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Theorem fids-in

⊦ fid ∈ tree.fids ↔ (∃ p. p ∈ tree ∧ tree[p] = fnode(fid))

Type: A user defined theorem
Validity: The proof is valid.
Features:
Comment:
Used by: fids-path, fids-str, fids-strpath,
Used by units in proved state:
Proof exists?: Yes

View the proof

Infos about the proof

Proved?: Yes
Interactions: 18
Proof steps: 30
Used lemmas: fids-ins, fids-emp, fids-dnode, fids-fnode,
Used spec theorems:
- directory-tree/tree-induction
- store/directory-tree-inst/exnotempty
- bag-union/fidbag/in-union
- store/directory-tree-inst/exin
Used simplifier rules:
- directory-tree-base/fnode-inj
- directory-tree/fnode-notin
- bag-union/fidbag/in-emp
- directory-tree-base/Size-insert
- nat-basic/s-04
- nat-basic/ref
- nat-basic/a
- nat-basic/c
- store/directory-tree-inst/At-in-out
- directory-tree-base/In-insert
- directory-tree-base/disj
- directory-tree/in-dnode
- store/directory-tree-inst/insin
- directory-tree/at-dnode-same
- directory-tree/nil-at
- directory-tree/nil-in
- bag-union/fidbag/in-one
- directory-tree/split-in
- list/path/c
- directory-tree/one-inf
- directory-tree/one-at
- list/path/a
- directory-tree/at-dnode
- directory-tree/one-ind
- directory-tree-base/At-same
- bag-union/fidbag/in-union
- bag-union/fidbag/union-ass
- bag-union/fidbag/union-com