Proof step 1 from proof add-preserves-reachable (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  h[r1].nxt = h[r2].nxt, ce.nxt = r2, r2 ≠ null, reachable(r, r0, h), reachable(r, r1, h), r2 ∈ h, ¬ reachable(r, r0, h[r1, ce]), ¬ reachable(r, r2, h) ⊦

Rule argument

Additional information

Proof step 2 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, reachable(r, r1, h), reachable(r, r0, h), ce.nxt ∈ h, ¬ reachable(r, r0, h[r1, ce]), ¬ reachable(r, ce.nxt, h) ⊦

Rule argument

Additional information

Proof step 3 from proof add-preserves-reachable (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)

• Proof step 5:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, reachable(r, r1, h), ce.nxt ∈ h, ¬ reachable(r, r0, h[r1, ce]), ¬ reachable(r, ce.nxt, h), ∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0 ⊦

Rule argument

Additional information

Used terms: h, r0, r

Proof step 4 from proof add-preserves-reachable (previous step) (back to tree)

Lemma reachable-def

Proof step 5 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 6:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, path(r ' + x, h), reachable(r, r1, h), ce.nxt ∈ h, ¬ reachable(r, (r ' + x).last, h[r1, ce]), ¬ reachable(r, ce.nxt, h) ⊦

Rule argument

Additional information

Proof step 6 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule structural induction

And obtained the following new goals

• Proof step 7:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, path(r ' + (r0 + x0), h), reachable(r, r1, h), ce.nxt ∈ h, ¬ reachable(r, (r ' + (r0 + x0)).last, h[r1, ce]), ¬ reachable(r, ce.nxt, h), ∀ h, r1, ce, r. ¬ ( h[r1].nxt = h[ce.nxt].nxt ∧ ce.nxt ≠ null ∧ path(r ' + x0, h) ∧ reachable(r, r1, h) ∧ ce.nxt ∈ h ∧ ¬ reachable(r, (r ' + x0).last, h[r1, ce]) ∧ ¬ reachable(r, ce.nxt, h)) ⊦

Rule argument

Additional information

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

Proof step 7 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 8:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r ≠ null, path(h[r].nxt ' + x0, h), reachable(r, r1, h), r ∈ h, ce.nxt ∈ h, ¬ reachable(r, (h[r].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r, ce.nxt, h), ∀ h0, r2, ce0, r3. h0[r2].nxt ≠ h0[ce0.nxt].nxt ∨ ce0.nxt = null ∨ ¬ path(r3 ' + x0, h0) ∨ ¬ reachable(r3, r2, h0) ∨ ¬ ce0.nxt ∈ h0 ∨ reachable(r3, (r3 ' + x0).last, h0[r2, ce0]) ∨ reachable(r3, ce0.nxt, h0) ⊦

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] ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) ⊦ a + x = a ' + x; (s)

Proof step 8 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 9:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r ≠ null, path(h[r].nxt ' + x0, h), reachable(r, r1, h), ce.nxt ∈ h, r ∈ h, ¬ reachable(r, (h[r].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r, ce.nxt, h), h[r1].nxt ≠ h[ce.nxt].nxt ∨ ce.nxt = null ∨ ¬ path(h[r].nxt ' + x0, h) ∨ ¬ reachable(h[r].nxt, r1, h) ∨ ¬ ce.nxt ∈ h ∨ reachable(h[r].nxt, (h[r].nxt ' + x0).last, h[r1, ce]) ∨ reachable(h[r].nxt, ce.nxt, h) ⊦

Rule argument

Additional information

Used terms: h, r1, ce, h[r].nxt

Proof step 9 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 10:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r ≠ null, path(h[r].nxt ' + x0, h), reachable(r, r1, h), r ∈ h, ce.nxt ∈ h, ¬ reachable(r, (h[r].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r, ce.nxt, h), ¬ reachable(h[r].nxt, r1, h) ∨ reachable(h[r].nxt, (h[r].nxt ' + x0).last, h[r1, ce]) ∨ reachable(h[r].nxt, ce.nxt, h) ⊦

Rule argument

Additional information

Proof step 10 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 11:

  h[r1].nxt = h[ce.nxt].nxt, r = r1, ce.nxt ≠ null, r ≠ null, path(h[r].nxt ' + x0, h), reachable(r, r1, h), ce.nxt ∈ h, r ∈ h, ¬ reachable(r, (h[r].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r, ce.nxt, h), ¬ reachable(h[r].nxt, r1, h) ∨ reachable(h[r].nxt, (h[r].nxt ' + x0).last, h[r1, ce]) ∨ reachable(h[r].nxt, ce.nxt, h) ⊦

• Proof step 46:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r ≠ r1, r ≠ null, path(h[r].nxt ' + x0, h), reachable(r, r1, h), ce.nxt ∈ h, r ∈ h, ¬ reachable(r, (h[r].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r, ce.nxt, h), ¬ reachable(h[r].nxt, r1, h) ∨ reachable(h[r].nxt, (h[r].nxt ' + x0).last, h[r1, ce]) ∨ reachable(h[r].nxt, ce.nxt, h) ⊦

Rule argument

Additional information

Used formulas: r = r1

Proof step 11 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 12:

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

Rule argument

Additional information

Used simplifier rules: self-reachable: ⊦ reachable(r, r, h) ↔ r ≠ null ∧ r ∈ h;

Proof step 12 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 13:

  ⊦ ¬ reachable(r, r0, H) ∧ reachable(r, r1, H) ∧ r0 ≠ null ∧ r0 ∈ H → (reachable(r, r1, H[r0, ce]) ↔ true)

• Proof step 14:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r1 ≠ null, path(h[r1].nxt ' + x0, h), ce.nxt ∈ h, r1 ∈ h, ¬ reachable(r1, (h[r1].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r1, ce.nxt, h), ¬ reachable(h[ce.nxt].nxt, r1, h) ∧ reachable(h[ce.nxt].nxt, (h[ce.nxt].nxt ' + x0).last, h) ∧ r1 ≠ null ∧ r1 ∈ h → (reachable(h[ce.nxt].nxt, (h[ce.nxt].nxt ' + x0).last, h[r1, ce]) ↔ true), ¬ reachable(h[r1].nxt, r1, h) ∨ reachable(h[r1].nxt, (h[r1].nxt ' + x0).last, h[r1, ce]) ∨ reachable(h[r1].nxt, ce.nxt, h) ⊦

Rule argument

Additional information

Used terms: ce, r1, h, h[ce.nxt].nxt, (h[ce.nxt].nxt ' + x0).last

Proof step 13 from proof add-preserves-reachable (previous step) (back to tree)

Lemma modnotreachable-reachable-weak

Proof step 14 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 15:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r1 ≠ null, path(h[r1].nxt ' + x0, h), r1 ∈ h, ce.nxt ∈ h, ¬ reachable(r1, (h[r1].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r1, ce.nxt, h), ¬ reachable(h[r1].nxt, r1, h) ∧ reachable(h[r1].nxt, (h[r1].nxt ' + x0).last, h) → reachable(h[r1].nxt, (h[r1].nxt ' + x0).last, h[r1, ce]), ¬ reachable(h[r1].nxt, r1, h) ∨ reachable(h[r1].nxt, (h[r1].nxt ' + x0).last, h[r1, ce]) ∨ reachable(h[r1].nxt, ce.nxt, h) ⊦

Rule argument

Additional information

Proof step 15 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 16:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r1 ≠ null, path(h[r1].nxt ' + x0, h), ce.nxt ∈ h, r1 ∈ h, ¬ reachable(r1, (h[r1].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r1, ce.nxt, h), ¬ reachable(h[r1].nxt, r1, h) ∨ reachable(h[r1].nxt, (h[r1].nxt ' + x0).last, h[r1, ce]) ∨ reachable(h[r1].nxt, ce.nxt, h) ⊦ ¬ reachable(h[r1].nxt, r1, h) ∧ reachable(h[r1].nxt, (h[r1].nxt ' + x0).last, h)

• Proof step 36:

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

Rule argument

Additional information

Proof step 16 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 17:

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

• Proof step 32:

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

Rule argument

Additional information

Proof step 17 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 18:

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

Rule argument

Additional information

Proof step 18 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 19:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r1 ≠ null, path(h[r1].nxt ' + x0, h), reachable(h[r1].nxt, r1, h), reachable(h[r1].nxt, (h[r1].nxt ' + x0).last, h[r1, ce]), r1 ∈ h, ce.nxt ∈ h, ¬ reachable(r1, (h[r1].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r1, ce.nxt, h) ⊦

• Proof step 29:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r1 ≠ null, path(h[r1].nxt ' + x0, h), reachable(h[r1].nxt, r1, h), reachable(h[r1].nxt, ce.nxt, h), r1 ∈ h, ce.nxt ∈ h, ¬ reachable(r1, (h[r1].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r1, ce.nxt, h) ⊦

Rule argument

Additional information

Proof step 19 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 20:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r1 ≠ null, path(h[r1].nxt ' + x0, h), reachable(h[r1].nxt, (h[r1].nxt ' + x0).last, h[r1, ce]), reachable(h[r1].nxt, r1, h), ce.nxt ∈ h, (h[r1].nxt ' + x0).last ∈ h[r1, ce], r1 ∈ h, ¬ reachable(r1, (h[r1].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r1, ce.nxt, h) ⊦

Rule argument

Additional information

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

Proof step 20 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 21:

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

• Proof step 22:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r1 ≠ null, path(h[r1].nxt ' + x0, h), reachable(h[r1].nxt, r1, h), reachable(h[r1].nxt, (h[r1].nxt ' + x0).last, h[r1, ce]), r1 ∈ h, (h[r1].nxt ' + x0).last ∈ h[r1, ce], ce.nxt ∈ h, ¬ reachable(r1, ce.nxt, h), ∀ x. ¬(path(r1 ' + x, h[r1, ce]) ∧ (r1 ' + x).last = (h[r1].nxt ' + x0).last) ⊦

Rule argument

Additional information

Used terms: h[r1, ce], (h[r1].nxt ' + x0).last, r1

Proof step 21 from proof add-preserves-reachable (previous step) (back to tree)

Lemma reachable-def

Proof step 22 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 23:

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

• Proof step 24:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r1 ≠ null, path(h[r1].nxt ' + x0, h), reachable(h[r1].nxt, r1, h), ce.nxt ∈ h, (h[r1].nxt ' + x0).last ∈ h[r1, ce], r1 ∈ h, ¬ reachable(r1, ce.nxt, h), ∀ x. ¬(path(r1 ' + x, h[r1, ce]) ∧ (r1 ' + x).last = (h[r1].nxt ' + x0).last), ∃ x. path(h[r1].nxt ' + x, h[r1, ce]) ∧ (h[r1].nxt ' + x).last = (h[r1].nxt ' + x0).last ⊦

Rule argument

Additional information

Used terms: h[r1, ce], (h[r1].nxt ' + x0).last, h[r1].nxt

Proof step 23 from proof add-preserves-reachable (previous step) (back to tree)

Lemma reachable-def

Proof step 24 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 25:

  (h[r1].nxt ' + x).last = (h[r1].nxt ' + x0).last, h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r1 ≠ null, path(h[r1].nxt ' + x, h[r1, ce]), path(h[r1].nxt ' + x0, h), reachable(h[r1].nxt, r1, h), r1 ∈ h, (h[r1].nxt ' + x0).last ∈ h[r1, ce], ce.nxt ∈ h, ¬ reachable(r1, ce.nxt, h), ∀ x1. ¬ path(r1 ' + x1, h[r1, ce]) ∨ (r1 ' + x1).last ≠ (h[r1].nxt ' + x0).last ⊦

Rule argument

Additional information

Proof step 25 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 26:

  (h[r1].nxt ' + x).last = (h[r1].nxt ' + x0).last, h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r1 ≠ null, path(h[r1].nxt ' + x0, h), path(h[r1].nxt ' + x, h[r1, ce]), reachable(h[r1].nxt, r1, h), ce.nxt ∈ h, (h[r1].nxt ' + x0).last ∈ h[r1, ce], r1 ∈ h, ¬ reachable(r1, ce.nxt, h), ¬ path(r1 ' + ce.nxt ' + h[r1].nxt ' + x, h[r1, ce]) ∨ (r1 ' + ce.nxt ' + h[r1].nxt ' + x).last ≠ (h[r1].nxt ' + x0).last ⊦

Rule argument

Additional information

Used terms: ce.nxt ' + h[r1].nxt ' + x

Proof step 26 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 27:

  (h[r1].nxt ' + x).last = (h[r1].nxt ' + x0).last, h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, h[r1, ce][ce.nxt].nxt ≠ h[r1].nxt, r1 ≠ null, path(h[r1].nxt ' + x0, h), path(h[r1].nxt ' + x, h[r1, ce]), reachable(h[r1].nxt, r1, h), ce.nxt ∈ h, (h[r1].nxt ' + x0).last ∈ h[r1, ce], r1 ∈ h, ce.nxt ∈ h[r1, ce], ¬ reachable(r1, ce.nxt, 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 ∈ st[b, d] ↔ a = b ∨ a ∈ st; (ws) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ (a ' + x + b ' + y).last = (b ' + y).last; (s) ⊦ st[a, d][a] = d; (s) ⊦ a ∈ st[a, d]; (s)

Proof step 27 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 28:

  ⊦ a ≠ b → st[b, d][a] = st[a]

Rule argument

Additional information

Used simplifier rules: self-reachable: ⊦ reachable(r, r, h) ↔ r ≠ null ∧ r ∈ h; reachable-notnullinH: ⊦ reachable(r, r1, h) ∧ r1 ≠ null → (r1 ∈ h ↔ true); ⊦ st[a, d][a] = d; (s) ⊦ a ∈ st[a, d]; (s) Used terms: h, r1, ce, ce.nxt

Proof step 28 from proof add-preserves-reachable (previous step) (back to tree)

A spec-lemma

Proof step 29 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 30:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r1 ≠ null, path(h[r1].nxt ' + x0, h), reachable(h[r1].nxt, ce.nxt, h), reachable(h[r1].nxt, r1, h), ce.nxt ∈ h, r1 ∈ h, ¬ reachable(r1, (h[r1].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r1, ce.nxt, h) ⊦

Rule argument

Additional information

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

Proof step 30 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 31:

  ⊦ reachable(h[r].nxt, r0, h) ∧ r ≠ r0 ∧ r ≠ null ∧ r ∈ h → (reachable(r, r0, h) ↔ true)

Rule argument

Additional information

Used simplifier rules: self-reachable: ⊦ reachable(r, r, h) ↔ r ≠ null ∧ r ∈ h; Used terms: h, ce.nxt, r1

Proof step 31 from proof add-preserves-reachable (previous step) (back to tree)

Lemma fromnext-reachable-rev

Proof step 32 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 33:

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

• Proof step 34:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r1 ≠ null, path(h[r1].nxt ' + x0, h), ce.nxt ∈ h, r1 ∈ h, ¬ reachable(r1, (h[r1].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r1, ce.nxt, h), ∀ x. ¬(path(h[r1].nxt ' + x, h) ∧ (h[r1].nxt ' + x).last = (h[r1].nxt ' + x0).last), ¬ reachable(h[r1].nxt, r1, h) ∨ reachable(h[r1].nxt, (h[r1].nxt ' + x0).last, h[r1, ce]) ∨ reachable(h[r1].nxt, ce.nxt, h) ⊦

Rule argument

Additional information

Used terms: h, (h[r1].nxt ' + x0).last, h[r1].nxt

Proof step 33 from proof add-preserves-reachable (previous step) (back to tree)

Lemma reachable-def

Proof step 34 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 35:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r1 ≠ null, path(h[r1].nxt ' + x0, h), r1 ∈ h, ce.nxt ∈ h, ¬ reachable(r1, (h[r1].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r1, ce.nxt, h), ∀ x. ¬ path(h[r1].nxt ' + x, h) ∨ (h[r1].nxt ' + x).last ≠ (h[r1].nxt ' + x0).last, ¬ reachable(h[r1].nxt, r1, h) ∨ reachable(h[r1].nxt, (h[r1].nxt ' + x0).last, h[r1, ce]) ∨ reachable(h[r1].nxt, ce.nxt, h) ⊦

Rule argument

Additional information

Proof step 35 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: x0

Proof step 36 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 37:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r1 ≠ null, path(h[r1].nxt ' + x0, h), reachable(h[r1].nxt, (h[r1].nxt ' + x0).last, h[r1, ce]), r1 ∈ h, ce.nxt ∈ h, ¬ reachable(r1, (h[r1].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r1, ce.nxt, h) ⊦

Rule argument

Additional information

Proof step 37 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 38:

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

• Proof step 39:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r1 ≠ null, path(h[r1].nxt ' + x0, h), reachable(h[r1].nxt, (h[r1].nxt ' + x0).last, h[r1, ce]), ce.nxt ∈ h, r1 ∈ h, ¬ reachable(r1, ce.nxt, h), ∀ x. ¬(path(r1 ' + x, h[r1, ce]) ∧ (r1 ' + x).last = (h[r1].nxt ' + x0).last) ⊦

Rule argument

Additional information

Used terms: h[r1, ce], (h[r1].nxt ' + x0).last, r1

Proof step 38 from proof add-preserves-reachable (previous step) (back to tree)

Lemma reachable-def

Proof step 39 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 40:

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

• Proof step 41:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r1 ≠ null, path(h[r1].nxt ' + x0, h), r1 ∈ h, ce.nxt ∈ h, ¬ reachable(r1, ce.nxt, h), ∀ x. ¬(path(r1 ' + x, h[r1, ce]) ∧ (r1 ' + x).last = (h[r1].nxt ' + x0).last), ∃ x. path(h[r1].nxt ' + x, h[r1, ce]) ∧ (h[r1].nxt ' + x).last = (h[r1].nxt ' + x0).last ⊦

Rule argument

Additional information

Used terms: h[r1, ce], (h[r1].nxt ' + x0).last, h[r1].nxt

Proof step 40 from proof add-preserves-reachable (previous step) (back to tree)

Lemma reachable-def

Proof step 41 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 42:

  (h[r1].nxt ' + x).last = (h[r1].nxt ' + x0).last, h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r1 ≠ null, path(h[r1].nxt ' + x, h[r1, ce]), path(h[r1].nxt ' + x0, h), ce.nxt ∈ h, r1 ∈ h, ¬ reachable(r1, ce.nxt, h), ∀ x1. ¬ path(r1 ' + x1, h[r1, ce]) ∨ (r1 ' + x1).last ≠ (h[r1].nxt ' + x0).last ⊦

Rule argument

Additional information

Proof step 42 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 43:

  (h[r1].nxt ' + x).last = (h[r1].nxt ' + x0).last, h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r1 ≠ null, path(h[r1].nxt ' + x0, h), path(h[r1].nxt ' + x, h[r1, ce]), r1 ∈ h, ce.nxt ∈ h, ¬ reachable(r1, ce.nxt, h), ¬ path(r1 ' + ce.nxt ' + h[r1].nxt ' + x, h[r1, ce]) ∨ (r1 ' + ce.nxt ' + h[r1].nxt ' + x).last ≠ (h[r1].nxt ' + x0).last ⊦

Rule argument

Additional information

Used terms: ce.nxt ' + h[r1].nxt ' + x

Proof step 43 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 44:

  (h[r1].nxt ' + x).last = (h[r1].nxt ' + x0).last, h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, h[r1, ce][ce.nxt].nxt ≠ h[r1].nxt, r1 ≠ null, path(h[r1].nxt ' + x0, h), path(h[r1].nxt ' + x, h[r1, ce]), r1 ∈ h, ce.nxt ∈ h, ce.nxt ∈ h[r1, ce], ¬ reachable(r1, ce.nxt, 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 ∈ st[b, d] ↔ a = b ∨ a ∈ st; (ws) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ (a ' + x + b ' + y).last = (b ' + y).last; (s) ⊦ st[a, d][a] = d; (s) ⊦ a ∈ st[a, d]; (s)

Proof step 44 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 45:

  ⊦ a ≠ b → st[b, d][a] = st[a]

Rule argument

Additional information

Used simplifier rules: self-reachable: ⊦ reachable(r, r, h) ↔ r ≠ null ∧ r ∈ h; ⊦ st[a, d][a] = d; (s) ⊦ a ∈ st[a, d]; (s) Used terms: h, r1, ce, ce.nxt

Proof step 45 from proof add-preserves-reachable (previous step) (back to tree)

A spec-lemma

Proof step 46 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 47:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r ≠ r1, r ≠ null, path(h[r].nxt ' + x0, h), reachable(r, r1, h), reachable(h[r].nxt, r1, h), r ∈ h, ce.nxt ∈ h, ¬ reachable(r, (h[r].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r, ce.nxt, h), ¬ reachable(h[r].nxt, r1, h) ∨ reachable(h[r].nxt, (h[r].nxt ' + x0).last, h[r1, ce]) ∨ reachable(h[r].nxt, ce.nxt, h) ⊦

Rule argument

Additional information

Used simplifier rules: fromnext-reachable: ⊦ reachable(r, r0, h) ∧ r ≠ r0 → (reachable(h[r].nxt, r0, h) ↔ true); Used formulas: reachable(h[r].nxt, r1, h)

Proof step 47 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 48:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r ≠ r1, r ≠ null, path(h[r].nxt ' + x0, h), reachable(h[r].nxt, r1, h), reachable(r, r1, h), ce.nxt ∈ h, r ∈ h, ¬ reachable(r, (h[r].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r, ce.nxt, h), reachable(h[r].nxt, (h[r].nxt ' + x0).last, h[r1, ce]) ∨ reachable(h[r].nxt, ce.nxt, h) ⊦

Rule argument

Additional information

Used simplifier rules: fromnext-reachable: ⊦ reachable(r, r0, h) ∧ r ≠ r0 → (reachable(h[r].nxt, r0, h) ↔ true);

Proof step 48 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 49:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r ≠ r1, r ≠ null, path(h[r].nxt ' + x0, h), reachable(r, r1, h), reachable(h[r].nxt, r1, h), reachable(h[r].nxt, (h[r].nxt ' + x0).last, h[r1, ce]), r ∈ h, ce.nxt ∈ h, ¬ reachable(r, (h[r].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r, ce.nxt, h) ⊦

• Proof step 57:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r ≠ r1, r ≠ null, path(h[r].nxt ' + x0, h), reachable(r, r1, h), reachable(h[r].nxt, r1, h), reachable(h[r].nxt, ce.nxt, h), r ∈ h, ce.nxt ∈ h, ¬ reachable(r, (h[r].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r, ce.nxt, h) ⊦

Rule argument

Additional information

Proof step 49 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 50:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r ≠ r1, r ≠ null, path(h[r].nxt ' + x0, h), reachable(h[r].nxt, (h[r].nxt ' + x0).last, h[r1, ce]), reachable(h[r].nxt, r1, h), reachable(r, r1, h), ce.nxt ∈ h, r ∈ h, (h[r].nxt ' + x0).last ∈ h[r1, ce], r1 ∈ h, ¬ reachable(r, (h[r].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r, ce.nxt, h) ⊦

Rule argument

Additional information

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

Proof step 50 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 51:

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

• Proof step 52:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r ≠ r1, r ≠ null, path(h[r].nxt ' + x0, h), reachable(r, r1, h), reachable(h[r].nxt, r1, h), r1 ∈ h, (h[r].nxt ' + x0).last ∈ h[r1, ce], r ∈ h, ce.nxt ∈ h, ¬ reachable(r, (h[r].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r, ce.nxt, h), ∃ x. path(h[r].nxt ' + x, h[r1, ce]) ∧ (h[r].nxt ' + x).last = (h[r].nxt ' + x0).last ⊦

Rule argument

Additional information

Used terms: h[r].nxt, (h[r].nxt ' + x0).last, h[r1, ce]

Proof step 51 from proof add-preserves-reachable (previous step) (back to tree)

Lemma reachable-def

Proof step 52 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 53:

  (h[r].nxt ' + x).last = (h[r].nxt ' + x0).last, h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r ≠ r1, r ≠ null, path(h[r].nxt ' + x, h[r1, ce]), path(h[r].nxt ' + x0, h), reachable(h[r].nxt, r1, h), reachable(r, r1, h), ce.nxt ∈ h, r ∈ h, (h[r].nxt ' + x0).last ∈ h[r1, ce], r1 ∈ h, ¬ reachable(r, (h[r].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r, ce.nxt, h) ⊦

Rule argument

Additional information

Proof step 53 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 54:

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

• Proof step 55:

  (h[r].nxt ' + x).last = (h[r].nxt ' + x0).last, h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r ≠ r1, r ≠ null, path(h[r].nxt ' + x0, h), path(h[r].nxt ' + x, h[r1, ce]), reachable(r, r1, h), reachable(h[r].nxt, r1, h), r1 ∈ h, (h[r].nxt ' + x0).last ∈ h[r1, ce], r ∈ h, ce.nxt ∈ h, ¬ reachable(r, ce.nxt, h), ∀ x. ¬(path(r ' + x, h[r1, ce]) ∧ (r ' + x).last = (h[r].nxt ' + x0).last) ⊦

Rule argument

Additional information

Used terms: r, (h[r].nxt ' + x0).last, h[r1, ce]

Proof step 54 from proof add-preserves-reachable (previous step) (back to tree)

Lemma reachable-def

Proof step 55 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 56:

  (h[r].nxt ' + x).last = (h[r].nxt ' + x0).last, h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r ≠ r1, r ≠ null, path(h[r].nxt ' + x, h[r1, ce]), path(h[r].nxt ' + x0, h), reachable(h[r].nxt, r1, h), reachable(r, r1, h), ce.nxt ∈ h, r ∈ h, (h[r].nxt ' + x0).last ∈ h[r1, ce], r1 ∈ h, ¬ reachable(r, ce.nxt, h), ∀ x1. ¬ path(r ' + x1, h[r1, ce]) ∨ (r ' + x1).last ≠ (h[r].nxt ' + x0).last ⊦

Rule argument

Additional information

Proof step 56 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule all left

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 ≠ b → (a ∈ st[b, d] ↔ a ∈ st); (s) ⊦ a ≠ b → st[b, d][a] = st[a]; (s) ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] Used terms: h[r].nxt ' + x

Proof step 57 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 58:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r ≠ r1, r ≠ null, path(h[r].nxt ' + x0, h), reachable(h[r].nxt, ce.nxt, h), reachable(h[r].nxt, r1, h), reachable(r, r1, h), r ∈ h, ce.nxt ∈ h, r1 ∈ h, ¬ reachable(r, (h[r].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r, ce.nxt, h) ⊦

Rule argument

Additional information

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

Proof step 58 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 59:

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

• Proof step 60:

  h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r ≠ r1, r ≠ null, path(h[r].nxt ' + x0, h), reachable(r, r1, h), reachable(h[r].nxt, r1, h), r1 ∈ h, ce.nxt ∈ h, r ∈ h, ¬ reachable(r, (h[r].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r, ce.nxt, h), ∃ x. path(h[r].nxt ' + x, h) ∧ (h[r].nxt ' + x).last = ce.nxt ⊦

Rule argument

Additional information

Used terms: h[r].nxt, ce.nxt, h

Proof step 59 from proof add-preserves-reachable (previous step) (back to tree)

Lemma reachable-def

Proof step 60 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 61:

  (h[r].nxt ' + x).last = ce.nxt, h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r ≠ r1, r ≠ null, path(h[r].nxt ' + x, h), path(h[r].nxt ' + x0, h), reachable(h[r].nxt, r1, h), reachable(r, r1, h), r ∈ h, ce.nxt ∈ h, r1 ∈ h, ¬ reachable(r, (h[r].nxt ' + x0).last, h[r1, ce]), ¬ reachable(r, ce.nxt, h) ⊦

Rule argument

Additional information

Proof step 61 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 62:

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

• Proof step 63:

  (h[r].nxt ' + x).last = ce.nxt, h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r ≠ r1, r ≠ null, path(h[r].nxt ' + x0, h), path(h[r].nxt ' + x, h), reachable(r, r1, h), reachable(h[r].nxt, r1, h), r1 ∈ h, ce.nxt ∈ h, r ∈ h, ¬ reachable(r, (h[r].nxt ' + x0).last, h[r1, ce]), ∀ x. ¬(path(r ' + x, h) ∧ (r ' + x).last = ce.nxt) ⊦

Rule argument

Additional information

Used terms: r, ce.nxt, h

Proof step 62 from proof add-preserves-reachable (previous step) (back to tree)

Lemma reachable-def

Proof step 63 from proof add-preserves-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 64:

  (h[r].nxt ' + x).last = ce.nxt, h[r1].nxt = h[ce.nxt].nxt, ce.nxt ≠ null, r ≠ r1, r ≠ null, path(h[r].nxt ' + x, h), path(h[r].nxt ' + x0, h), reachable(h[r].nxt, r1, h), reachable(r, r1, h), r ∈ h, ce.nxt ∈ h, r1 ∈ h, ¬ reachable(r, (h[r].nxt ' + x0).last, h[r1, ce]), ∀ x1. ¬ path(r ' + x1, h) ∨ (r ' + x1).last ≠ ce.nxt ⊦

Rule argument

Additional information

Proof step 64 from proof add-preserves-reachable (previous step) (back to tree)

User applied rule all left

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