \ datakeys(key, ri0)) -- key) ∧ ¬ key0 ∈ datakeys(key, (ri0 \ datakeys(key, ri0)) -- key) ⊃ (∅)[key0] ;(key0 ∈ ks0 ⊃ (∅)[key0];(key0 ∈ ri0 ∧ key0.data? ∧ key0.ino = key.ino ⊃ (∅)[key0];ri0[key0])) ) ) = (key0 ∈ ri1 ∧ key0.data? ∧ inodekey(key0.ino) ∈ ro1 ⊃ (∅)[key0];(key0 ∈ ro1 ⊃ (∅)[key0];ri1[key0])), key0 ∈ ri0 ∧ ¬ key0 ∈ ks0 ∧ ¬ key0.data?

Rule argument

spec: set-union, inst: node-index name: Difference seq: ⊦ key ∈ ks \ ks0 ↔ key ∈ ks ∧ ¬ key ∈ ks0 subst: [ks0, ks, key] → [datakeys(key, (ri0 \ datakeys(key, ri0)) -- key), datakeys(ks0, (ri0 \ datakeys(key, ri0)) -- key), key0] flags: <all paths>

Additional information

Used terms: datakeys(key, (ri0 \ datakeys(key, ri0)) -- key), datakeys(ks0, (ri0 \ datakeys(key, ri0)) -- key), key0