Proof step 1 from proof add-preserves-reachable-rev (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, ¬(reachable(r, r0, h[r1 := ce]) ↔ false), r0 ≠ r2, r2 ≠ null, reachable(r, r1, h), r2 ∈ h, ¬ reachable(r, r2, h), ¬ reachable(r, r0, h) ⊦

Rule argument

Additional information

Proof step 2 from proof add-preserves-reachable-rev (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, r0 ≠ ce.nxt, reachable(r, r1, h), reachable(r, r0, h[r1 := ce]), ce.nxt ∈ h, r1 ∈ h, r0 ∈ h[r1 := ce], ¬ reachable(r, ce.nxt, h), ¬ reachable(r, r0, h) ⊦

Rule argument

Additional information

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

Proof step 3 from proof add-preserves-reachable-rev (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, r0 ≠ ce.nxt, reachable(r, r1, h), r0 ∈ h[r1 := ce], r1 ∈ h, ce.nxt ∈ h, ¬ reachable(r, ce.nxt, h), ¬ reachable(r, r0, h), ∃ x. path(r ' + x, h[r1 := ce]) ∧ (r ' + x).last = r0 ⊦

Rule argument

Additional information

Used terms: h[r1 := ce], r0, r

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

Lemma reachable-def

Proof step 5 from proof add-preserves-reachable-rev (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, (r ' + x).last ≠ ce.nxt, ce.nxt ≠ null, path(r ' + x, h[r1 := ce]), reachable(r, r1, h), ce.nxt ∈ h, r1 ∈ h, (r ' + x).last ∈ h[r1 := ce], ¬ reachable(r, ce.nxt, h), ¬ reachable(r, (r ' + x).last, h) ⊦

Rule argument

Additional information

Proof step 6 from proof add-preserves-reachable-rev (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, (r ' + (r0 + x0)).last ≠ ce.nxt, ce.nxt ≠ null, path(r ' + (r0 + x0), h[r1 := ce]), reachable(r, r1, h), (r ' + (r0 + x0)).last ∈ h[r1 := ce], r1 ∈ h, ce.nxt ∈ h, ¬ reachable(r, ce.nxt, h), ¬ reachable(r, (r ' + (r0 + x0)).last, h), ∀ h, r1, ce, r. ¬ ( h[r1].nxt = h[ce.nxt].nxt ∧ (r ' + x0).last ≠ ce.nxt ∧ ce.nxt ≠ null ∧ path(r ' + x0, h[r1 := ce]) ∧ reachable(r, r1, h) ∧ ce.nxt ∈ h ∧ r1 ∈ h ∧ (r ' + x0).last ∈ h[r1 := ce] ∧ ¬ reachable(r, ce.nxt, h) ∧ ¬ reachable(r, (r ' + x0).last, 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; list('a) freely generated by [], +;; ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a '.last = a; (s) ⊦ a ∈ st[b := d] ↔ a = b ∨ a ∈ st; (ws)

Proof step 7 from proof add-preserves-reachable-rev (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, ((h[r1 := ce])[r].nxt ' + x0).last ≠ ce.nxt, ce.nxt ≠ null, r ≠ null, path((h[r1 := ce])[r].nxt ' + x0, h[r1 := ce]), reachable(r, r1, h), r ∈ h[r1 := ce], ((h[r1 := ce])[r].nxt ' + x0).last ∈ h[r1 := ce], r1 ∈ h, ce.nxt ∈ h, ¬ reachable(r, ((h[r1 := ce])[r].nxt ' + x0).last, h), ¬ reachable(r, ce.nxt, h), ∀ h0, r2, ce0, r3. h0[r2].nxt ≠ h0[ce0.nxt].nxt ∨ (r3 ' + x0).last = ce0.nxt ∨ ce0.nxt = null ∨ ¬ path(r3 ' + x0, h0[r2 := ce0]) ∨ ¬ reachable(r3, r2, h0) ∨ ¬ ce0.nxt ∈ h0 ∨ ¬ r2 ∈ h0 ∨ ¬ (r3 ' + x0).last ∈ h0[r2 := ce0] ∨ reachable(r3, ce0.nxt, h0) ∨ reachable(r3, (r3 ' + x0).last, 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 polylist ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) ⊦ a + x = a ' + x; (s)

Proof step 8 from proof add-preserves-reachable-rev (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, ((h[r1 := ce])[r].nxt ' + x0).last ≠ ce.nxt, ce.nxt ≠ null, r ≠ null, path((h[r1 := ce])[r].nxt ' + x0, h[r1 := ce]), reachable(r, r1, h), ce.nxt ∈ h, r1 ∈ h, ((h[r1 := ce])[r].nxt ' + x0).last ∈ h[r1 := ce], r ∈ h[r1 := ce], ¬ reachable(r, ((h[r1 := ce])[r].nxt ' + x0).last, h), ¬ reachable(r, ce.nxt, h), h[r1].nxt ≠ h[ce.nxt].nxt ∨ ((h[r1 := ce])[r].nxt ' + x0).last = ce.nxt ∨ ce.nxt = null ∨ ¬ path((h[r1 := ce])[r].nxt ' + x0, h[r1 := ce]) ∨ ¬ reachable((h[r1 := ce])[r].nxt, r1, h) ∨ ¬ ce.nxt ∈ h ∨ ¬ r1 ∈ h ∨ ¬ ((h[r1 := ce])[r].nxt ' + x0).last ∈ h[r1 := ce] ∨ reachable((h[r1 := ce])[r].nxt, ce.nxt, h) ∨ reachable((h[r1 := ce])[r].nxt, ((h[r1 := ce])[r].nxt ' + x0).last, h) ⊦

Rule argument

Additional information

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

Proof step 9 from proof add-preserves-reachable-rev (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, ((h[r1 := ce])[r].nxt ' + x0).last ≠ ce.nxt, ce.nxt ≠ null, r ≠ null, path((h[r1 := ce])[r].nxt ' + x0, h[r1 := ce]), reachable(r, r1, h), r ∈ h[r1 := ce], ((h[r1 := ce])[r].nxt ' + x0).last ∈ h[r1 := ce], r1 ∈ h, ce.nxt ∈ h, ¬ reachable(r, ((h[r1 := ce])[r].nxt ' + x0).last, h), ¬ reachable(r, ce.nxt, h), ¬ reachable((h[r1 := ce])[r].nxt, r1, h) ∨ reachable((h[r1 := ce])[r].nxt, ce.nxt, h) ∨ reachable((h[r1 := ce])[r].nxt, ((h[r1 := ce])[r].nxt ' + x0).last, h) ⊦

Rule argument

Additional information

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 11:

  ⊦ a ≠ b → (st[b := d])[a] = st[a]

• Proof step 12:

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

• Proof step 59:

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

Rule argument

Additional information

Used terms: r, ce, r1, h

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

A spec-lemma

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 13:

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

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)

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

User applied rule cut formula

And obtained the following new goals

• Proof step 14:

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

• Proof step 43:

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

Rule argument

Additional information

Used formulas: ∃ r. reachable(r1, h[r].nxt, h) ∧ path(r ' + x0, h[r1 := ce]) ∧ ¬ reachable(r1, (r ' + x0).last, h)

Proof step 14 from proof add-preserves-reachable-rev (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 ' + x0).last ≠ ce.nxt, ce.nxt ≠ null, r1 ≠ null, path(r ' + x0, h[r1 := ce]), path(ce.nxt ' + x0, h[r1 := ce]), reachable(r1, h[r].nxt, h), (ce.nxt ' + x0).last ∈ h[r1 := ce], r1 ∈ h, ce.nxt ∈ h, h[r].nxt ∈ h, ¬ reachable(r1, ce.nxt, h), ¬ reachable(r1, (ce.nxt ' + x0).last, h), ¬ reachable(r1, (r ' + x0).last, h) ⊦

Rule argument

Additional information

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

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

User applied rule weakening

And obtained the following new goals

• Proof step 16:

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

Rule argument

Additional information

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

User applied rule structural induction

And obtained the following new goals

• Proof step 17:

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

Rule argument

Additional information

Used simplifier rules: list('a) freely generated by [], +;; ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a '.last = a; (s)

Proof step 17 from proof add-preserves-reachable-rev (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, ((h[r1 := ce])[r].nxt ' + x).last ≠ ce.nxt, ce.nxt ≠ null, r ≠ null, r1 ≠ null, path((h[r1 := ce])[r].nxt ' + x, h[r1 := ce]), reachable(r1, h[r].nxt, h), r ∈ h[r1 := ce], r1 ∈ h, ce.nxt ∈ h, ¬ reachable(r1, ((h[r1 := ce])[r].nxt ' + x).last, h), ¬ reachable(r1, ce.nxt, h), ∀ h0, r2, ce0, r3. h0[r2].nxt ≠ h0[ce0.nxt].nxt ∨ (ce0.nxt ' + x).last = ce0.nxt ∨ ce0.nxt = null ∨ r2 = null ∨ ¬ path(r3 ' + x, h0[r2 := ce0]) ∨ ¬ reachable(r2, h0[r3].nxt, h0) ∨ ¬ ce0.nxt ∈ h0 ∨ ¬ r2 ∈ h0 ∨ reachable(r2, ce0.nxt, h0) ∨ reachable(r2, (ce0.nxt ' + x).last, h0) ∨ reachable(r2, (r3 ' + x).last, 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 polylist ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) ⊦ a + x = a ' + x; (s)

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 19:

  ⊦ a ≠ b → (st[b := d])[a] = st[a]

• Proof step 20:

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

• Proof step 22:

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

Rule argument

Additional information

Used terms: r, ce, r1, h

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

A spec-lemma

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 21:

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

Rule argument

Additional information

Used simplifier rules: reach-alloc: ⊦ reachable(r, r1, h) → r1 ∈ h; ⊦ (st[a := d])[a] = d; (s) ⊦ a ∈ st[a := d]; (s)

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

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: reachnext: ⊦ reachable(r0, h[r0].nxt, h) ↔ r0 ≠ null ∧ r0 ∈ h ∧ h[r0].nxt ∈ h ∧ h[r0].nxt ≠ null; Used terms: h, r1, ce, ce.nxt

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 23:

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

Rule argument

Additional information

Used simplifier rules: ⊦ a ≠ b → (a ∈ st[b := d] ↔ a ∈ st); (s)

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

Heuristic weak cut applied rule cut formula

And obtained the following new goals

• Proof step 24:

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

Rule argument

Additional information

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 25:

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

Rule argument

Additional information

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

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

User applied rule all left

And obtained the following new goals

• Proof step 26:

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

Rule argument

Additional information

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 27:

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

Rule argument

Additional information

Used simplifier rules: reach-reachnext: ⊦ reachable(r, r1, h) → (reachable(r, h[r1].nxt, h) ↔ ¬(h[r1].nxt = null ∨ ¬ h[r1].nxt ∈ h));

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

User applied rule constructor cut

And obtained the following new goals

• Proof step 28:

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

Rule argument

Additional information

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 29:

  h[r1].nxt = h[ce.nxt].nxt, (h[r1 := ce])[h[r].nxt].nxt ' + x0 ≠ [], ((h[r1 := ce])[h[r].nxt].nxt ' + x0).last ≠ ce.nxt, h[r].nxt ≠ null, ce.nxt ≠ null, r ≠ null, r ≠ r1, r1 ≠ null, path((h[r1 := ce])[h[r].nxt].nxt ' + x0, h[r1 := ce]), reachable(r1, h[r].nxt, h), h[r].nxt ∈ h[r1 := ce], r ∈ h, ce.nxt ∈ h, r1 ∈ h, ¬ reachable(r1, ((h[r1 := ce])[h[r].nxt].nxt ' + x0).last, h), ¬ reachable(r1, ce.nxt, h) ⊦ h[h[r].nxt].nxt ≠ null ∧ h[h[r].nxt].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 + x = a ' + x; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 30:

  ⊦ a ≠ b → (st[b := d])[a] = st[a]

• Proof step 31:

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

• Proof step 41:

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

Rule argument

Additional information

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

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

A spec-lemma

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 32:

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

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)

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

User applied rule constructor cut

And obtained the following new goals

• Proof step 33:

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

Rule argument

Additional information

Used simplifier rules: list('a) freely generated by [], +;; ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a '.last = a; (s)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 34:

  h[h[r].nxt].nxt = h[ce.nxt].nxt, ce.nxt ' + (h[h[r].nxt := ce])[ce.nxt].nxt ' + x ≠ [], ((h[h[r].nxt := ce])[ce.nxt].nxt ' + x).last ≠ ce.nxt, h[r].nxt ≠ null, ce.nxt ≠ null, r ≠ h[r].nxt, r ≠ null, path((h[h[r].nxt := ce])[ce.nxt].nxt ' + x, h[h[r].nxt := ce]), ce.nxt ∈ h[h[r].nxt := ce], h[r].nxt ∈ h, ce.nxt ∈ h, r ∈ h, ¬ reachable(h[r].nxt, ((h[h[r].nxt := ce])[ce.nxt].nxt ' + x).last, h), ¬ reachable(h[r].nxt, ce.nxt, h) ⊦ h[ce.nxt].nxt ≠ null ∧ h[ce.nxt].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: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) ⊦ a + x = a ' + x; (s)

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

User applied rule constructor cut

And obtained the following new goals

• Proof step 35:

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

• Proof step 38:

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

Rule argument

Additional information

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 36:

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

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)

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 37:

  ⊦ a ≠ b → (st[b := d])[a] = st[a]

Rule argument

Additional information

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

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

A spec-lemma

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 39:

  h[h[r].nxt].nxt = h[ce.nxt].nxt, ce.nxt ' + (h[h[r].nxt := ce])[ce.nxt].nxt ' + (h[h[r].nxt := ce])[(h[h[r].nxt := ce])[ce.nxt].nxt].nxt ' + x1 ≠ [], ((h[h[r].nxt := ce])[(h[h[r].nxt := ce])[ce.nxt].nxt].nxt ' + x1).last ≠ ce.nxt, (h[h[r].nxt := ce])[ce.nxt].nxt ≠ null, ce.nxt ≠ null, h[r].nxt ≠ null, r ≠ null, r ≠ h[r].nxt, path((h[h[r].nxt := ce])[(h[h[r].nxt := ce])[ce.nxt].nxt].nxt ' + x1, h[h[r].nxt := ce]), (h[h[r].nxt := ce])[ce.nxt].nxt ∈ h[h[r].nxt := ce], r ∈ h, ce.nxt ∈ h, h[r].nxt ∈ h, ce.nxt ∈ h[h[r].nxt := ce], ¬ reachable(h[r].nxt, ((h[h[r].nxt := ce])[(h[h[r].nxt := ce])[ce.nxt].nxt].nxt ' + x1).last, h), ¬ reachable(h[r].nxt, ce.nxt, h) ⊦ h[ce.nxt].nxt ≠ null ∧ h[ce.nxt].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: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) ⊦ a + x = a ' + x; (s)

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 40:

  ⊦ 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; ⊦ a ∈ st[b := d] ↔ a = b ∨ a ∈ st; (ws) ⊦ a = b → (st[b := d])[a] = d; (s) ⊦ a ∈ st[a := d]; (s) Used terms: ce.nxt, ce, h[r].nxt, h

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

A spec-lemma

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 42:

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

Rule argument

Additional information

Used simplifier rules: ⊦ a ∈ st[b := d] ↔ a = b ∨ a ∈ st; (ws)

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

User applied rule constructor cut

And closed the goal!

Rule argument

Additional information

Used simplifier rules: path-one: ⊦ path(r ', h) ↔ r ∈ h ∧ r ≠ null; reach-reachnext: ⊦ reachable(r, r1, h) → (reachable(r, h[r1].nxt, h) ↔ ¬(h[r1].nxt = null ∨ ¬ h[r1].nxt ∈ h)); path-cons: ⊦ path(r ' + r0 ' + x, h) ↔ r ∈ h ∧ r ≠ null ∧ h[r].nxt = r0 ∧ path(r0 ' + x, h); list('a) freely generated by [], +;; ⊦ a + x = a ' + x; (s) ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a ∈ st[b := d] ↔ a = b ∨ a ∈ st; (ws) ⊦ a = b → (st[b := d])[a] = d; (s) ⊦ x + [] = x; (s) ⊦ a '.last = a; (s)

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

User applied rule exists right

And obtained the following new goals

• Proof step 44:

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

Rule argument

Additional information

Used terms: ce.nxt

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 45:

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

Rule argument

Additional information

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 46:

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

• Proof step 47:

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

Rule argument

Additional information

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

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

Lemma reachable-def

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

User applied rule all left

And obtained the following new goals

• Proof step 48:

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

Rule argument

Additional information

Used terms: h[r1].nxt '

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 49:

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

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

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

User applied rule constructor cut

And obtained the following new goals

• Proof step 50:

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

Rule argument

Additional information

Used simplifier rules: list('a) freely generated by [], +;; ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a '.last = a; (s)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 51:

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

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

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 52:

  ⊦ a ≠ b → (st[b := d])[a] = st[a]

• Proof step 53:

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

Rule argument

Additional information

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

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

A spec-lemma

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 54:

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

Rule argument

Additional information

Used simplifier rules: ⊦ a ∈ st[b := d] ↔ a = b ∨ a ∈ st; (ws)

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 55:

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

• Proof step 56:

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

Rule argument

Additional information

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

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

Lemma reachable-def

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

User applied rule all left

And obtained the following new goals

• Proof step 57:

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

Rule argument

Additional information

Used terms: h[r1].nxt '

Proof step 57 from proof add-preserves-reachable-rev (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, (h[r1].nxt ' + x).last ≠ ce.nxt, ce.nxt ≠ r1, ce.nxt ≠ null, r1 ≠ null, path(h[r1].nxt ' + x, h[r1 := ce]), ce.nxt ∈ h, (h[r1].nxt ' + x).last ∈ h[r1 := ce], r1 ∈ h, ¬ reachable(r1, ce.nxt, h) ⊦ h[r1].nxt ∈ h ∧ h[r1].nxt ≠ null

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: ⊦ (x + y) + z = x + y + z; from specification polylist

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

User applied rule constructor cut

And closed the goal!

Rule argument

Additional information

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

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

User applied rule case distinction

And obtained the following new goals

• Proof step 60:

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

• Proof step 65:

  h[r1].nxt = h[ce.nxt].nxt, (h[r].nxt ' + x0).last ≠ ce.nxt, ce.nxt ≠ null, r ≠ r1, r ≠ null, path(h[r].nxt ' + x0, h[r1 := ce]), reachable(r, r1, h), reachable(h[r].nxt, (h[r].nxt ' + x0).last, h), r ∈ h[r1 := ce], (h[r].nxt ' + x0).last ∈ h[r1 := ce], r1 ∈ h, ce.nxt ∈ h, ¬ reachable(r, (h[r].nxt ' + x0).last, h), ¬ reachable(r, 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); ⊦ a ≠ b → (a ∈ st[b := d] ↔ a ∈ st); (s)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 61:

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

Rule argument

Additional information

Used simplifier rules: reach-alloc: ⊦ reachable(r, r1, h) → r1 ∈ h; ⊦ a ≠ b → (a ∈ st[b := d] ↔ a ∈ st); (s)

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

Heuristic weak cut applied rule cut formula

And obtained the following new goals

• Proof step 62:

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

Rule argument

Additional information

Used simplifier rules: path-one: ⊦ path(r ', h) ↔ r ∈ h ∧ r ≠ null; reachnext: ⊦ reachable(r0, h[r0].nxt, h) ↔ r0 ≠ null ∧ r0 ∈ h ∧ h[r0].nxt ∈ h ∧ h[r0].nxt ≠ null; startnotinH-notreachable: ⊦ ¬ r ∈ h → (reachable(r, r0, h) ↔ false); ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a '.last = a; (s) Used formulas: x0 = []

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 63:

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

Rule argument

Additional information

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

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 64:

  ⊦ 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, r

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

Lemma fromnext-reachable-rev

Proof step 65 from proof add-preserves-reachable-rev (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 66:

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

Rule argument

Additional information

Used simplifier rules: reach-alloc: ⊦ reachable(r, r1, h) → r1 ∈ h; self-reachable: ⊦ reachable(r, r, h) ↔ r ≠ null ∧ r ∈ h; ⊦ a ∈ st[b := d] ↔ a = b ∨ a ∈ st; (ws) Used terms: h, (h[r].nxt ' + x0).last, r

Proof step 66 from proof add-preserves-reachable-rev (previous step) (back to tree)

Lemma fromnext-reachable-rev