[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 54 from proof acyclic-rem-not-reachable (previous step)

ce.nxt = H[H[(r ' + x).last].nxt].nxt, path(r ' + x, H), path(r ' + x1 + H[(r ' + x).last].nxt ', H), (r ' + x).last ∈ H, H[(r ' + x).last].nxt ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ (r ' + x).last = r, (r ' + x).last = H[(r ' + x).last].nxt, H[(r ' + x).last].nxt = null, r = H[(r ' + x).last].nxt, dups(x1), dups(x), H[(r ' + x).last].nxt ∈ x1, r ∈ x1, (r ' + x).last ∈ x1, r ∈ x

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 55:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 56:
ce.nxt = H[H[x.last].nxt].nxt, path(r ' + x1 + H[x.last].nxt ', H), path(r ' + x, H), H[x.last].nxt ∈ H, x.last ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ x.last = H[x.last].nxt, x.last = r, H[x.last].nxt = null, r = H[x.last].nxt, x = [], dups(x), dups(x1), r ∈ x, x.last ∈ x1, r ∈ x1, H[x.last].nxt ∈ x1

Rule argument

spec: list-last, inst: reflist name: lapp seq: ⊦ x0 ≠ [] → (x1 + x0).last = x0.last subst: [x0, x1] → [x, r '] flags: <all paths>

Additional information

Used simplifier rules: ⊦ x0 + [] = x0; (s)%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]% ⊦ r '.last = r; (s) Used terms: x, r '