Proof step 1 from proof rem-stays-reachable-aux1 (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

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

Rule argument

Additional information

Proof step 2 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

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

Rule argument

Additional information

Proof step 3 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 4:

  h[r0].nxt ≠ null, h[r0].nxt ≠ h[h[r0].nxt].nxt, reachable(h[h[r0].nxt].nxt, h[r0].nxt, h), h[r0].nxt ∈ h, r0 ∈ h, ¬ reachable(h[h[r0].nxt].nxt, r0, h) ⊦ ∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x

• Proof step 27:

  h[r0].nxt ≠ null, h[r0].nxt ≠ h[h[r0].nxt].nxt, h[r0].nxt ∈ h, r0 ∈ h, ∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x ⊦ reachable(h[h[r0].nxt].nxt, h[r0].nxt, h) ∧ ¬ reachable(h[h[r0].nxt].nxt, r0, h)

Rule argument

Additional information

Proof step 4 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 5:

  h[r0].nxt ≠ null, h[r0].nxt ≠ h[h[r0].nxt].nxt, reachable(h[h[r0].nxt].nxt, h[r0].nxt, h), r0 ∈ h, h[r0].nxt ∈ h, ¬ reachable(h[h[r0].nxt].nxt, r0, h) ⊦ ∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x

Rule argument

Additional information

Proof step 5 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 6:

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

• Proof step 7:

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

Rule argument

Additional information

Used terms: h, h[r0].nxt, h[h[r0].nxt].nxt

Proof step 6 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Lemma reachable-nodups-def

Proof step 7 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 8:

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

Rule argument

Additional information

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

Proof step 8 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 9:

  ⊦ y ≠ [] → (x + y).last = y.last

• Proof step 10:

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

Rule argument

Additional information

Used simplifier rules: ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ a '.last = a; (s) Used terms: x, h[h[r0].nxt].nxt '

Proof step 9 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

A spec-lemma

Proof step 10 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 11:

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

Rule argument

Additional information

Proof step 11 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule apply elim lemma

And obtained the following new goals

• Proof step 12:

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

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 12 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 13:

  h[r0].nxt ≠ null, h[r0].nxt ≠ h[h[r0].nxt].nxt, path(h[h[r0].nxt].nxt ' + x0 + h[r0].nxt ', h), r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(x0), ¬ reachable(h[h[r0].nxt].nxt, r0, h), ¬ h[r0].nxt ∈ x0, ¬ h[h[r0].nxt].nxt ∈ x0 ⊦ ∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x

Rule argument

Additional information

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

Proof step 13 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule exists right

And obtained the following new goals

• Proof step 14:

  h[r0].nxt ≠ null, h[r0].nxt ≠ h[h[r0].nxt].nxt, path(h[h[r0].nxt].nxt ' + x0 + h[r0].nxt ', h), h[r0].nxt ∈ h, r0 ∈ h, ¬ dups(x0), ¬ reachable(h[h[r0].nxt].nxt, r0, h), ¬ h[r0].nxt ∈ x0, ¬ h[h[r0].nxt].nxt ∈ x0 ⊦ ∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x, path(h[r0].nxt ' + (h[h[r0].nxt].nxt ' + x0) + h[r0].nxt ', h) ∧ ¬ r0 ∈ h[h[r0].nxt].nxt ' + x0

Rule argument

Additional information

Used terms: h[h[r0].nxt].nxt ' + x0

Proof step 14 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 15:

  h[r0].nxt ≠ null, h[r0].nxt ≠ h[h[r0].nxt].nxt, path(h[h[r0].nxt].nxt ' + x0 + h[r0].nxt ', h), r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(x0), ¬ reachable(h[h[r0].nxt].nxt, r0, h), ¬ h[r0].nxt ∈ x0, ¬ h[h[r0].nxt].nxt ∈ x0 ⊦ ∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x, r0 ≠ h[h[r0].nxt].nxt ∧ ¬ 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 list[reflist] ⊦ a ∈ b ' + x ↔ a = b ∨ a ∈ x; (s)

Proof step 15 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 16:

  h[r0].nxt ≠ null, h[r0].nxt ≠ h[h[r0].nxt].nxt, path(h[h[r0].nxt].nxt ' + x0 + h[r0].nxt ', h), r0 ∈ x0, h[r0].nxt ∈ h, r0 ∈ h, ¬ dups(x0), ¬ reachable(h[h[r0].nxt].nxt, r0, h), ¬ h[r0].nxt ∈ x0, ¬ h[h[r0].nxt].nxt ∈ x0 ⊦ ∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x

Rule argument

Additional information

Used simplifier rules: self-reachable: ⊦ reachable(r, r, h) ↔ r ≠ null ∧ r ∈ h; path-first-in-H: ⊦ path(r ' + x, h) → (r ∈ h ↔ true); null-not-start-of-path: ⊦ ¬ path(null ' + x, h); a: ⊦ (x + y) + z = x + y + z; from specification list[reflist]

Proof step 16 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 17:

  h[r0].nxt ≠ null, h[r0].nxt ≠ h[h[r0].nxt].nxt, path(h[h[r0].nxt].nxt ' + x0 + h[r0].nxt ', h), r0 ∈ h, h[r0].nxt ∈ h, r0 ∈ x0, ¬ dups(x0), ¬ reachable(h[h[r0].nxt].nxt, r0, h), ¬ h[r0].nxt ∈ x0, ¬ h[h[r0].nxt].nxt ∈ x0 ⊦ ∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x

Rule argument

Additional information

Proof step 17 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 18:

  ⊦ a ∈ x ↔ (∃ y, z. x = y + (a + z))

• Proof step 19:

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

Rule argument

Additional information

Used terms: x0, r0

Proof step 18 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

A spec-lemma

Proof step 19 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 20:

  h[r0].nxt ≠ null, h[r0].nxt ≠ h[h[r0].nxt].nxt, h[r0].nxt ≠ r0, h[h[r0].nxt].nxt ≠ r0, path(h[h[r0].nxt].nxt ' + x + (r0 ' + x1) + h[r0].nxt ', h), r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(x + x1), ¬ reachable(h[h[r0].nxt].nxt, r0, h), ¬ r0 ∈ x, ¬ r0 ∈ x1, ¬ h[r0].nxt ∈ x, ¬ h[r0].nxt ∈ x1, ¬ h[h[r0].nxt].nxt ∈ x, ¬ h[h[r0].nxt].nxt ∈ x1 ⊦ ∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x

Rule argument

Additional information

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

Proof step 20 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 21:

  h[r0].nxt ≠ null, h[r0].nxt ≠ h[h[r0].nxt].nxt, h[r0].nxt ≠ r0, h[h[r0].nxt].nxt ≠ r0, path(h[h[r0].nxt].nxt ' + x + (r0 ' + x1) + h[r0].nxt ', h), path((r0 ' + x1) + h[r0].nxt ', h), h[r0].nxt ∈ h, r0 ∈ h, ¬ dups(x + x1), ¬ reachable(h[h[r0].nxt].nxt, r0, h), ¬ r0 ∈ x, ¬ r0 ∈ x1, ¬ h[r0].nxt ∈ x, ¬ h[r0].nxt ∈ x1, ¬ h[h[r0].nxt].nxt ∈ x, ¬ h[h[r0].nxt].nxt ∈ x1 ⊦ ∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x

Rule argument

Additional information

Used simplifier rules: path-postfix: ⊦ path(x + r ' + y, h) → (path(r ' + y, h) ↔ true); a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] Used formulas: path((r0 ' + x1) + h[r0].nxt ', h)

Proof step 21 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule constructor cut

And obtained the following new goals

• Proof step 22:

  x1 = [], h[r0].nxt ≠ null, h[r0].nxt ≠ h[h[r0].nxt].nxt, h[r0].nxt ≠ r0, h[h[r0].nxt].nxt ≠ r0, path((r0 ' + x1) + h[r0].nxt ', h), path(h[h[r0].nxt].nxt ' + x + (r0 ' + x1) + h[r0].nxt ', h), r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(x + x1), ¬ reachable(h[h[r0].nxt].nxt, r0, h), ¬ r0 ∈ x, ¬ r0 ∈ x1, ¬ h[r0].nxt ∈ x, ¬ h[r0].nxt ∈ x1, ¬ h[h[r0].nxt].nxt ∈ x, ¬ h[h[r0].nxt].nxt ∈ x1 ⊦ ∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ 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); reflist freely generated by [], +;; ⊦ a + x = a ' + x; (s) ⊦ a ∈ b ' + x ↔ ¬(a ≠ b ∧ ¬ a ∈ x); (ss) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist]

Proof step 22 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 23:

  h[r0].nxt ≠ null, h[r0].nxt ≠ h[h[r0].nxt].nxt, h[r0].nxt ≠ r0, h[h[r0].nxt].nxt ≠ r0, r0 ≠ null, path(h[h[r0].nxt].nxt ' + x + r0 ' + h[r0].nxt ', h), h[r0].nxt ∈ h, r0 ∈ h, ¬ dups(x), ¬ reachable(h[h[r0].nxt].nxt, r0, h), ¬ r0 ∈ x, ¬ h[r0].nxt ∈ x, ¬ h[h[r0].nxt].nxt ∈ x ⊦ ∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x

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 list[reflist] ⊦ x + [] = x; (s) ⊦ [] + x = x; (s)

Proof step 23 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 24:

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

• Proof step 25:

  h[r0].nxt ≠ null, h[r0].nxt ≠ h[h[r0].nxt].nxt, h[r0].nxt ≠ r0, h[h[r0].nxt].nxt ≠ r0, r0 ≠ null, path(h[h[r0].nxt].nxt ' + x + r0 ' + h[r0].nxt ', h), r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(x), ¬ r0 ∈ x, ¬ h[r0].nxt ∈ x, ¬ h[h[r0].nxt].nxt ∈ x, ∀ x. ¬(path(h[h[r0].nxt].nxt ' + x, h) ∧ (h[h[r0].nxt].nxt ' + x).last = r0) ⊦ ∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x

Rule argument

Additional information

Used terms: h[h[r0].nxt].nxt, r0, h

Proof step 24 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Lemma reachable-def

Proof step 25 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 26:

  h[r0].nxt ≠ null, h[r0].nxt ≠ h[h[r0].nxt].nxt, h[r0].nxt ≠ r0, h[h[r0].nxt].nxt ≠ r0, r0 ≠ null, path(h[h[r0].nxt].nxt ' + x + r0 ' + h[r0].nxt ', h), h[r0].nxt ∈ h, r0 ∈ h, ¬ dups(x), ¬ r0 ∈ x, ¬ h[r0].nxt ∈ x, ¬ h[h[r0].nxt].nxt ∈ x, ∀ x0. ¬ path(h[h[r0].nxt].nxt ' + x0, h) ∨ (h[h[r0].nxt].nxt ' + x0).last ≠ r0 ⊦ ∃ x. path(h[r0].nxt ' + x + h[r0].nxt ', h) ∧ ¬ r0 ∈ x

Rule argument

Additional information

Proof step 26 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: path-butlast: ⊦ path(r ' + x + r0 ', h) → (path(r ' + x, h) ↔ true); ⊦ (a ' + x + b ').last = b; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] Used terms: x + r0 '

Proof step 27 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 28:

  h[r0].nxt ≠ null, h[r0].nxt ≠ h[h[r0].nxt].nxt, path(h[r0].nxt ' + x + h[r0].nxt ', h), r0 ∈ h, h[r0].nxt ∈ h, ¬ r0 ∈ x ⊦ reachable(h[h[r0].nxt].nxt, h[r0].nxt, h) ∧ ¬ reachable(h[h[r0].nxt].nxt, r0, h)

Rule argument

Additional information

Proof step 28 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule constructor cut

And obtained the following new goals

• Proof step 29:

  x = r + x0, h[r0].nxt ≠ null, h[r0].nxt ≠ h[h[r0].nxt].nxt, path(h[r0].nxt ' + x + h[r0].nxt ', h), h[r0].nxt ∈ h, r0 ∈ h, ¬ r0 ∈ x ⊦ reachable(h[h[r0].nxt].nxt, h[r0].nxt, h) ∧ ¬ reachable(h[h[r0].nxt].nxt, r0, h)

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; reflist freely generated by [], +;; ⊦ [] + x = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist]

Proof step 29 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 30:

  h[r0].nxt ≠ h[h[r0].nxt].nxt, h[r0].nxt ≠ null, r0 ≠ h[h[r0].nxt].nxt, path(h[h[r0].nxt].nxt ' + x0 + h[r0].nxt ', h), h[r0].nxt ∈ h, r0 ∈ h, ¬ r0 ∈ x0 ⊦ reachable(h[h[r0].nxt].nxt, h[r0].nxt, h) ∧ ¬ reachable(h[h[r0].nxt].nxt, r0, h)

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 list[reflist] ⊦ a ∈ b ' + x ↔ ¬(a ≠ b ∧ ¬ a ∈ x); (ss) ⊦ a + x = a ' + x; (s)

Proof step 30 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 31:

  h[r0].nxt ≠ h[h[r0].nxt].nxt, h[r0].nxt ≠ null, r0 ≠ h[h[r0].nxt].nxt, path(h[h[r0].nxt].nxt ' + x0 + h[r0].nxt ', h), r0 ∈ h, h[r0].nxt ∈ h, ¬ reachable(h[h[r0].nxt].nxt, h[r0].nxt, h), ¬ r0 ∈ x0 ⊦

• Proof step 35:

  h[r0].nxt ≠ h[h[r0].nxt].nxt, h[r0].nxt ≠ null, r0 ≠ h[h[r0].nxt].nxt, path(h[h[r0].nxt].nxt ' + x0 + h[r0].nxt ', h), reachable(h[h[r0].nxt].nxt, r0, h), r0 ∈ h, h[r0].nxt ∈ h, ¬ r0 ∈ x0 ⊦

Rule argument

Additional information

Proof step 31 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 32:

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

• Proof step 33:

  h[r0].nxt ≠ h[h[r0].nxt].nxt, h[r0].nxt ≠ null, r0 ≠ h[h[r0].nxt].nxt, path(h[h[r0].nxt].nxt ' + x0 + h[r0].nxt ', h), h[r0].nxt ∈ h, r0 ∈ h, ¬ r0 ∈ x0, ∀ x. ¬(path(h[h[r0].nxt].nxt ' + x, h) ∧ (h[h[r0].nxt].nxt ' + x).last = h[r0].nxt) ⊦

Rule argument

Additional information

Used terms: h, h[r0].nxt, h[h[r0].nxt].nxt

Proof step 32 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Lemma reachable-def

Proof step 33 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 34:

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

Rule argument

Additional information

Proof step 34 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ (a ' + x + b ').last = b; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] Used terms: x0 + h[r0].nxt '

Proof step 35 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 36:

  h[r0].nxt ≠ h[h[r0].nxt].nxt, h[r0].nxt ≠ null, r0 ≠ h[h[r0].nxt].nxt, path(h[h[r0].nxt].nxt ' + x0 + h[r0].nxt ', h), reachable(h[h[r0].nxt].nxt, r0, h), h[r0].nxt ∈ h, r0 ∈ h, ¬ r0 ∈ x0 ⊦

Rule argument

Additional information

Proof step 36 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 37:

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

• Proof step 38:

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

Rule argument

Additional information

Used terms: h, r0, h[h[r0].nxt].nxt

Proof step 37 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Lemma reachable-nodups-def

Proof step 38 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 39:

  (h[h[r0].nxt].nxt ' + x).last = r0, h[r0].nxt ≠ h[h[r0].nxt].nxt, h[r0].nxt ≠ null, r0 ≠ h[h[r0].nxt].nxt, path(h[h[r0].nxt].nxt ' + x, h), path(h[h[r0].nxt].nxt ' + x0 + h[r0].nxt ', h), h[r0].nxt ∈ h, r0 ∈ h, ¬ dups(x), ¬ r0 ∈ x0, ¬ h[h[r0].nxt].nxt ∈ x ⊦

Rule argument

Additional information

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

Proof step 39 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 40:

  ⊦ y ≠ [] → (x + y).last = y.last

• Proof step 41:

  x.last = r0, h[r0].nxt ≠ h[h[r0].nxt].nxt, h[r0].nxt ≠ null, r0 ≠ h[h[r0].nxt].nxt, x ≠ [], path(h[h[r0].nxt].nxt ' + x0 + h[r0].nxt ', h), path(h[h[r0].nxt].nxt ' + x, h), r0 ∈ h, h[r0].nxt ∈ h, ¬ dups(x), ¬ r0 ∈ x0, ¬ h[h[r0].nxt].nxt ∈ x ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ a '.last = a; (s) Used terms: x, h[h[r0].nxt].nxt '

Proof step 40 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

A spec-lemma

Proof step 41 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 42:

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

Rule argument

Additional information

Proof step 42 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule apply elim lemma

And obtained the following new goals

• Proof step 43:

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

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 43 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 44:

  h[r].nxt ≠ h[h[r].nxt].nxt, h[r].nxt ≠ null, r ≠ h[h[r].nxt].nxt, path(h[h[r].nxt].nxt ' + x1 + r ', h), path(h[h[r].nxt].nxt ' + x0 + h[r].nxt ', h), h[r].nxt ∈ h, r ∈ h, ¬ dups(x1), ¬ r ∈ x1, ¬ r ∈ x0, ¬ h[h[r].nxt].nxt ∈ x1 ⊦

Rule argument

Additional information

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

Proof step 44 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 45:

  path(r ' + x, h), path(r ' + y, h) ⊦ x = y ∨ (∃ r0, y0. x = y + r0 ' + y0) ∨ (∃ r0, x0. y = x + r0 ' + x0)

• Proof step 46:

  h[r].nxt ≠ h[h[r].nxt].nxt, h[r].nxt ≠ null, r ≠ h[h[r].nxt].nxt, path(h[h[r].nxt].nxt ' + x0 + h[r].nxt ', h), path(h[h[r].nxt].nxt ' + x1 + r ', h), r ∈ h, h[r].nxt ∈ h, ¬ dups(x1), ¬ r ∈ x1, ¬ r ∈ x0, ¬ h[h[r].nxt].nxt ∈ x1, x0 + h[r].nxt ' = x1 + r ' ∨ (∃ r0, y0. x0 + h[r].nxt ' = (x1 + r ') + r0 ' + y0) ∨ (∃ r0, x2. x1 + r ' = (x0 + h[r].nxt ') + r0 ' + x2) ⊦

Rule argument

Additional information

Used terms: h[h[r].nxt].nxt, x0 + h[r].nxt ', h, x1 + r '

Proof step 45 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Lemma linearity-path

Proof step 46 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 47:

  h[r].nxt ≠ h[h[r].nxt].nxt, h[r].nxt ≠ null, r ≠ h[h[r].nxt].nxt, path(h[h[r].nxt].nxt ' + x1 + r ', h), path(h[h[r].nxt].nxt ' + x0 + h[r].nxt ', h), h[r].nxt ∈ h, r ∈ h, ¬ dups(x1), ¬ r ∈ x1, ¬ r ∈ x0, ¬ h[h[r].nxt].nxt ∈ x1, x0 + h[r].nxt ' = x1 + r ' ∨ x0 + h[r].nxt ' = (x1 + r ') + r0 ' + x ∨ x1 + r ' = (x0 + h[r].nxt ') + r1 ' + x2 ⊦

Rule argument

Additional information

Proof step 47 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 48:

  x1 + r ' = (x0 + h[r].nxt ') + r1 ' + x2, h[r].nxt ≠ h[h[r].nxt].nxt, h[r].nxt ≠ null, r ≠ h[h[r].nxt].nxt, path(h[h[r].nxt].nxt ' + x0 + h[r].nxt ', h), path(h[h[r].nxt].nxt ' + x1 + r ', h), r ∈ h, h[r].nxt ∈ h, ¬ dups(x1), ¬ r ∈ x1, ¬ r ∈ x0, ¬ h[h[r].nxt].nxt ∈ x1 ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ a ∈ b ' + x ↔ a = b ∨ a ∈ x; (s) ⊦ ¬ c ∈ y ∧ c ≠ b → y + b ' ≠ y0 + c ' + z; (s) in-15: ⊦ x + a ' = y → a ∈ y; (forward) from specification list[reflist] a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ x + a ' = y + b ' ↔ x = y ∧ a = b; (ws)

Proof step 48 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 49:

  x1 + r ' = (x0 + h[r].nxt ') + r1 ' + x2, h[r].nxt ≠ null, h[r].nxt ≠ h[h[r].nxt].nxt, r ≠ h[h[r].nxt].nxt, path(h[h[r].nxt].nxt ' + x0 + h[r].nxt ', h), path(h[h[r].nxt].nxt ' + x1 + r ', h), r ∈ h, h[r].nxt ∈ h, ¬ dups(x1), ¬ h[h[r].nxt].nxt ∈ x1, ¬ r ∈ x0, ¬ r ∈ x1, r = h[r].nxt ∨ r = r1 ∨ r ∈ x2 ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ a ∈ b ' + x ↔ a = b ∨ a ∈ x; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) in-15: ⊦ x + a ' = y → a ∈ y; (forward) from specification list[reflist]

Proof step 49 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 50:

  y ≠ [] ⊦ x + a ' = y ↔ x = y.butlast ∧ a = y.last

• Proof step 51:

  r = ((x0 + h[r].nxt ') + r1 ' + x2).last, x1 = ((x0 + h[r].nxt ') + r1 ' + x2).butlast, (x0 + h[r].nxt ') + r1 ' + x2 ≠ [], h[r].nxt ≠ null, h[r].nxt ≠ h[h[r].nxt].nxt, r ≠ h[h[r].nxt].nxt, path(h[h[r].nxt].nxt ' + x1 + r ', h), path(h[h[r].nxt].nxt ' + x0 + h[r].nxt ', h), h[r].nxt ∈ h, r ∈ h, ¬ dups(x1), ¬ h[h[r].nxt].nxt ∈ x1, ¬ r ∈ x0, ¬ r ∈ x1, r = h[r].nxt ∨ r = r1 ∨ r ∈ x2 ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ x + y = [] ↔ x = [] ∧ y = []; (as) ⊦ a ' + x ≠ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] Used terms: (x0 + h[r].nxt ') + r1 ' + x2, r, x1

Proof step 50 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

A spec-lemma

Proof step 51 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 52:

  (r1 ' + x2).last ≠ h[(r1 ' + x2).last].nxt, (r1 ' + x2).last ≠ h[h[(r1 ' + x2).last].nxt].nxt, h[(r1 ' + x2).last].nxt ≠ h[h[(r1 ' + x2).last].nxt].nxt, h[(r1 ' + x2).last].nxt ≠ null, path(h[h[(r1 ' + x2).last].nxt].nxt ' + x0 + h[(r1 ' + x2).last].nxt ', h), path(h[h[(r1 ' + x2).last].nxt].nxt ' + x0 + h[(r1 ' + x2).last].nxt ' + r1 ' + x2, h), h[(r1 ' + x2).last].nxt ∈ h, (r1 ' + x2).last ∈ h, ¬ dups(x0 + (r1 ' + x2).butlast), ¬ (r1 ' + x2).last ∈ (r1 ' + x2).butlast, ¬ (r1 ' + x2).last ∈ x0, ¬ h[h[(r1 ' + x2).last].nxt].nxt ∈ (r1 ' + x2).butlast, ¬ h[h[(r1 ' + x2).last].nxt].nxt ∈ x0, ¬ h[(r1 ' + x2).last].nxt ∈ (r1 ' + x2).butlast, ¬ h[(r1 ' + x2).last].nxt ∈ x0, (r1 ' + x2).last = r1 ∨ (r1 ' + x2).last ∈ x2 ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] x ≠ [] ⊦ x.butlast + x.last ' = x; (s) ⊦ a ' + x ≠ []; (s) ⊦ a ∈ b ' + x ↔ a = b ∨ a ∈ x; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ (a ' + b ' + x).butlast = a ' + (b ' + x).butlast; (s) ⊦ (x + a ' + y).butlast = x + (a ' + y).butlast; (s) ⊦ a ≠ b → (a ∈ b ' + x ↔ a ∈ x); (s) ⊦ dups(x + a ' + y) ↔ a ∈ x ∨ a ∈ y ∨ dups(x + y); (ws) ⊦ x + a ' + y ≠ []; (s) ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) ⊦ (x + a ' + y).last = (a ' + y).last; (s)

Proof step 52 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 53:

  ⊦ y ≠ [] → (x + y).last = y.last

• Proof step 54:

  x2 = [], (r1 ' + x2).last ≠ h[(r1 ' + x2).last].nxt, (r1 ' + x2).last ≠ h[h[(r1 ' + x2).last].nxt].nxt, h[(r1 ' + x2).last].nxt ≠ h[h[(r1 ' + x2).last].nxt].nxt, h[(r1 ' + x2).last].nxt ≠ null, path(h[h[(r1 ' + x2).last].nxt].nxt ' + x0 + h[(r1 ' + x2).last].nxt ' + r1 ' + x2, h), path(h[h[(r1 ' + x2).last].nxt].nxt ' + x0 + h[(r1 ' + x2).last].nxt ', h), (r1 ' + x2).last ∈ h, h[(r1 ' + x2).last].nxt ∈ h, ¬ dups(x0 + (r1 ' + x2).butlast), ¬ (r1 ' + x2).last ∈ (r1 ' + x2).butlast, ¬ (r1 ' + x2).last ∈ x0, ¬ h[h[(r1 ' + x2).last].nxt].nxt ∈ (r1 ' + x2).butlast, ¬ h[h[(r1 ' + x2).last].nxt].nxt ∈ x0, ¬ h[(r1 ' + x2).last].nxt ∈ (r1 ' + x2).butlast, ¬ h[(r1 ' + x2).last].nxt ∈ x0, (r1 ' + x2).last = r1 ∨ (r1 ' + x2).last ∈ x2 ⊦

• Proof step 57:

  x2.last ≠ h[x2.last].nxt, x2.last ≠ h[h[x2.last].nxt].nxt, h[x2.last].nxt ≠ h[h[x2.last].nxt].nxt, h[x2.last].nxt ≠ null, x2 ≠ [], path(h[h[x2.last].nxt].nxt ' + x0 + h[x2.last].nxt ' + r1 ' + x2, h), path(h[h[x2.last].nxt].nxt ' + x0 + h[x2.last].nxt ', h), x2.last ∈ h, h[x2.last].nxt ∈ h, ¬ dups(x0 + (r1 ' + x2).butlast), ¬ x2.last ∈ (r1 ' + x2).butlast, ¬ x2.last ∈ x0, ¬ h[h[x2.last].nxt].nxt ∈ (r1 ' + x2).butlast, ¬ h[h[x2.last].nxt].nxt ∈ x0, ¬ h[x2.last].nxt ∈ (r1 ' + x2).butlast, ¬ h[x2.last].nxt ∈ x0, x2.last = r1 ∨ x2.last ∈ x2 ⊦

Rule argument

Additional information

Used terms: x2, r1 '

Proof step 53 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

A spec-lemma

Proof step 54 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 55:

  h[r1].nxt ≠ h[h[r1].nxt].nxt, h[r1].nxt ≠ null, r1 ≠ h[r1].nxt, r1 ≠ h[h[r1].nxt].nxt, path(h[h[r1].nxt].nxt ' + x0 + h[r1].nxt ', h), path(h[h[r1].nxt].nxt ' + x0 + h[r1].nxt ' + r1 ', h), h[r1].nxt ∈ h, r1 ∈ h, ¬ dups(x0), ¬ r1 ∈ x0, ¬ h[h[r1].nxt].nxt ∈ x0, ¬ h[r1].nxt ∈ x0 ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ a '.last = a; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ x + [] = x; (s) ⊦ ¬ a ∈ []; (s) ⊦ a '.butlast = []; (s)

Proof step 55 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 56:

  ⊦ path(r ' + x + r0 ', h) ↔ path(r ' + x, h) ∧ h[(r ' + x).last].nxt = r0 ∧ r0 ∈ h ∧ r0 ≠ null

Rule argument

Additional information

Used simplifier rules: ⊦ (a ' + x + b ').last = b; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] Used terms: h, r1, x0 + h[r1].nxt ', h[h[r1].nxt].nxt

Proof step 56 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Lemma path-split-last

Proof step 57 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 58:

  x2.last ≠ h[x2.last].nxt, x2.last ≠ h[h[x2.last].nxt].nxt, x2.last ≠ r1, h[x2.last].nxt ≠ h[h[x2.last].nxt].nxt, h[x2.last].nxt ≠ null, h[h[x2.last].nxt].nxt ≠ r1, h[x2.last].nxt ≠ r1, x2 ≠ [], path(h[h[x2.last].nxt].nxt ' + x0 + h[x2.last].nxt ', h), path(h[h[x2.last].nxt].nxt ' + x0 + h[x2.last].nxt ' + r1 ' + x2, h), h[x2.last].nxt ∈ h, x2.last ∈ h, ¬ dups(x0 + x2.butlast), ¬ h[x2.last].nxt ∈ x0, ¬ r1 ∈ x0, ¬ r1 ∈ x2.butlast, ¬ x2.last ∈ x2.butlast, ¬ x2.last ∈ x0, ¬ h[h[x2.last].nxt].nxt ∈ x2.butlast, ¬ h[h[x2.last].nxt].nxt ∈ x0, ¬ h[x2.last].nxt ∈ x2.butlast ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ x.last ∈ x ↔ x ≠ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ a ∈ b ' + x ↔ ¬(a ≠ b ∧ ¬ a ∈ x); (ss) ⊦ x ≠ [] → (a ' + x).butlast = a ' + x.butlast; (s) ⊦ dups(x + a ' + y) ↔ a ∈ x ∨ a ∈ y ∨ dups(x + y); (ws)

Proof step 58 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule apply elim lemma

And obtained the following new goals

• Proof step 59:

  x2 = x1 + r ', h[r].nxt ≠ h[h[r].nxt].nxt, h[r].nxt ≠ null, h[h[r].nxt].nxt ≠ r1, h[r].nxt ≠ r1, r ≠ h[r].nxt, r ≠ h[h[r].nxt].nxt, r ≠ r1, x2 ≠ [], path(h[h[r].nxt].nxt ' + x0 + h[r].nxt ' + r1 ' + x2, h), path(h[h[r].nxt].nxt ' + x0 + h[r].nxt ', h), r ∈ h, h[r].nxt ∈ h, ¬ dups(x0 + x1), ¬ h[r].nxt ∈ x0, ¬ r1 ∈ x0, ¬ r1 ∈ x1, ¬ r ∈ x1, ¬ r ∈ x0, ¬ h[h[r].nxt].nxt ∈ x1, ¬ h[h[r].nxt].nxt ∈ x0, ¬ h[r].nxt ∈ x1 ⊦

Rule argument

Additional information

Used simplifier rules: x ≠ [] ⊦ a = x.last ∧ y = x.butlast ↔ x = y + a '; (elim) Used terms: x2, a, y

Proof step 59 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 60:

  h[r].nxt ≠ h[h[r].nxt].nxt, h[r].nxt ≠ null, h[h[r].nxt].nxt ≠ r1, h[r].nxt ≠ r1, r ≠ h[r].nxt, r ≠ h[h[r].nxt].nxt, r ≠ r1, path(h[h[r].nxt].nxt ' + x0 + h[r].nxt ', h), path(h[h[r].nxt].nxt ' + x0 + h[r].nxt ' + r1 ' + x1 + r ', h), h[r].nxt ∈ h, r ∈ h, ¬ dups(x0 + x1), ¬ h[r].nxt ∈ x0, ¬ r1 ∈ x0, ¬ r1 ∈ x1, ¬ r ∈ x1, ¬ r ∈ x0, ¬ h[h[r].nxt].nxt ∈ x1, ¬ h[h[r].nxt].nxt ∈ x0, ¬ h[r].nxt ∈ x1 ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ x + a ' ≠ []; (s)

Proof step 60 from proof rem-stays-reachable-aux1 (previous step) (back to tree)

User applied rule cut formula

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); path-postfix: ⊦ path(x + r ' + y, h) → (path(r ' + y, h) ↔ true); a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] Used formulas: path(h[r].nxt ' + r1 ' + x1 + r ', h)