[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 108 from proof rem-stays-reachable-aux2 (previous step)

x1 + r2 ' + x2 = x + H[(r0 ' + y).last].nxt ' + (r0 ' + y).butlast, ((r0 ' + y).last ' + x0).last = (r ' + x).last, (r ' + x1).last = (r ' + x).last, (r ' + x).last ∈ H, path(r ' + x1, H), path(r ' + x + H[(r0 ' + y).last].nxt ', H), path(r ' + x1 + r2 ' + x2 + (r0 ' + y).last ', H), H[(r0 ' + y).last].nxt ∈ H, (r0 ' + y).last ∈ H, path((r0 ' + y).last ' + x0, H), (r0 ' + y).last = H[(r0 ' + y).last].nxt ∨ (r0 ' + y).last = r0 ∨ (r0 ' + y).last ∈ y ⊦ H[(r0 ' + y).last].nxt = null, (r0 ' + y).last = r, r = r2, r2 = (r0 ' + y).last, (r0 ' + y).last ∈ x, (r0 ' + y).last ∈ x0, dups(x0), r ∈ x1, r ∈ x2, r2 ∈ x1, r2 ∈ x2, (r0 ' + y).last ∈ x1, (r0 ' + y).last ∈ x2, dups(x1 + x2), dups(x1), dups(x2)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 109:
⊦ x0 ≠ [] → (x1 + x0).butlast = x1 + x0.butlast
Proof step 110:
¬ y ≠ [], x1 + r2 ' + x2 = x + H[(r0 ' + y).last].nxt ' + (r0 ' + y).butlast, ((r0 ' + y).last ' + x0).last = (r ' + x).last, (r ' + x1).last = (r ' + x).last, (r ' + x).last ∈ H, path(r ' + x1, H), path(r ' + x + H[(r0 ' + y).last].nxt ', H), path(r ' + x1 + r2 ' + x2 + (r0 ' + y).last ', H), H[(r0 ' + y).last].nxt ∈ H, (r0 ' + y).last ∈ H, path((r0 ' + y).last ' + x0, H), (r0 ' + y).last = H[(r0 ' + y).last].nxt ∨ (r0 ' + y).last = r0 ∨ (r0 ' + y).last ∈ y ⊦ H[(r0 ' + y).last].nxt = null, (r0 ' + y).last = r, r = r2, r2 = (r0 ' + y).last, (r0 ' + y).last ∈ x, (r0 ' + y).last ∈ x0, dups(x0), r ∈ x1, r ∈ x2, r2 ∈ x1, r2 ∈ x2, (r0 ' + y).last ∈ x1, (r0 ' + y).last ∈ x2, dups(x1 + x2), dups(x1), dups(x2)
Proof step 155:
y ≠ [], x1 + r2 ' + x2 = x + H[(r0 ' + y).last].nxt ' + r0 ' + y.butlast, ((r0 ' + y).last ' + x0).last = (r ' + x).last, (r ' + x1).last = (r ' + x).last, (r ' + x).last ∈ H, path(r ' + x1, H), path(r ' + x + H[(r0 ' + y).last].nxt ', H), path(r ' + x1 + r2 ' + x2 + (r0 ' + y).last ', H), H[(r0 ' + y).last].nxt ∈ H, (r0 ' + y).last ∈ H, path((r0 ' + y).last ' + x0, H), (r0 ' + y).last = H[(r0 ' + y).last].nxt ∨ (r0 ' + y).last = r0 ∨ (r0 ' + y).last ∈ y ⊦ H[(r0 ' + y).last].nxt = null, (r0 ' + y).last = r, r = r2, r2 = (r0 ' + y).last, (r0 ' + y).last ∈ x, (r0 ' + y).last ∈ x0, dups(x0), r ∈ x1, r ∈ x2, r2 ∈ x1, r2 ∈ x2, (r0 ' + y).last ∈ x1, (r0 ' + y).last ∈ x2, dups(x1 + x2), dups(x1), dups(x2)

Rule argument

spec: list-last inst: reflist name: bapp seq: ⊦ x0 ≠ [] → (x1 + x0).butlast = x1 + x0.butlast subst: [x0, x1] → [y, r0 '] flags: rewrite

Additional information

Used terms: y, r0 '