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

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  ¬(reachable(r, r1, h[r0 := ce]) ↔ false), r0 ∈ h, ¬ reachable(r, r1, h), ¬ reachable(r, r0, h) ⊦

Rule argument

Additional information

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  reachable(r, r1, h[r0 := ce]), r0 ∈ h, ¬ reachable(r, r1, h), ¬ reachable(r, r0, h) ⊦

Rule argument

Additional information

Proof step 3 from proof modnotreachable-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:

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

Rule argument

Additional information

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

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

Lemma reachable-def

Proof step 5 from proof modnotreachable-reachable-rev (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 6:

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

Rule argument

Additional information

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

User applied rule structural induction

And obtained the following new goals

• Proof step 7:

  path(r ' + (r1 + x0), h[r0 := ce]), r0 ∈ h, ¬ reachable(r, (r ' + (r1 + x0)).last, h), ¬ reachable(r, r0, h), ∀ r, h, r0, ce. ¬(path(r ' + x0, h[r0 := ce]) ∧ r0 ∈ h ∧ ¬ reachable(r, (r ' + x0).last, h) ∧ ¬ reachable(r, r0, 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 modnotreachable-reachable-rev (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 8:

  r ≠ null, path((h[r0 := ce])[r].nxt ' + x0, h[r0 := ce]), r ∈ h[r0 := ce], r0 ∈ h, ¬ reachable(r, ((h[r0 := ce])[r].nxt ' + x0).last, h), ¬ reachable(r, r0, h), ∀ r2, h0, r3, ce0. ¬ path(r2 ' + x0, h0[r3 := ce0]) ∨ ¬ r3 ∈ h0 ∨ reachable(r2, (r2 ' + x0).last, h0) ∨ reachable(r2, r3, 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 modnotreachable-reachable-rev (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 9:

  ⊦ a ≠ b → (st[b := d])[a] = st[a]

• Proof step 10:

  r ≠ r0, r ≠ null, path(h[r].nxt ' + x0, h[r0 := ce]), r0 ∈ h, r ∈ h[r0 := ce], ¬ reachable(r, (h[r].nxt ' + x0).last, h), ¬ reachable(r, r0, h), ∀ r2, h0, r3, ce0. ¬ path(r2 ' + x0, h0[r3 := ce0]) ∨ ¬ r3 ∈ h0 ∨ reachable(r2, (r2 ' + x0).last, h0) ∨ reachable(r2, r3, h0) ⊦

Rule argument

Additional information

Used simplifier rules: self-reachable: ⊦ reachable(r, r, h) ↔ r ≠ null ∧ r ∈ h; ⊦ a ∈ st[a := d]; (s) ⊦ (st[a := d])[a] = d; (s) Used terms: r, ce, r0, h

Proof step 9 from proof modnotreachable-reachable-rev (previous step) (back to tree)

A spec-lemma

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 11:

  r ≠ r0, r ≠ null, path(h[r].nxt ' + x0, h[r0 := ce]), r0 ∈ h, r ∈ h, ¬ reachable(r, (h[r].nxt ' + x0).last, h), ¬ reachable(r, r0, h), ∀ r2, h0, r3, ce0. ¬ path(r2 ' + x0, h0[r3 := ce0]) ∨ ¬ r3 ∈ h0 ∨ reachable(r2, (r2 ' + x0).last, h0) ∨ reachable(r2, r3, h0) ⊦

Rule argument

Additional information

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

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

User applied rule all left

And obtained the following new goals

• Proof step 12:

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

Rule argument

Additional information

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 13:

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

Rule argument

Additional information

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

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 14:

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

• Proof step 15:

  r ≠ null, r ≠ r0, path(h[r].nxt ' + x0, h[r0 := ce]), r0 ∈ h, r ∈ h, ¬ reachable(r, r0, h), ¬ reachable(r, (h[r].nxt ' + x0).last, h), ¬ reachable(h[r].nxt, r0, 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 14 from proof modnotreachable-reachable-rev (previous step) (back to tree)

Lemma reachable-def

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 16:

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

Rule argument

Additional information

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 17:

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

• Proof step 18:

  (h[r].nxt ' + x).last = (h[r].nxt ' + x0).last, r ≠ null, r ≠ r0, path(h[r].nxt ' + x0, h[r0 := ce]), path(h[r].nxt ' + x, h), r0 ∈ h, r ∈ h, ¬ reachable(r, r0, h), ¬ reachable(h[r].nxt, r0, 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 17 from proof modnotreachable-reachable-rev (previous step) (back to tree)

Lemma reachable-def

Proof step 18 from proof modnotreachable-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