Proof step 1 from proof rem-stays-reachable-aux (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ h[h[r0].nxt].nxt, reachable(r, r0, h), h[r0].nxt ∈ h ⊦ reachable(r, h[r0].nxt, h[r0 := ce]) ↔ h[h[r0].nxt].nxt = h[r0].nxt ∨ r ≠ r0 ∧ r = h[r0].nxt ∨ (∃ x. path(r ' + x + h[r0].nxt ', h) ∧ r0 ≠ r ∧ ¬ r0 ∈ x) ∨ (∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x)

Rule argument

Additional information

Proof step 2 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, reachable(r, r0, h), h[r0].nxt ∈ h, r0 ∈ h ⊦ reachable(r, h[r0].nxt, h[r0 := ce]) ↔ h[r0].nxt ≠ r0 ∧ r = h[r0].nxt ∨ (∃ x. path(r ' + x + h[r0].nxt ', h) ∧ r0 ≠ r ∧ ¬ r0 ∈ x) ∨ (∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x)

Rule argument

Additional information

Used simplifier rules: reach-alloc: ⊦ reachable(r, r1, h) → r1 ∈ h;

Proof step 3 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 4:

  ⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0 ∧ ¬ dups(r ' + x))

• Proof step 5:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r0 ∈ h, h[r0].nxt ∈ h, ∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0 ∧ ¬ dups(r ' + x) ⊦ reachable(r, h[r0].nxt, h[r0 := ce]) ↔ h[r0].nxt ≠ r0 ∧ r = h[r0].nxt ∨ (∃ x. path(r ' + x + h[r0].nxt ', h) ∧ r0 ≠ r ∧ ¬ r0 ∈ x) ∨ (∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x)

Rule argument

Additional information

Used terms: h, r0, r

Proof step 4 from proof rem-stays-reachable-aux (previous step) (back to tree)

Lemma reachable-nodups-def

Proof step 5 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 6:

  ce.nxt = h[h[(r ' + x).last].nxt].nxt, h[(r ' + x).last].nxt ≠ null, h[(r ' + x).last].nxt ≠ ce.nxt, path(r ' + x, h), h[(r ' + x).last].nxt ∈ h, (r ' + x).last ∈ h, ¬ dups(x), ¬ r ∈ x ⊦ reachable(r, h[(r ' + x).last].nxt, h[(r ' + x).last := ce]) ↔ h[(r ' + x).last].nxt ≠ (r ' + x).last ∧ r = h[(r ' + x).last].nxt ∨ (∃ x0. path(r ' + x0 + h[(r ' + x).last].nxt ', h) ∧ (r ' + x).last ≠ r ∧ ¬ (r ' + x).last ∈ x0) ∨ (∃ x0. path(h[(r ' + x).last].nxt ' + x0 + h[(r ' + x).last].nxt ', h) ∧ ¬ (r ' + x).last ∈ x0)

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ dups(a ' + x) ↔ a ∈ x ∨ dups(x); (s)

Proof step 6 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 7:

  ce.nxt = h[h[(r ' + x).last].nxt].nxt, h[(r ' + x).last].nxt ≠ null, h[(r ' + x).last].nxt ≠ ce.nxt, path(r ' + x, h), reachable(r, h[(r ' + x).last].nxt, h[(r ' + x).last := ce]), (r ' + x).last ∈ h, h[(r ' + x).last].nxt ∈ h, ¬ dups(x), ¬ r ∈ x ⊦ ∃ x0. path(r ' + x0 + h[(r ' + x).last].nxt ', h) ∧ (r ' + x).last ≠ r ∧ ¬ (r ' + x).last ∈ x0, ∃ x0. path(h[(r ' + x).last].nxt ' + x0 + h[(r ' + x).last].nxt ', h) ∧ ¬ (r ' + x).last ∈ x0, h[(r ' + x).last].nxt ≠ (r ' + x).last ∧ r = h[(r ' + x).last].nxt

• Proof step 50:

  ce.nxt = h[h[(r ' + x).last].nxt].nxt, h[(r ' + x).last].nxt ≠ null, h[(r ' + x).last].nxt ≠ ce.nxt, path(r ' + x, h), (r ' + x).last ∈ h, h[(r ' + x).last].nxt ∈ h, ¬ dups(x), ¬ reachable(r, h[(r ' + x).last].nxt, h[(r ' + x).last := ce]), ¬ r ∈ x, h[(r ' + x).last].nxt ≠ (r ' + x).last ∧ r = h[(r ' + x).last].nxt ∨ (∃ x0. path(r ' + x0 + h[(r ' + x).last].nxt ', h) ∧ (r ' + x).last ≠ r ∧ ¬ (r ' + x).last ∈ x0) ∨ (∃ x0. path(h[(r ' + x).last].nxt ' + x0 + h[(r ' + x).last].nxt ', h) ∧ ¬ (r ' + x).last ∈ x0) ⊦

Rule argument

Additional information

Proof step 7 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 8:

  ce.nxt = h[h[(r ' + x).last].nxt].nxt, h[(r ' + x).last].nxt ≠ null, h[(r ' + x).last].nxt ≠ ce.nxt, path(r ' + x, h), reachable(r, h[(r ' + x).last].nxt, h[(r ' + x).last := ce]), h[(r ' + x).last].nxt ∈ h, (r ' + x).last ∈ h, ¬ dups(x), ¬ r ∈ x ⊦ ∃ x0. path(h[(r ' + x).last].nxt ' + x0 + h[(r ' + x).last].nxt ', h) ∧ ¬ (r ' + x).last ∈ x0, ∃ x0. path(r ' + x0 + h[(r ' + x).last].nxt ', h) ∧ (r ' + x).last ≠ r ∧ ¬ (r ' + x).last ∈ x0, h[(r ' + x).last].nxt ≠ (r ' + x).last ∧ r = h[(r ' + x).last].nxt

Rule argument

Additional information

Proof step 8 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 9:

  ⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0 ∧ ¬ dups(r ' + x))

• Proof step 10:

  ce.nxt = h[h[(r ' + x).last].nxt].nxt, h[(r ' + x).last].nxt ≠ null, h[(r ' + x).last].nxt ≠ ce.nxt, path(r ' + x, h), (r ' + x).last ∈ h, h[(r ' + x).last].nxt ∈ h, ¬ dups(x), ¬ r ∈ x, ∃ x0. path(r ' + x0, h[(r ' + x).last := ce]) ∧ (r ' + x0).last = h[(r ' + x).last].nxt ∧ ¬ dups(r ' + x0) ⊦ ∃ x0. path(r ' + x0 + h[(r ' + x).last].nxt ', h) ∧ (r ' + x).last ≠ r ∧ ¬ (r ' + x).last ∈ x0, ∃ x0. path(h[(r ' + x).last].nxt ' + x0 + h[(r ' + x).last].nxt ', h) ∧ ¬ (r ' + x).last ∈ x0, h[(r ' + x).last].nxt ≠ (r ' + x).last ∧ r = h[(r ' + x).last].nxt

Rule argument

Additional information

Used terms: h[(r ' + x).last := ce], h[(r ' + x).last].nxt, r

Proof step 9 from proof rem-stays-reachable-aux (previous step) (back to tree)

Lemma reachable-nodups-def

Proof step 10 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 11:

  (r ' + x0).last = h[(r ' + x).last].nxt, ce.nxt = h[(r ' + x0).last].nxt, (r ' + x0).last ≠ null, (r ' + x0).last ≠ ce.nxt, path(r ' + x0, h[(r ' + x).last := ce]), path(r ' + x, h), (r ' + x0).last ∈ h, (r ' + x).last ∈ h, ¬ dups(x), ¬ dups(x0), ¬ r ∈ x, ¬ r ∈ x0 ⊦ ∃ x0. path(h[(r ' + x).last].nxt ' + x0 + h[(r ' + x).last].nxt ', h) ∧ ¬ (r ' + x).last ∈ x0, ∃ x0. path(r ' + x0 + h[(r ' + x).last].nxt ', h) ∧ (r ' + x).last ≠ r ∧ ¬ (r ' + x).last ∈ x0, (r ' + x0).last ≠ (r ' + x).last ∧ r = (r ' + x0).last

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ dups(a ' + x) ↔ a ∈ x ∨ dups(x); (s)

Proof step 11 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 12:

  ⊦ y ≠ [] → (x + y).last = y.last

• Proof step 13:

  (r ' + x0).last = h[(r ' + x).last].nxt, ce.nxt = h[(r ' + x0).last].nxt, x = [], (r ' + x0).last ≠ null, (r ' + x0).last ≠ ce.nxt, path(r ' + x, h), path(r ' + x0, h[(r ' + x).last := ce]), (r ' + x).last ∈ h, (r ' + x0).last ∈ h, ¬ dups(x), ¬ dups(x0), ¬ r ∈ x, ¬ r ∈ x0 ⊦ ∃ x0. path(r ' + x0 + h[(r ' + x).last].nxt ', h) ∧ (r ' + x).last ≠ r ∧ ¬ (r ' + x).last ∈ x0, ∃ x0. path(h[(r ' + x).last].nxt ' + x0 + h[(r ' + x).last].nxt ', h) ∧ ¬ (r ' + x).last ∈ x0, (r ' + x0).last ≠ (r ' + x).last ∧ r = (r ' + x0).last

• Proof step 25:

  (r ' + x0).last = h[x.last].nxt, ce.nxt = h[(r ' + x0).last].nxt, (r ' + x0).last ≠ null, (r ' + x0).last ≠ ce.nxt, x ≠ [], path(r ' + x, h), path(r ' + x0, h[x.last := ce]), x.last ∈ h, (r ' + x0).last ∈ h, ¬ dups(x), ¬ dups(x0), ¬ r ∈ x, ¬ r ∈ x0 ⊦ ∃ x0. path(r ' + x0 + h[x.last].nxt ', h) ∧ x.last ≠ r ∧ ¬ x.last ∈ x0, ∃ x0. path(h[x.last].nxt ' + x0 + h[x.last].nxt ', h) ∧ ¬ x.last ∈ x0, (r ' + x0).last ≠ x.last ∧ r = (r ' + x0).last

Rule argument

Additional information

Used terms: x, r '

Proof step 12 from proof rem-stays-reachable-aux (previous step) (back to tree)

A spec-lemma

Proof step 13 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 14:

  (r ' + x0).last = h[r].nxt, ce.nxt = h[h[r].nxt].nxt, h[r].nxt ≠ null, h[r].nxt ≠ ce.nxt, r ≠ null, path(r ' + x0, h[r := ce]), h[r].nxt ∈ h, r ∈ h, ¬ dups(x0), ¬ r ∈ x0 ⊦ ∃ x. path(h[r].nxt ' + x + h[r].nxt ', h) ∧ ¬ r ∈ x

Rule argument

Additional information

Used simplifier rules: path-one: ⊦ path(r ', h) ↔ r ∈ h ∧ r ≠ null; ⊦ a '.last = a; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ x + [] = x; (s) ⊦ ¬ a ∈ []; (s) ⊦ ¬ dups([]); (s)

Proof step 14 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule constructor cut

And obtained the following new goals

• Proof step 15:

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

Rule argument

Additional information

Used simplifier rules: path-one: ⊦ path(r ', h) ↔ r ∈ h ∧ r ≠ null; list('a) freely generated by [], +;; ⊦ a '.last = a; (s) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ st[a := d]; (s) ⊦ ¬ dups([]); (s) ⊦ ¬ a ∈ []; (s)

Proof step 15 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 16:

  (ce.nxt ' + x1).last = h[r].nxt, ce.nxt = h[h[r].nxt].nxt, h[r].nxt ≠ ce.nxt, h[r].nxt ≠ null, r ≠ ce.nxt, r ≠ null, path(ce.nxt ' + x1, h[r := ce]), r ∈ h, h[r].nxt ∈ h, ¬ dups(x1), ¬ r ∈ x1, ¬ ce.nxt ∈ x1 ⊦ ∃ x. path(h[r].nxt ' + x + h[r].nxt ', h) ∧ ¬ r ∈ x

Rule argument

Additional information

Used simplifier rules: path-cons: ⊦ path(r ' + r0 ' + x, h) ↔ r ∈ h ∧ r ≠ null ∧ h[r].nxt = r0 ∧ path(r0 ' + x, h); a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ a0 ' + x ↔ ¬(a ≠ a0 ∧ ¬ a ∈ x); (ss) ⊦ a + x = a ' + x; (s) ⊦ dups(a ' + x) ↔ a ∈ x ∨ dups(x); (s) ⊦ (st[a := d])[a] = d; (s) ⊦ a ∈ st[a := d]; (s) ⊦ (b ' + a ' + x).last = (a ' + x).last; (s)

Proof step 16 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 17:

  ⊦ y ≠ [] → (x + y).last = y.last

• Proof step 18:

  x1.last = h[r].nxt, ce.nxt = h[h[r].nxt].nxt, h[r].nxt ≠ ce.nxt, h[r].nxt ≠ null, r ≠ ce.nxt, r ≠ null, x1 ≠ [], path(ce.nxt ' + x1, h[r := ce]), h[r].nxt ∈ h, r ∈ h, ¬ dups(x1), ¬ r ∈ x1, ¬ ce.nxt ∈ x1 ⊦ ∃ x. path(h[r].nxt ' + x + h[r].nxt ', h) ∧ ¬ r ∈ x

Rule argument

Additional information

Used simplifier rules: ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a '.last = a; (s) Used terms: x1, ce.nxt '

Proof step 17 from proof rem-stays-reachable-aux (previous step) (back to tree)

A spec-lemma

Proof step 18 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 19:

  x1.last = h[r].nxt, ce.nxt = h[x1.last].nxt, x1.last ≠ ce.nxt, x1.last ≠ null, r ≠ ce.nxt, r ≠ null, x1 ≠ [], path(ce.nxt ' + x1, h[r := ce]), r ∈ h, x1.last ∈ h, ¬ dups(x1), ¬ r ∈ x1, ¬ ce.nxt ∈ x1 ⊦ ∃ x. path(x1.last ' + x + x1.last ', h) ∧ ¬ r ∈ x

Rule argument

Additional information

Proof step 19 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply elim lemma

And obtained the following new goals

• Proof step 20:

  ce.nxt = h[r0].nxt, r0 = h[r].nxt, x1 = y + r0 ', r ≠ ce.nxt, r ≠ null, r0 ≠ ce.nxt, r0 ≠ null, x1 ≠ [], path(ce.nxt ' + x1, h[r := ce]), r0 ∈ h, r ∈ h, ¬ dups(x1), ¬ r ∈ x1, ¬ ce.nxt ∈ x1 ⊦ ∃ x. path(r0 ' + x + r0 ', h) ∧ ¬ r ∈ x

Rule argument

Additional information

Used simplifier rules: x ≠ [] ⊦ a = x.last ∧ y = x.butlast ↔ x = y + a '; (elim) Used terms: x1, a, y

Proof step 20 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 21:

  ce.nxt = h[h[r].nxt].nxt, h[r].nxt ≠ ce.nxt, h[r].nxt ≠ null, r ≠ ce.nxt, r ≠ null, r ≠ h[r].nxt, path(ce.nxt ' + y + h[r].nxt ', h[r := ce]), r ∈ h, h[r].nxt ∈ h, ¬ dups(y), ¬ h[r].nxt ∈ y, ¬ r ∈ y, ¬ ce.nxt ∈ y ⊦ ∃ x. path(h[r].nxt ' + x + h[r].nxt ', h) ∧ ¬ r ∈ x

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ dups(x + a ') ↔ a ∈ x ∨ dups(x); (ws) ⊦ x + a ' ≠ []; (s)

Proof step 21 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 22:

  ⊦ ¬ r0 ∈ x → (path(x, h[r0 := ce]) ↔ path(x, h))

• Proof step 23:

  ce.nxt = h[h[r].nxt].nxt, h[r].nxt ≠ ce.nxt, h[r].nxt ≠ null, r ≠ ce.nxt, r ≠ null, r ≠ h[r].nxt, path(ce.nxt ' + y + h[r].nxt ', h), h[r].nxt ∈ h, r ∈ h, ¬ dups(y), ¬ r ∈ ce.nxt ' + y + h[r].nxt ', ¬ h[r].nxt ∈ y, ¬ r ∈ y, ¬ ce.nxt ∈ y ⊦ ∃ x. path(h[r].nxt ' + x + h[r].nxt ', h) ∧ ¬ r ∈ x

Rule argument

Additional information

Used simplifier rules: ⊦ a ≠ a0 → (a ∈ a0 ' + x ↔ a ∈ x); (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a ∈ a0 ' ↔ a = a0; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist Used terms: ce, r, h, ce.nxt ' + y + h[r].nxt '

Proof step 22 from proof rem-stays-reachable-aux (previous step) (back to tree)

Lemma path-modify

Proof step 23 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 24:

  ce.nxt = h[h[r].nxt].nxt, h[r].nxt ≠ ce.nxt, h[r].nxt ≠ null, r ≠ ce.nxt, r ≠ null, r ≠ h[r].nxt, path(ce.nxt ' + y + h[r].nxt ', h), r ∈ h, h[r].nxt ∈ h, ¬ dups(y), ¬ h[r].nxt ∈ y, ¬ r ∈ y, ¬ ce.nxt ∈ y ⊦ ∃ x. path(h[r].nxt ' + x + h[r].nxt ', h) ∧ ¬ r ∈ x

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a ≠ a0 → (a ∈ a0 ' + x ↔ a ∈ x); (s)

Proof step 24 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule exists right

And closed the goal!

Rule argument

Additional information

Used simplifier rules: path-cons: ⊦ path(r ' + r0 ' + x, h) ↔ r ∈ h ∧ r ≠ null ∧ h[r].nxt = r0 ∧ path(r0 ' + x, h); ⊦ a ≠ a0 → (a ∈ a0 ' + x ↔ a ∈ x); (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist Used terms: ce.nxt ' + y

Proof step 25 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 26:

  (r ' + x0).last = h[x.last].nxt, ce.nxt = h[h[x.last].nxt].nxt, h[x.last].nxt ≠ null, h[x.last].nxt ≠ ce.nxt, x ≠ [], path(r ' + x0, h[x.last := ce]), path(r ' + x, h), h[x.last].nxt ∈ h, x.last ∈ h, ¬ dups(x), ¬ dups(x0), ¬ r ∈ x, ¬ r ∈ x0 ⊦ ∃ x0. path(h[x.last].nxt ' + x0 + h[x.last].nxt ', h) ∧ ¬ x.last ∈ x0, ∃ x0. path(r ' + x0 + h[x.last].nxt ', h) ∧ x.last ≠ r ∧ ¬ x.last ∈ x0, h[x.last].nxt ≠ x.last ∧ r = h[x.last].nxt

Rule argument

Additional information

Proof step 26 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply elim lemma

And obtained the following new goals

• Proof step 27:

  (r ' + x0).last = h[r0].nxt, ce.nxt = h[h[r0].nxt].nxt, x = y + r0 ', h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, x ≠ [], path(r ' + x, h), path(r ' + x0, h[r0 := ce]), r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(x), ¬ dups(x0), ¬ r ∈ x, ¬ r ∈ x0 ⊦ ∃ x0. path(r ' + x0 + h[r0].nxt ', h) ∧ r0 ≠ r ∧ ¬ r0 ∈ x0, ∃ x0. path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0, h[r0].nxt ≠ r0 ∧ r = h[r0].nxt

Rule argument

Additional information

Used simplifier rules: x ≠ [] ⊦ a = x.last ∧ y = x.butlast ↔ x = y + a '; (elim) Used terms: x, a, y

Proof step 27 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 28:

  (r ' + x0).last = h[r0].nxt, ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, r ≠ h[r0].nxt, path(r ' + x0, h[r0 := ce]), path(r ' + y + r0 ', h), h[r0].nxt ∈ h, r0 ∈ h, ¬ dups(y), ¬ dups(x0), ¬ r ∈ x0, ¬ r0 ∈ y, ¬ r ∈ y ⊦ ∃ x0. path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0, ∃ x. path(r ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ dups(x + a ') ↔ a ∈ x ∨ dups(x); (ws) ⊦ x + a ' ≠ []; (s)

Proof step 28 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 29:

  ⊦ y ≠ [] → (x + y).last = y.last

• Proof step 30:

  x0.last = h[r0].nxt, ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, r ≠ h[r0].nxt, x0 ≠ [], path(r ' + y + r0 ', h), path(r ' + x0, h[r0 := ce]), r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(y), ¬ dups(x0), ¬ r ∈ x0, ¬ r0 ∈ y, ¬ r ∈ y ⊦ ∃ x. path(r ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x, ∃ x0. path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0

Rule argument

Additional information

Used simplifier rules: ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a '.last = a; (s) Used terms: x0, r '

Proof step 29 from proof rem-stays-reachable-aux (previous step) (back to tree)

A spec-lemma

Proof step 30 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 31:

  x0.last = h[r0].nxt, ce.nxt = h[x0.last].nxt, x0.last ≠ null, x0.last ≠ ce.nxt, r ≠ r0, r ≠ x0.last, x0 ≠ [], path(r ' + x0, h[r0 := ce]), path(r ' + y + r0 ', h), x0.last ∈ h, r0 ∈ h, ¬ dups(y), ¬ dups(x0), ¬ r ∈ x0, ¬ r0 ∈ y, ¬ r ∈ y ⊦ ∃ x0. path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0, ∃ x. path(r ' + x + x0.last ', h) ∧ ¬ r0 ∈ x

Rule argument

Additional information

Proof step 31 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply elim lemma

And obtained the following new goals

• Proof step 32:

  ce.nxt = h[r1].nxt, r1 = h[r0].nxt, x0 = y0 + r1 ', r ≠ r0, r ≠ r1, r1 ≠ null, r1 ≠ ce.nxt, x0 ≠ [], path(r ' + y + r0 ', h), path(r ' + x0, h[r0 := ce]), r0 ∈ h, r1 ∈ h, ¬ dups(y), ¬ dups(x0), ¬ r ∈ x0, ¬ r0 ∈ y, ¬ r ∈ y ⊦ ∃ x. path(r ' + x + r1 ', h) ∧ ¬ r0 ∈ x, ∃ x0. path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0

Rule argument

Additional information

Used simplifier rules: x ≠ [] ⊦ a = x.last ∧ y = x.butlast ↔ x = y + a '; (elim) Used terms: x0, a, y0

Proof step 32 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 33:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, r ≠ h[r0].nxt, path(r ' + y0 + h[r0].nxt ', h[r0 := ce]), path(r ' + y + r0 ', h), h[r0].nxt ∈ h, r0 ∈ h, ¬ dups(y), ¬ dups(y0), ¬ r0 ∈ y, ¬ r ∈ y, ¬ h[r0].nxt ∈ y0, ¬ r ∈ y0 ⊦ ∃ x0. path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0, ∃ x. path(r ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ dups(x + a ') ↔ a ∈ x ∨ dups(x); (ws) ⊦ x + a ' ≠ []; (s)

Proof step 33 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 34:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, r ≠ h[r0].nxt, path(r ' + y + r0 ', h), path(r ' + y0 + h[r0].nxt ', h[r0 := ce]), r0 ∈ h, h[r0].nxt ∈ h, r0 ∈ y0, ¬ dups(y), ¬ dups(y0), ¬ r0 ∈ y, ¬ r ∈ y, ¬ h[r0].nxt ∈ y0, ¬ r ∈ y0 ⊦ ∃ x. path(r ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x, ∃ x0. path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0

• Proof step 46:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, r ≠ h[r0].nxt, path(r ' + y + r0 ', h), path(r ' + y0 + h[r0].nxt ', h[r0 := ce]), r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(y), ¬ dups(y0), ¬ r0 ∈ y0, ¬ r0 ∈ y, ¬ r ∈ y, ¬ h[r0].nxt ∈ y0, ¬ r ∈ y0 ⊦ ∃ x. path(r ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x, ∃ x0. path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0

Rule argument

Additional information

Used formulas: r0 ∈ y0

Proof step 34 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 35:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, r ≠ h[r0].nxt, path(r ' + y0 + h[r0].nxt ', h[r0 := ce]), path(r ' + y + r0 ', h), r0 ∈ y0, h[r0].nxt ∈ h, r0 ∈ h, ¬ dups(y), ¬ dups(y0), ¬ r0 ∈ y, ¬ r ∈ y, ¬ h[r0].nxt ∈ y0, ¬ r ∈ y0 ⊦ ∃ x0. path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0, ∃ x. path(r ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x

Rule argument

Additional information

Proof step 35 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 36:

  ⊦ a ∈ x ↔ (∃ y, z. x = y + a ' + z ∧ ¬ a ∈ y)

• Proof step 37:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, r ≠ h[r0].nxt, path(r ' + y + r0 ', h), path(r ' + y0 + h[r0].nxt ', h[r0 := ce]), r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(y), ¬ dups(y0), ¬ r0 ∈ y, ¬ r ∈ y, ¬ h[r0].nxt ∈ y0, ¬ r ∈ y0, ∃ y, z. y0 = y + r0 ' + z ∧ ¬ r0 ∈ y ⊦ ∃ x. path(r ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x, ∃ x0. path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0

Rule argument

Additional information

Used terms: y0, r0

Proof step 36 from proof rem-stays-reachable-aux (previous step) (back to tree)

A spec-lemma

Proof step 37 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 38:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, h[r0].nxt ≠ r0, r ≠ r0, r ≠ h[r0].nxt, path(r ' + (y1 + r0 ' + y2) + h[r0].nxt ', h[r0 := ce]), path(r ' + y + r0 ', h), h[r0].nxt ∈ h, r0 ∈ h, ¬ dups(y), ¬ dups(y1 + y2), ¬ r0 ∈ y1, ¬ r0 ∈ y2, ¬ r0 ∈ y, ¬ r ∈ y, ¬ h[r0].nxt ∈ y1, ¬ h[r0].nxt ∈ y2, ¬ r ∈ y1, ¬ r ∈ y2 ⊦ ∃ x0. path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0, ∃ x. path(r ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ dups(x + a ' + y) ↔ a ∈ x ∨ a ∈ y ∨ dups(x + y); (ws)

Proof step 38 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 39:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, h[r0].nxt ≠ r0, r ≠ r0, r ≠ h[r0].nxt, path(r ' + y + r0 ', h), path(r ' + (y1 + r0 ' + y2) + h[r0].nxt ', h[r0 := ce]), path(r ' + y1 + r0 ', h[r0 := ce]), path(r0 ' + y2 + h[r0].nxt ', h[r0 := ce]), r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(y), ¬ dups(y1 + y2), ¬ r0 ∈ y1, ¬ r0 ∈ y2, ¬ r0 ∈ y, ¬ r ∈ y, ¬ h[r0].nxt ∈ y1, ¬ h[r0].nxt ∈ y2, ¬ r ∈ y1, ¬ r ∈ y2 ⊦ ∃ x. path(r ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x, ∃ x0. path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0

Rule argument

Additional information

Used simplifier rules: path-prefix: ⊦ path(r ' + x + y, h) → (path(r ' + x, h) ↔ true); path-postfix: ⊦ path(x + r ' + y, h) → (path(r ' + y, h) ↔ true); a: ⊦ (x + y) + z = x + y + z; from specification polylist Used formulas: path(r ' + y1 + r0 ', h[r0 := ce]) ∧ path(r0 ' + y2 + h[r0].nxt ', h[r0 := ce])

Proof step 39 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 40:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, h[r0].nxt ≠ r0, r ≠ r0, r ≠ h[r0].nxt, path(r0 ' + y2 + h[r0].nxt ', h[r0 := ce]), path(r ' + y1 + r0 ', h[r0 := ce]), path(r ' + (y1 + r0 ' + y2) + h[r0].nxt ', h[r0 := ce]), path(r ' + y + r0 ', h), h[r0].nxt ∈ h, r0 ∈ h, ¬ dups(y), ¬ dups(y1 + y2), ¬ r0 ∈ y1, ¬ r0 ∈ y2, ¬ r0 ∈ y, ¬ r ∈ y, ¬ h[r0].nxt ∈ y1, ¬ h[r0].nxt ∈ y2, ¬ r ∈ y1, ¬ r ∈ y2 ⊦ ∃ x0. path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0, ∃ x. path(r ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x

Rule argument

Additional information

Proof step 40 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule constructor cut

And obtained the following new goals

• Proof step 41:

  ce.nxt = h[h[r0].nxt].nxt, y2 = r1 + y0, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, h[r0].nxt ≠ r0, r ≠ r0, r ≠ h[r0].nxt, path(r ' + y + r0 ', h), path(r ' + (y1 + r0 ' + y2) + h[r0].nxt ', h[r0 := ce]), path(r ' + y1 + r0 ', h[r0 := ce]), path(r0 ' + y2 + h[r0].nxt ', h[r0 := ce]), r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(y), ¬ dups(y1 + y2), ¬ r0 ∈ y1, ¬ r0 ∈ y2, ¬ r0 ∈ y, ¬ r ∈ y, ¬ h[r0].nxt ∈ y1, ¬ h[r0].nxt ∈ y2, ¬ r ∈ y1, ¬ r ∈ y2 ⊦ ∃ x. path(r ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x, ∃ x0. path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0

Rule argument

Additional information

Used simplifier rules: path-two: ⊦ path(r ' + r0 ', h) ↔ r ∈ h ∧ r0 ∈ h ∧ r ≠ null ∧ r0 ≠ null ∧ h[r].nxt = r0; list('a) freely generated by [], +;; ⊦ [] + x = x; (s) ⊦ a ∈ st[a := d]; (s) ⊦ a ≠ b → (a ∈ st[b := d] ↔ a ∈ st); (s) ⊦ (st[a := d])[a] = d; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist

Proof step 41 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 42:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ r0, h[r0].nxt ≠ ce.nxt, h[r0].nxt ≠ null, r ≠ ce.nxt, r ≠ h[r0].nxt, r ≠ r0, r0 ≠ ce.nxt, r0 ≠ null, path(r ' + y1 + r0 ' + (ce.nxt ' + y0) + h[r0].nxt ', h[r0 := ce]), path(r ' + y1 + r0 ', h[r0 := ce]), path(ce.nxt ' + y0 + h[r0].nxt ', h[r0 := ce]), path(r ' + y + r0 ', h), r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(y1 + y0), ¬ dups(y), ¬ r ∈ y0, ¬ r ∈ y1, ¬ h[r0].nxt ∈ y0, ¬ h[r0].nxt ∈ y1, ¬ r ∈ y, ¬ r0 ∈ y, ¬ r0 ∈ y0, ¬ r0 ∈ y1, ¬ ce.nxt ∈ y0, ¬ ce.nxt ∈ y1 ⊦ ∃ x. path(r ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x, ∃ x0. path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0

Rule argument

Additional information

Used simplifier rules: path-cons: ⊦ path(r ' + r0 ' + x, h) ↔ r ∈ h ∧ r ≠ null ∧ h[r].nxt = r0 ∧ path(r0 ' + x, h); a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ a0 ' + x ↔ ¬(a ≠ a0 ∧ ¬ a ∈ x); (ss) ⊦ a + x = a ' + x; (s) ⊦ dups(x + a ' + y) ↔ a ∈ x ∨ a ∈ y ∨ dups(x + y); (ws) ⊦ (st[a := d])[a] = d; (s) ⊦ a ∈ st[a := d]; (s)

Proof step 42 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 43:

  ⊦ ¬ r0 ∈ x ∧ r0 ∈ h → (path(x + r1 ', h[r0 := ce]) ↔ path(x + r1 ', h))

• Proof step 44:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ r0, h[r0].nxt ≠ ce.nxt, h[r0].nxt ≠ null, r ≠ ce.nxt, r ≠ h[r0].nxt, r ≠ r0, r0 ≠ ce.nxt, r0 ≠ null, path(r ' + y + r0 ', h), path((ce.nxt ' + y0) + h[r0].nxt ', h), path(r ' + y1 + r0 ', h[r0 := ce]), path(r ' + y1 + r0 ' + (ce.nxt ' + y0) + h[r0].nxt ', h[r0 := ce]), h[r0].nxt ∈ h, r0 ∈ h, ¬ dups(y1 + y0), ¬ dups(y), ¬ r0 ∈ ce.nxt ' + y0, ¬ r ∈ y0, ¬ r ∈ y1, ¬ h[r0].nxt ∈ y0, ¬ h[r0].nxt ∈ y1, ¬ r ∈ y, ¬ r0 ∈ y, ¬ r0 ∈ y0, ¬ r0 ∈ y1, ¬ ce.nxt ∈ y0, ¬ ce.nxt ∈ y1 ⊦ ∃ x0. path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0, ∃ x. path(r ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x

Rule argument

Additional information

Used simplifier rules: ⊦ a ≠ a0 → (a ∈ a0 ' + x ↔ a ∈ x); (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist Used terms: ce, r0, ce.nxt ' + y0, h, h[r0].nxt

Proof step 43 from proof rem-stays-reachable-aux (previous step) (back to tree)

Lemma path-modify-strong

Proof step 44 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 45:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ r0, h[r0].nxt ≠ ce.nxt, h[r0].nxt ≠ null, r ≠ ce.nxt, r ≠ h[r0].nxt, r ≠ r0, r0 ≠ ce.nxt, r0 ≠ null, path(r ' + y1 + r0 ' + (ce.nxt ' + y0) + h[r0].nxt ', h[r0 := ce]), path(r ' + y1 + r0 ', h[r0 := ce]), path((ce.nxt ' + y0) + h[r0].nxt ', h), path(r ' + y + r0 ', h), r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(y1 + y0), ¬ dups(y), ¬ r ∈ y0, ¬ r ∈ y1, ¬ h[r0].nxt ∈ y0, ¬ h[r0].nxt ∈ y1, ¬ r ∈ y, ¬ r0 ∈ y, ¬ r0 ∈ y1, ¬ ce.nxt ∈ y0, ¬ ce.nxt ∈ y1, ¬ r0 ∈ y0 ⊦ ∃ x. path(r ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x, ∃ x0. path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ≠ a0 → (a ∈ a0 ' + x ↔ a ∈ x); (s)

Proof step 45 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule exists right

And closed the goal!

Rule argument

Additional information

Used simplifier rules: path-cons: ⊦ path(r ' + r0 ' + x, h) ↔ r ∈ h ∧ r ≠ null ∧ h[r].nxt = r0 ∧ path(r0 ' + x, h); ⊦ a ≠ a0 → (a ∈ a0 ' + x ↔ a ∈ x); (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist Used terms: ce.nxt ' + y0

Proof step 46 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 47:

  ⊦ ¬ r0 ∈ x ∧ r0 ∈ h → (path(x + r1 ', h[r0 := ce]) ↔ path(x + r1 ', h))

• Proof step 48:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, r ≠ h[r0].nxt, path((r ' + y0) + h[r0].nxt ', h), path(r ' + y + r0 ', h), h[r0].nxt ∈ h, r0 ∈ h, ¬ dups(y), ¬ dups(y0), ¬ r0 ∈ r ' + y0, ¬ r0 ∈ y0, ¬ r0 ∈ y, ¬ r ∈ y, ¬ h[r0].nxt ∈ y0, ¬ r ∈ y0 ⊦ ∃ x0. path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0, ∃ x. path(r ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x

Rule argument

Additional information

Used simplifier rules: ⊦ a ≠ a0 → (a ∈ a0 ' + x ↔ a ∈ x); (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist Used terms: ce, r0, r ' + y0, h, h[r0].nxt

Proof step 47 from proof rem-stays-reachable-aux (previous step) (back to tree)

Lemma path-modify-strong

Proof step 48 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 49:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, r ≠ h[r0].nxt, path(r ' + y + r0 ', h), path((r ' + y0) + h[r0].nxt ', h), r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(y), ¬ dups(y0), ¬ r0 ∈ y, ¬ r ∈ y, ¬ h[r0].nxt ∈ y0, ¬ r ∈ y0, ¬ r0 ∈ y0 ⊦ ∃ x. path(r ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x, ∃ x0. path(h[r0].nxt ' + x0 + h[r0].nxt ', h) ∧ ¬ r0 ∈ x0

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ≠ a0 → (a ∈ a0 ' + x ↔ a ∈ x); (s)

Proof step 49 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist Used terms: y0

Proof step 50 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 51:

  ce.nxt = h[h[(r ' + x).last].nxt].nxt, h[(r ' + x).last].nxt ≠ null, h[(r ' + x).last].nxt ≠ ce.nxt, path(r ' + x, h), h[(r ' + x).last].nxt ∈ h, (r ' + x).last ∈ h, ¬ dups(x), ¬ reachable(r, h[(r ' + x).last].nxt, h[(r ' + x).last := ce]), ¬ r ∈ x, h[(r ' + x).last].nxt ≠ (r ' + x).last ∧ r = h[(r ' + x).last].nxt ∨ path(r ' + x0 + h[(r ' + x).last].nxt ', h) ∧ (r ' + x).last ≠ r ∧ ¬ (r ' + x).last ∈ x0 ∨ path(h[(r ' + x).last].nxt ' + x1 + h[(r ' + x).last].nxt ', h) ∧ ¬ (r ' + x).last ∈ x1 ⊦

Rule argument

Additional information

Proof step 51 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 52:

  ⊦ reachable(r, r0, h) ↔ (∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0)

• Proof step 53:

  ce.nxt = h[h[(r ' + x).last].nxt].nxt, h[(r ' + x).last].nxt ≠ null, h[(r ' + x).last].nxt ≠ ce.nxt, path(r ' + x, h), (r ' + x).last ∈ h, h[(r ' + x).last].nxt ∈ h, ¬ dups(x), ¬ r ∈ x, ∀ x0. ¬(path(r ' + x0, h[(r ' + x).last := ce]) ∧ (r ' + x0).last = h[(r ' + x).last].nxt), h[(r ' + x).last].nxt ≠ (r ' + x).last ∧ r = h[(r ' + x).last].nxt ∨ path(r ' + x0 + h[(r ' + x).last].nxt ', h) ∧ (r ' + x).last ≠ r ∧ ¬ (r ' + x).last ∈ x0 ∨ path(h[(r ' + x).last].nxt ' + x1 + h[(r ' + x).last].nxt ', h) ∧ ¬ (r ' + x).last ∈ x1 ⊦

Rule argument

Additional information

Used terms: h[(r ' + x).last := ce], h[(r ' + x).last].nxt, r

Proof step 52 from proof rem-stays-reachable-aux (previous step) (back to tree)

Lemma reachable-def

Proof step 53 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 54:

  ce.nxt = h[h[(r ' + x).last].nxt].nxt, r = h[(r ' + x).last].nxt, h[(r ' + x).last].nxt ≠ (r ' + x).last, h[(r ' + x).last].nxt ≠ null, h[(r ' + x).last].nxt ≠ ce.nxt, path(r ' + x, h), h[(r ' + x).last].nxt ∈ h, (r ' + x).last ∈ h, ¬ dups(x), ¬ r ∈ x, ∀ x0. ¬(path(r ' + x0, h[(r ' + x).last := ce]) ∧ (r ' + x0).last = h[(r ' + x).last].nxt) ⊦

• Proof step 56:

  ce.nxt = h[h[(r ' + x).last].nxt].nxt, (r ' + x).last ≠ r, h[(r ' + x).last].nxt ≠ null, h[(r ' + x).last].nxt ≠ ce.nxt, path(r ' + x, h), path(r ' + x0 + h[(r ' + x).last].nxt ', h), h[(r ' + x).last].nxt ∈ h, (r ' + x).last ∈ h, ¬ dups(x), ¬ (r ' + x).last ∈ x0, ¬ r ∈ x, ∀ x0. ¬(path(r ' + x0, h[(r ' + x).last := ce]) ∧ (r ' + x0).last = h[(r ' + x).last].nxt) ⊦

• Proof step 61:

  ce.nxt = h[h[(r ' + x).last].nxt].nxt, h[(r ' + x).last].nxt ≠ null, h[(r ' + x).last].nxt ≠ ce.nxt, path(r ' + x, h), path(h[(r ' + x).last].nxt ' + x1 + h[(r ' + x).last].nxt ', h), h[(r ' + x).last].nxt ∈ h, (r ' + x).last ∈ h, ¬ dups(x), ¬ (r ' + x).last ∈ x1, ¬ r ∈ x, ∀ x0. ¬(path(r ' + x0, h[(r ' + x).last := ce]) ∧ (r ' + x0).last = h[(r ' + x).last].nxt) ⊦

Rule argument

Additional information

Proof step 54 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 55:

  ce.nxt = h[r].nxt, r = h[(r ' + x).last].nxt, r ≠ (r ' + x).last, r ≠ null, r ≠ ce.nxt, path(r ' + x, h), (r ' + x).last ∈ h, r ∈ h, ¬ dups(x), ¬ r ∈ x, ∀ x0. ¬ path(r ' + x0, h[(r ' + x).last := ce]) ∨ (r ' + x0).last ≠ r ⊦

Rule argument

Additional information

Proof step 55 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: path-one: ⊦ path(r ', h) ↔ r ∈ h ∧ r ≠ null; path-first-in-H: ⊦ path(r ' + x, h) → (r ∈ h ↔ true); ⊦ a ≠ b → (a ∈ st[b := d] ↔ a ∈ st); (s) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a '.last = a; (s) Used terms: []

Proof step 56 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 57:

  ce.nxt = h[h[(r ' + x).last].nxt].nxt, (r ' + x).last ≠ r, h[(r ' + x).last].nxt ≠ null, h[(r ' + x).last].nxt ≠ ce.nxt, path(r ' + x0 + h[(r ' + x).last].nxt ', h), path(r ' + x, h), (r ' + x).last ∈ h, h[(r ' + x).last].nxt ∈ h, ¬ dups(x), ¬ (r ' + x).last ∈ x0, ¬ r ∈ x, ∀ x1. ¬ path(r ' + x1, h[(r ' + x).last := ce]) ∨ (r ' + x1).last ≠ h[(r ' + x).last].nxt ⊦

Rule argument

Additional information

Proof step 57 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 58:

  ce.nxt = h[h[(r ' + x).last].nxt].nxt, (r ' + x).last ≠ r, h[(r ' + x).last].nxt ≠ null, h[(r ' + x).last].nxt ≠ ce.nxt, path(r ' + x, h), path(r ' + x0 + h[(r ' + x).last].nxt ', h), h[(r ' + x).last].nxt ∈ h, (r ' + x).last ∈ h, ¬ dups(x), ¬ (r ' + x).last ∈ x0, ¬ r ∈ x, ¬ path(r ' + x0 + h[(r ' + x).last].nxt ', h[(r ' + x).last := ce]) ∨ (r ' + x0 + h[(r ' + x).last].nxt ').last ≠ h[(r ' + x).last].nxt ⊦

Rule argument

Additional information

Used terms: x0 + h[(r ' + x).last].nxt '

Proof step 58 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 59:

  ce.nxt = h[h[(r ' + x).last].nxt].nxt, (r ' + x).last ≠ r, h[(r ' + x).last].nxt ≠ null, h[(r ' + x).last].nxt ≠ ce.nxt, path(r ' + x0 + h[(r ' + x).last].nxt ', h), path(r ' + x, h), (r ' + x).last ∈ h, h[(r ' + x).last].nxt ∈ h, ¬ dups(x), ¬ path(r ' + x0 + h[(r ' + x).last].nxt ', h[(r ' + x).last := ce]), ¬ (r ' + x).last ∈ x0, ¬ r ∈ x ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ (a ' + x + b ').last = b; (s)

Proof step 59 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 60:

  ⊦ ¬ r0 ∈ x → (path(x, h[r0 := ce]) ↔ path(x, h))

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ≠ a0 → (a ∈ a0 ' + x ↔ a ∈ x); (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a ∈ a0 ' ↔ a = a0; (s) Used terms: r ' + x0 + h[(r ' + x).last].nxt ', h, (r ' + x).last, ce

Proof step 60 from proof rem-stays-reachable-aux (previous step) (back to tree)

Lemma path-modify

Proof step 61 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 62:

  ce.nxt = h[h[(r ' + x).last].nxt].nxt, h[(r ' + x).last].nxt ≠ null, h[(r ' + x).last].nxt ≠ ce.nxt, path(h[(r ' + x).last].nxt ' + x1 + h[(r ' + x).last].nxt ', h), path(r ' + x, h), (r ' + x).last ∈ h, h[(r ' + x).last].nxt ∈ h, ¬ dups(x), ¬ (r ' + x).last ∈ x1, ¬ r ∈ x, ∀ x0. ¬ path(r ' + x0, h[(r ' + x).last := ce]) ∨ (r ' + x0).last ≠ h[(r ' + x).last].nxt ⊦

Rule argument

Additional information

Proof step 62 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 63:

  ⊦ y ≠ [] → (x + y).last = y.last

• Proof step 64:

  ce.nxt = h[h[(r ' + x).last].nxt].nxt, x = [], h[(r ' + x).last].nxt ≠ null, h[(r ' + x).last].nxt ≠ ce.nxt, path(r ' + x, h), path(h[(r ' + x).last].nxt ' + x1 + h[(r ' + x).last].nxt ', h), h[(r ' + x).last].nxt ∈ h, (r ' + x).last ∈ h, ¬ dups(x), ¬ (r ' + x).last ∈ x1, ¬ r ∈ x, ∀ x0. ¬ path(r ' + x0, h[(r ' + x).last := ce]) ∨ (r ' + x0).last ≠ h[(r ' + x).last].nxt ⊦

• Proof step 71:

  ce.nxt = h[h[x.last].nxt].nxt, h[x.last].nxt ≠ null, h[x.last].nxt ≠ ce.nxt, x ≠ [], path(r ' + x, h), path(h[x.last].nxt ' + x1 + h[x.last].nxt ', h), h[x.last].nxt ∈ h, x.last ∈ h, ¬ dups(x), ¬ x.last ∈ x1, ¬ r ∈ x, ∀ x0. ¬ path(r ' + x0, h[x.last := ce]) ∨ (r ' + x0).last ≠ h[x.last].nxt ⊦

Rule argument

Additional information

Used terms: x, r '

Proof step 63 from proof rem-stays-reachable-aux (previous step) (back to tree)

A spec-lemma

Proof step 64 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 65:

  ce.nxt = h[h[r].nxt].nxt, h[r].nxt ≠ null, h[r].nxt ≠ ce.nxt, r ≠ null, path(h[r].nxt ' + x1 + h[r].nxt ', h), r ∈ h, h[r].nxt ∈ h, ¬ r ∈ x1, ∀ x0. ¬ path(r ' + x0, h[r := ce]) ∨ (r ' + x0).last ≠ h[r].nxt ⊦

Rule argument

Additional information

Used simplifier rules: path-one: ⊦ path(r ', h) ↔ r ∈ h ∧ r ≠ null; ⊦ a '.last = a; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ x + [] = x; (s) ⊦ ¬ a ∈ []; (s) ⊦ ¬ dups([]); (s)

Proof step 65 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule constructor cut

And obtained the following new goals

• Proof step 66:

  ce.nxt = h[h[r].nxt].nxt, x1 = r0 + x, h[r].nxt ≠ null, h[r].nxt ≠ ce.nxt, r ≠ null, path(h[r].nxt ' + x1 + h[r].nxt ', h), h[r].nxt ∈ h, r ∈ h, ¬ r ∈ x1, ∀ x0. ¬ path(r ' + x0, h[r := ce]) ∨ (r ' + x0).last ≠ h[r].nxt ⊦

Rule argument

Additional information

Used simplifier rules: path-two: ⊦ path(r ' + r0 ', h) ↔ r ∈ h ∧ r0 ∈ h ∧ r ≠ null ∧ r0 ≠ null ∧ h[r].nxt = r0; list('a) freely generated by [], +;; ⊦ [] + x = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ ¬ a ∈ []; (s)

Proof step 66 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 67:

  ce.nxt = h[h[r].nxt].nxt, h[r].nxt ≠ ce.nxt, h[r].nxt ≠ null, r ≠ ce.nxt, r ≠ null, path(ce.nxt ' + x + h[r].nxt ', h), h[r].nxt ∈ h, r ∈ h, ¬ r ∈ x, ∀ x0. ¬ path(r ' + x0, h[r := ce]) ∨ (r ' + x0).last ≠ h[r].nxt ⊦

Rule argument

Additional information

Used simplifier rules: path-cons: ⊦ path(r ' + r0 ' + x, h) ↔ r ∈ h ∧ r ≠ null ∧ h[r].nxt = r0 ∧ path(r0 ' + x, h); a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ a0 ' + x ↔ ¬(a ≠ a0 ∧ ¬ a ∈ x); (ss) ⊦ a + x = a ' + x; (s)

Proof step 67 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 68:

  ce.nxt = h[h[r].nxt].nxt, h[r].nxt ≠ ce.nxt, h[r].nxt ≠ null, r ≠ ce.nxt, r ≠ null, path(ce.nxt ' + x + h[r].nxt ', h), r ∈ h, h[r].nxt ∈ h, ¬ r ∈ x, ¬ path(r ' + ce.nxt ' + x + h[r].nxt ', h[r := ce]) ∨ (r ' + ce.nxt ' + x + h[r].nxt ').last ≠ h[r].nxt ⊦

Rule argument

Additional information

Used terms: ce.nxt ' + x + h[r].nxt '

Proof step 68 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 69:

  ce.nxt = h[h[r].nxt].nxt, h[r].nxt ≠ ce.nxt, h[r].nxt ≠ null, r ≠ ce.nxt, r ≠ null, path(ce.nxt ' + x + h[r].nxt ', h), h[r].nxt ∈ h, r ∈ h, ¬ path(ce.nxt ' + x + h[r].nxt ', h[r := ce]), ¬ r ∈ x ⊦

Rule argument

Additional information

Used simplifier rules: path-cons: ⊦ path(r ' + r0 ' + x, h) ↔ r ∈ h ∧ r ≠ null ∧ h[r].nxt = r0 ∧ path(r0 ' + x, h); a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ (a ' + x + b ').last = b; (s) ⊦ (st[a := d])[a] = d; (s) ⊦ a ∈ st[a := d]; (s)

Proof step 69 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 70:

  ⊦ ¬ r0 ∈ x → (path(x, h[r0 := ce]) ↔ path(x, h))

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ≠ a0 → (a ∈ a0 ' + x ↔ a ∈ x); (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a ∈ a0 ' ↔ a = a0; (s) Used terms: ce.nxt ' + x + h[r].nxt ', h, r, ce

Proof step 70 from proof rem-stays-reachable-aux (previous step) (back to tree)

Lemma path-modify

Proof step 71 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 72:

  ce.nxt = h[h[x.last].nxt].nxt, h[x.last].nxt ≠ null, h[x.last].nxt ≠ ce.nxt, x ≠ [], path(h[x.last].nxt ' + x1 + h[x.last].nxt ', h), path(r ' + x, h), x.last ∈ h, h[x.last].nxt ∈ h, ¬ dups(x), ¬ x.last ∈ x1, ¬ r ∈ x, ∀ x0. ¬ path(r ' + x0, h[x.last := ce]) ∨ (r ' + x0).last ≠ h[x.last].nxt ⊦

Rule argument

Additional information

Proof step 72 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply elim lemma

And obtained the following new goals

• Proof step 73:

  ce.nxt = h[h[r0].nxt].nxt, x = y + r0 ', h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, x ≠ [], path(r ' + x, h), path(h[r0].nxt ' + x1 + h[r0].nxt ', h), h[r0].nxt ∈ h, r0 ∈ h, ¬ dups(x), ¬ r0 ∈ x1, ¬ r ∈ x, ∀ x0. ¬ path(r ' + x0, h[r0 := ce]) ∨ (r ' + x0).last ≠ h[r0].nxt ⊦

Rule argument

Additional information

Used simplifier rules: x ≠ [] ⊦ a = x.last ∧ y = x.butlast ↔ x = y + a '; (elim) Used terms: x, a, y

Proof step 73 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 74:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, path(h[r0].nxt ' + x1 + h[r0].nxt ', h), path(r ' + y + r0 ', h), r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(y), ¬ r0 ∈ y, ¬ r0 ∈ x1, ¬ r ∈ y, ∀ x0. ¬ path(r ' + x0, h[r0 := ce]) ∨ (r ' + x0).last ≠ h[r0].nxt ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ dups(x + a ') ↔ a ∈ x ∨ dups(x); (ws) ⊦ x + a ' ≠ []; (s)

Proof step 74 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 75:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, path(r ' + y + r0 ', h), path(h[r0].nxt ' + x1 + h[r0].nxt ', h), path(h[r0].nxt ' + x1 + h[r0].nxt ', h[r0 := ce]), h[r0].nxt ∈ h, r0 ∈ h, ¬ dups(y), ¬ r0 ∈ y, ¬ r0 ∈ x1, ¬ r ∈ y, ∀ x0. ¬ path(r ' + x0, h[r0 := ce]) ∨ (r ' + x0).last ≠ h[r0].nxt ⊦

• Proof step 91:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, path(r ' + y + r0 ', h), path(h[r0].nxt ' + x1 + h[r0].nxt ', h), h[r0].nxt ∈ h, r0 ∈ h, ¬ dups(y), ¬ path(h[r0].nxt ' + x1 + h[r0].nxt ', h[r0 := ce]), ¬ r0 ∈ y, ¬ r0 ∈ x1, ¬ r ∈ y, ∀ x0. ¬ path(r ' + x0, h[r0 := ce]) ∨ (r ' + x0).last ≠ h[r0].nxt ⊦

Rule argument

Additional information

Used formulas: path(h[r0].nxt ' + x1 + h[r0].nxt ', h[r0 := ce])

Proof step 75 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule weakening

And obtained the following new goals

• Proof step 76:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, path(h[r0].nxt ' + x1 + h[r0].nxt ', h[r0 := ce]), path(r ' + y + r0 ', h), r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(y), ¬ r0 ∈ y, ¬ r0 ∈ x1, ¬ r ∈ y, ∀ x0. ¬ path(r ' + x0, h[r0 := ce]) ∨ (r ' + x0).last ≠ h[r0].nxt ⊦

Rule argument

Additional information

Proof step 76 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 77:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, path(r ' + y + r0 ', h), path(h[r0].nxt ' + x1 + h[r0].nxt ', h[r0 := ce]), path(r ' + y + r0 ', h[r0 := ce]), h[r0].nxt ∈ h, r0 ∈ h, ¬ dups(y), ¬ r0 ∈ y, ¬ r0 ∈ x1, ¬ r ∈ y, ∀ x0. ¬ path(r ' + x0, h[r0 := ce]) ∨ (r ' + x0).last ≠ h[r0].nxt ⊦

• Proof step 89:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, path(r ' + y + r0 ', h), path(h[r0].nxt ' + x1 + h[r0].nxt ', h[r0 := ce]), h[r0].nxt ∈ h, r0 ∈ h, ¬ dups(y), ¬ path(r ' + y + r0 ', h[r0 := ce]), ¬ r0 ∈ y, ¬ r0 ∈ x1, ¬ r ∈ y, ∀ x0. ¬ path(r ' + x0, h[r0 := ce]) ∨ (r ' + x0).last ≠ h[r0].nxt ⊦

Rule argument

Additional information

Used formulas: path(r ' + y + r0 ', h[r0 := ce])

Proof step 77 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 78:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, path(r ' + y + r0 ', h[r0 := ce]), path(h[r0].nxt ' + x1 + h[r0].nxt ', h[r0 := ce]), path(r ' + y + r0 ', h), r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(y), ¬ r0 ∈ y, ¬ r0 ∈ x1, ¬ r ∈ y, ¬ path(r ' + y + (r0 + (x1 + h[r0].nxt)), h[r0 := ce]) ∨ (r ' + y + (r0 + (x1 + h[r0].nxt))).last ≠ h[r0].nxt ⊦

Rule argument

Additional information

Used terms: y + (r0 + (x1 + h[r0].nxt))

Proof step 78 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 79:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, path(r ' + y + r0 ', h), path(h[r0].nxt ' + x1 + h[r0].nxt ', h[r0 := ce]), path(r ' + y + r0 ', h[r0 := ce]), h[r0].nxt ∈ h, r0 ∈ h, ¬ dups(y), ¬ path(r ' + y + r0 ' + x1 + h[r0].nxt ', h[r0 := ce]), ¬ r0 ∈ y, ¬ r0 ∈ x1, ¬ r ∈ y ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ (a ' + x + b ').last = b; (s) ⊦ a + x = a ' + x; (s) ⊦ x + a = x + a '; (s)

Proof step 79 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 80:

  path(x, h) ∧ x.last = r ∧ path(r ' + y, h) ⊦ path(x + y, h)

• Proof step 81:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, path(r ' + y + r0 ', h[r0 := ce]), path(h[r0].nxt ' + x1 + h[r0].nxt ', h[r0 := ce]), path(r ' + y + r0 ', h), r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(y), ¬ path(r ' + y + r0 ' + x1 + h[r0].nxt ', h[r0 := ce]), ¬ r0 ∈ y, ¬ r0 ∈ x1, ¬ r ∈ y ⊦ (r ' + y + r0 ').last = r0 ∧ path(r0 ' + x1 + h[r0].nxt ', h[r0 := ce])

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist Used terms: r ' + y + r0 ', h[r0 := ce], r0, x1 + h[r0].nxt '

Proof step 80 from proof rem-stays-reachable-aux (previous step) (back to tree)

Lemma path-trans

Proof step 81 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 82:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, path(r ' + y + r0 ', h), path(h[r0].nxt ' + x1 + h[r0].nxt ', h[r0 := ce]), path(r ' + y + r0 ', h[r0 := ce]), h[r0].nxt ∈ h, r0 ∈ h, ¬ dups(y), ¬ path(r ' + y + r0 ' + x1 + h[r0].nxt ', h[r0 := ce]), ¬ path(r0 ' + x1 + h[r0].nxt ', h[r0 := ce]), ¬ r0 ∈ y, ¬ r0 ∈ x1, ¬ r ∈ y ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ (a ' + x + b ').last = b; (s)

Proof step 82 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule constructor cut

And obtained the following new goals

• Proof step 83:

  ce.nxt = h[h[r0].nxt].nxt, x1 = [], h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, path(r ' + y + r0 ', h[r0 := ce]), path(h[r0].nxt ' + x1 + h[r0].nxt ', h[r0 := ce]), path(r ' + y + r0 ', h), r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(y), ¬ path(r ' + y + r0 ' + x1 + h[r0].nxt ', h[r0 := ce]), ¬ path(r0 ' + x1 + h[r0].nxt ', h[r0 := ce]), ¬ r0 ∈ y, ¬ r0 ∈ x1, ¬ r ∈ y ⊦

• Proof step 86:

  ce.nxt = h[h[r0].nxt].nxt, x1 = r1 + x, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, path(r ' + y + r0 ', h[r0 := ce]), path(h[r0].nxt ' + x1 + h[r0].nxt ', h[r0 := ce]), path(r ' + y + r0 ', h), r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(y), ¬ path(r ' + y + r0 ' + x1 + h[r0].nxt ', h[r0 := ce]), ¬ path(r0 ' + x1 + h[r0].nxt ', h[r0 := ce]), ¬ r0 ∈ y, ¬ r0 ∈ x1, ¬ r ∈ y ⊦

Rule argument

Additional information

Used simplifier rules: list('a) freely generated by [], +;;

Proof step 83 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 84:

  ce.nxt = h[h[r0].nxt].nxt, (h[r0 := ce])[h[r0].nxt].nxt = h[r0].nxt, h[r0].nxt ≠ null, h[r0].nxt ≠ ce.nxt, r ≠ r0, path(r ' + y + r0 ', h[r0 := ce]), path(r ' + y + r0 ', h), h[r0].nxt ∈ h, r0 ∈ h, h[r0].nxt ∈ h[r0 := ce], ¬ dups(y), ¬ path(r ' + y + r0 ' + h[r0].nxt ', h[r0 := ce]), ¬ r0 ∈ y, ¬ r ∈ y ⊦

Rule argument

Additional information

Used simplifier rules: path-two: ⊦ path(r ' + r0 ', h) ↔ r ∈ h ∧ r0 ∈ h ∧ r ≠ null ∧ r0 ≠ null ∧ h[r].nxt = r0; ⊦ ¬ a ∈ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ (st[a := d])[a] = d; (s) ⊦ a ∈ st[a := d]; (s) ⊦ [] + x = x; (s)

Proof step 84 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 85:

  ⊦ a ≠ b → (st[b := d])[a] = st[a]

Rule argument

Additional information

Used simplifier rules: ⊦ a ≠ b → (a ∈ st[b := d] ↔ a ∈ st); (s) ⊦ a = b → (st[b := d])[a] = d; (s) Used terms: h[r0].nxt, ce, r0, h

Proof step 85 from proof rem-stays-reachable-aux (previous step) (back to tree)

A spec-lemma

Proof step 86 from proof rem-stays-reachable-aux (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 87:

  ce.nxt = h[h[r0].nxt].nxt, h[r0].nxt ≠ ce.nxt, h[r0].nxt ≠ null, r ≠ r0, r0 ≠ (h[r0 := ce])[h[r0].nxt].nxt, r0 ≠ null, path((h[r0 := ce])[h[r0].nxt].nxt ' + x + h[r0].nxt ', h[r0 := ce]), path(r ' + y + r0 ', h), path(r ' + y + r0 ', h[r0 := ce]), h[r0].nxt ∈ h[r0 := ce], r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(y), ¬ path(r ' + y + r0 ' + ((h[r0 := ce])[h[r0].nxt].nxt ' + x) + h[r0].nxt ', h[r0 := ce]), ¬ r0 ∈ x, ¬ r0 ∈ y, ¬ r ∈ y ⊦ path(ce.nxt ' + x + h[r0].nxt ', h[r0 := ce]) ∧ ce.nxt = (h[r0 := ce])[h[r0].nxt].nxt

Rule argument

Additional information

Used simplifier rules: path-cons: ⊦ path(r ' + r0 ' + x, h) ↔ r ∈ h ∧ r ≠ null ∧ h[r].nxt = r0 ∧ path(r0 ' + x, h); path-last-not-null: ⊦ ¬ path(r ' + x + null ', h); a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ≠ b → (st[b := d])[a] = st[a]; (s) ⊦ a ∈ a0 ' + x ↔ ¬(a ≠ a0 ∧ ¬ a ∈ x); (ss) ⊦ a + x = a ' + x; (s) ⊦ (st[a := d])[a] = d; (s) ⊦ a ∈ st[a := d]; (s)

Proof step 87 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 88:

  ⊦ a ≠ b → (st[b := d])[a] = st[a]

Rule argument

Additional information

Used simplifier rules: ⊦ a ≠ b → (a ∈ st[b := d] ↔ a ∈ st); (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a = b → (st[b := d])[a] = d; (s) Used terms: h[r0].nxt, ce, r0, h

Proof step 88 from proof rem-stays-reachable-aux (previous step) (back to tree)

A spec-lemma

Proof step 89 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 90:

  ⊦ ¬ r0 ∈ x ∧ r0 ∈ h → (path(x + r1 ', h[r0 := ce]) ↔ path(x + r1 ', h))

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ≠ a0 → (a ∈ a0 ' + x ↔ a ∈ x); (s) Used terms: ce, r0, r ' + y, h, r0

Proof step 90 from proof rem-stays-reachable-aux (previous step) (back to tree)

Lemma path-modify-strong

Proof step 91 from proof rem-stays-reachable-aux (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 92:

  ⊦ ¬ r0 ∈ x → (path(x, h[r0 := ce]) ↔ path(x, h))

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a ∈ a0 ' ↔ a = a0; (s) Used terms: ce, r0, h, h[r0].nxt ' + x1 + h[r0].nxt '

Proof step 92 from proof rem-stays-reachable-aux (previous step) (back to tree)

Lemma path-modify