[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 45 from proof rem-stays-reachable-aux (previous step)

ce.nxt = H[H[r0].nxt].nxt, path(r ' + x2 + H[r0].nxt ', H[r0, ce]), path(r ' + x1 + r0 ', H), H[r0].nxt ∈ H, r0 ∈ H, H[r0].nxt ∈ H[r0, ce] ⊦ r0 ∈ x2, H[r0].nxt = ce.nxt, H[r0].nxt = null, r = r0, r = H[r0].nxt, r ∈ x1, r0 ∈ x1, dups(x1), r ∈ x2, H[r0].nxt ∈ x2, dups(x2), ∃ x0. path(H[r0].nxt ' + x0 + H[r0].nxt ', H) ∧ ¬ r0 ∈ x0, ∃ x. path(r ' + x + H[r0].nxt ', H) ∧ ¬ r0 ∈ x

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 46:
⊦ ¬ r0 ∈ x ∧ r0 ∈ H → (path(x + r1 ', H[r0, ce]) ↔ path(x + r1 ', H))
Proof step 47:
¬ r0 ∈ r ' + x2 ∧ r0 ∈ H, ce.nxt = H[H[r0].nxt].nxt, path((r ' + x2) + H[r0].nxt ', H), path(r ' + x1 + r0 ', H), H[r0].nxt ∈ H, r0 ∈ H, H[r0].nxt ∈ H[r0, ce] ⊦ r0 ∈ x2, H[r0].nxt = ce.nxt, H[r0].nxt = null, r = r0, r = H[r0].nxt, r ∈ x1, r0 ∈ x1, dups(x1), r ∈ x2, H[r0].nxt ∈ x2, dups(x2), ∃ x0. path(H[r0].nxt ' + x0 + H[r0].nxt ', H) ∧ ¬ r0 ∈ x0, ∃ x. path(r ' + x + H[r0].nxt ', H) ∧ ¬ r0 ∈ x

Rule argument

spec: inst: name: path-modify-strong seq: ⊦ ¬ r0 ∈ x ∧ r0 ∈ H → (path(x + r1 ', H[r0, ce]) ↔ path(x + r1 ', H)) subst: [ce, r0, H, r1, x] → [ce, r0, H, H[r0].nxt, r ' + x2] flags: rewrite, current

Additional information

Used simplifier rules: in-01: ⊦ r ≠ r0 → (r ∈ r0 ' + x0 ↔ r ∈ x0); from specification list[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist] Used terms: ce, r0, H, H[r0].nxt, r ' + x2