Proof step 1 from proof frommodified (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

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

Rule argument

Additional information

Proof step 2 from proof frommodified (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

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

Rule argument

Additional information

Proof step 3 from proof frommodified (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 4:

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

• Proof step 17:

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

Rule argument

Additional information

Proof step 4 from proof frommodified (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 5:

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

Rule argument

Additional information

Proof step 5 from proof frommodified (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)

• Proof step 7:

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

Rule argument

Additional information

Used terms: h(r := ce), r0, r

Proof step 6 from proof frommodified (previous step) (back to tree)

Lemma reachable-def

Proof step 7 from proof frommodified (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 8:

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

Rule argument

Additional information

Proof step 8 from proof frommodified (previous step) (back to tree)

User applied rule constructor cut

And obtained the following new goals

• Proof step 9:

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

Rule argument

Additional information

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

Proof step 9 from proof frommodified (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 10:

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

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) ⊦ st(a := d)(a) = d; (s) ⊦ a ∈ st(a := d); (s)

Proof step 10 from proof frommodified (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 11:

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

• Proof step 16:

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

Rule argument

Additional information

Used formulas: ce.nxt ≠ null ∧ ce.nxt ∈ h

Proof step 11 from proof frommodified (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 12:

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

Rule argument

Additional information

Proof step 12 from proof frommodified (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 13:

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

• Proof step 14:

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

Rule argument

Additional information

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

Proof step 13 from proof frommodified (previous step) (back to tree)

Lemma reachable-def

Proof step 14 from proof frommodified (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 15:

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

Rule argument

Additional information

Proof step 15 from proof frommodified (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 16 from proof frommodified (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); Reflist freely generated by [], +; ⊦ a ≠ b → (a ∈ st(b := d) ↔ a ∈ st); (s) ⊦ a ≠ b → st(b := d)(a) = st(a); (s) ⊦ a + x = a ' + x; (s) ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] ⊦ x + [] = x; (s) ⊦ a '.last = a; (s)

Proof step 17 from proof frommodified (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 18:

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

Rule argument

Additional information

Used simplifier rules: self-reachable: ⊦ reachable(r, r, h) ↔ r ≠ null ∧ r ∈ h; ⊦ a ∈ st(a := d); (s)

Proof step 18 from proof frommodified (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 19:

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

• Proof step 20:

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

Rule argument

Additional information

Used terms: h(r := ce), r0, ce.nxt

Proof step 19 from proof frommodified (previous step) (back to tree)

Lemma reachable-def

Proof step 20 from proof frommodified (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 21:

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

Rule argument

Additional information

Proof step 21 from proof frommodified (previous step) (back to tree)

Heuristic weak cut applied rule cut formula

And obtained the following new goals

• Proof step 22:

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

• Proof step 27:

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

Rule argument

Additional information

Used formulas: x = []

Proof step 22 from proof frommodified (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 23:

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

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 list[reflist] ⊦ x + [] = x; (s) ⊦ a ≠ b → (a ∈ st(b := d) ↔ a ∈ st); (s)

Proof step 23 from proof frommodified (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:

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

Rule argument

Additional information

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

Proof step 24 from proof frommodified (previous step) (back to tree)

Lemma reachable-def

Proof step 25 from proof frommodified (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 26:

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

Rule argument

Additional information

Proof step 26 from proof frommodified (previous step) (back to tree)

User applied rule all left

And closed the goal!

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

Proof step 27 from proof frommodified (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 28:

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

Rule argument

Additional information

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

Proof step 28 from proof frommodified (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 29:

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

• Proof step 30:

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

Rule argument

Additional information

Used terms: h(r := ce), x.last, r

Proof step 29 from proof frommodified (previous step) (back to tree)

Lemma reachable-def

Proof step 30 from proof frommodified (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 31:

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

Rule argument

Additional information

Proof step 31 from proof frommodified (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 ∈ st(a := d); (s) ⊦ st(a := d)(a) = d; (s) ⊦ a + x = a ' + x; (s) ⊦ y ≠ [] → (a ' + x + y).last = y.last; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] Used terms: ce.nxt + x