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

System applied rule normalize

And obtained the following new goals

• Proof step 2:

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

Rule argument

Additional information

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

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

Rule argument

Additional information

Proof step 3 from proof changenonnxt-reachable (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:

  ce.nxt = (h(r1)).nxt, ¬ 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 changenonnxt-reachable (previous step) (back to tree)

Lemma reachable-def

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 6:

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

Rule argument

Additional information

Proof step 6 from proof changenonnxt-reachable (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:

  ce.nxt = (h(r1)).nxt, path(r ' + x, 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 changenonnxt-reachable (previous step) (back to tree)

Lemma reachable-def

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

User applied rule all left

And obtained the following new goals

• Proof step 9:

  ce.nxt = (h(r1)).nxt, path(r ' + x, 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 changenonnxt-reachable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 10:

  ce.nxt = (h(r1)).nxt, path(r ' + x, h), ¬ path(r ' + x, h(r1 := ce)) ⊦

Rule argument

Additional information

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

User applied rule structural induction

And obtained the following new goals

• Proof step 11:

  ce.nxt = (h(r1)).nxt, path(r ' + (r0 + x0), h), ¬ path(r ' + (r0 + x0), h(r1 := ce)), ∀ ce, h, r1, r. ¬(ce.nxt = (h(r1)).nxt ∧ path(r ' + x0, h) ∧ ¬ path(r ' + x0, h(r1 := ce))) ⊦

Rule argument

Additional information

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 12:

  ce.nxt = (h(r1)).nxt, r ≠ null, path((h(r)).nxt ' + x0, h), r ∈ h, r ∈ h(r1 := ce), ∀ ce0, h0, r2, r3. ce0.nxt ≠ (h0(r2)).nxt ∨ ¬ path(r3 ' + x0, h0) ∨ path(r3 ' + x0, h0(r2 := ce0)) ⊦ 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 list[reflist] ⊦ a + x = a ' + x; (s)

Proof step 12 from proof changenonnxt-reachable (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:

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

• Proof step 16:

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

A spec-lemma

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 15:

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

Rule argument

Additional information

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

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

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ a ∈ st(a := d); (s) Used terms: ce, h, r1, ce.nxt

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 17:

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

Rule argument

Additional information

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

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ a ≠ b → (a ∈ st(b := d) ↔ a ∈ st); (s) Used terms: ce, h, r1, (h(r)).nxt