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

System applied rule normalize

And obtained the following new goals

• Proof step 2:

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

Rule argument

Additional information

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

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

Rule argument

Additional information

Proof step 3 from proof reachable-modbehind (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:

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

Rule argument

Additional information

Used terms: h, r0, r

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

Lemma reachable-def

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 6:

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

Rule argument

Additional information

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 7:

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

• Proof step 8:

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

Rule argument

Additional information

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

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

Lemma reachable-def

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

User applied rule all left

And obtained the following new goals

• Proof step 9:

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

Rule argument

Additional information

Used terms: x

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 10:

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

Rule argument

Additional information

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

User applied rule structural induction

And obtained the following new goals

• Proof step 11:

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

Rule argument

Additional information

Used simplifier rules: path-one: ⊦ path(r ', h) ↔ r ∈ h ∧ r ≠ null; 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 11 from proof reachable-modbehind (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 12:

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

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

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 13:

  ⊦ a ≠ b → (st[b := d])[a] = st[a]

• Proof step 14:

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

• Proof step 19:

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

Rule argument

Additional information

Used terms: r, ce, r1, h

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

A spec-lemma

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 15:

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

Rule argument

Additional information

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

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 16:

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

• Proof step 17:

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

Rule argument

Additional information

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

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

Lemma reachable-def

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 18:

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

Rule argument

Additional information

Proof step 18 from proof reachable-modbehind (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 = a ' + x; (s) ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist Used terms: h[r1].nxt + x0

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 20:

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

Rule argument

Additional information

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

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

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

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