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

System applied rule normalize

And obtained the following new goals

• Proof step 2:

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

Rule argument

Additional information

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

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

Rule argument

Additional information

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

User applied rule case distinction

And obtained the following new goals

• Proof step 4:

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

Rule argument

Additional information

Used simplifier rules: rem-reachable: ⊦ reachable(r, r0, h) ∧ ce.nxt = (h((h(r1)).nxt)).nxt ∧ r0 ≠ (h(r1)).nxt ∧ (h(r1)).nxt ∈ h ∧ (h(r1)).nxt ≠ null ∧ r1 ≠ null ∧ r1 ∈ h → (reachable(r, r0, h(r1 := ce)) ↔ true);

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 5:

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

• Proof step 6:

  ce.nxt = (h((h(r1)).nxt)).nxt, (h(r1)).nxt ≠ null, r0 ≠ (h(r1)).nxt, r1 ≠ null, r1 ∈ h, (h(r1)).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 5 from proof rem-reachable-rev (previous step) (back to tree)

Lemma reachable-def

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 7:

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

Rule argument

Additional information

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

User applied rule structural induction

And obtained the following new goals

• Proof step 8:

  ce.nxt = (h((h(r1)).nxt)).nxt, (r ' + (r0 + x0)).last ≠ (h(r1)).nxt, (h(r1)).nxt ≠ null, r1 ≠ null, path(r ' + (r0 + x0), h(r1 := ce)), r1 ∈ h, (h(r1)).nxt ∈ h, ¬ reachable(r, (r ' + (r0 + x0)).last, h), ∀ ce, h, r1, r. ¬ ( ce.nxt = (h((h(r1)).nxt)).nxt ∧ (r ' + x0).last ≠ (h(r1)).nxt ∧ (h(r1)).nxt ≠ null ∧ r1 ≠ null ∧ path(r ' + x0, h(r1 := ce)) ∧ (h(r1)).nxt ∈ h ∧ r1 ∈ 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 8 from proof rem-reachable-rev (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 9:

  ce.nxt = (h((h(r1)).nxt)).nxt, ((h(r1 := ce)(r)).nxt ' + x0).last ≠ (h(r1)).nxt, (h(r1)).nxt ≠ null, r ≠ null, r1 ≠ null, path((h(r1 := ce)(r)).nxt ' + x0, h(r1 := ce)), r ∈ h(r1 := ce), r1 ∈ h, (h(r1)).nxt ∈ h, ¬ reachable(r, ((h(r1 := ce)(r)).nxt ' + x0).last, h), ∀ ce0, h0, r2, r3. ce0.nxt ≠ (h0((h0(r2)).nxt)).nxt ∨ (r3 ' + x0).last = (h0(r2)).nxt ∨ (h0(r2)).nxt = null ∨ r2 = null ∨ ¬ path(r3 ' + x0, h0(r2 := ce0)) ∨ ¬ (h0(r2)).nxt ∈ h0 ∨ ¬ r2 ∈ 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 9 from proof rem-reachable-rev (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 10:

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

Rule argument

Additional information

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 11:

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

Rule argument

Additional information

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 12:

  ⊦ a ≠ b → st(b := d)(a) = st(a)

• Proof step 13:

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

• Proof step 21:

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

Rule argument

Additional information

Used terms: r, ce, r1, h

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

A spec-lemma

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 14:

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

Rule argument

Additional information

Used simplifier rules: ⊦ st(a := d)(a) = d; (s) ⊦ a ∈ st(a := d); (s)

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 15:

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

• Proof step 16:

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

Rule argument

Additional information

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

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

Lemma reachable-def

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 17:

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

Rule argument

Additional information

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 18:

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

• Proof step 19:

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

Rule argument

Additional information

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

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

Lemma reachable-def

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 20:

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

Rule argument

Additional information

Proof step 20 from proof rem-reachable-rev (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 ' + x + b ' + y).last = (b ' + y).last; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist Used terms: (h(r1)).nxt ' + (h((h(r1)).nxt)).nxt ' + x

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 22:

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

Rule argument

Additional information

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

Proof step 22 from proof rem-reachable-rev (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:

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

Rule argument

Additional information

Used terms: h, ((h(r)).nxt ' + x0).last, (h(r)).nxt

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

Lemma reachable-def

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 25:

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

Rule argument

Additional information

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 26:

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

• Proof step 27:

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

Rule argument

Additional information

Used terms: h, ((h(r)).nxt ' + x0).last, r

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

Lemma reachable-def

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 28:

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

Rule argument

Additional information

Proof step 28 from proof rem-reachable-rev (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 polylist Used terms: (h(r)).nxt ' + x