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

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  ce.nxt = null, ¬(reachable(r, r1, h) ↔ true), reachable(r, r1, h(r2 := ce)), r1 ∈ h, ¬ r2 ∈ h ⊦

Rule argument

Additional information

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  ce.nxt = null, reachable(r, r1, h(r2 := ce)), r1 ∈ h, ¬ reachable(r, r1, h), ¬ r2 ∈ h ⊦

Rule argument

Additional information

Proof step 3 from proof reachable-addnewtonull-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:

  ce.nxt = null, reachable(r, r1, h(r2 := ce)), r1 ∈ h, ¬ r2 ∈ h, ∀ x. ¬(path(r ' + x, h) ∧ (r ' + x).last = r1) ⊦

Rule argument

Additional information

Used terms: h, r1, r

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

Lemma reachable-def

Proof step 5 from proof reachable-addnewtonull-rev (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 = null, r1 ∈ h, ¬ r2 ∈ h, ∀ x. ¬(path(r ' + x, h) ∧ (r ' + x).last = r1), ∃ x. path(r ' + x, h(r2 := ce)) ∧ (r ' + x).last = r1 ⊦

Rule argument

Additional information

Used terms: h(r2 := ce), r1, r

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

Lemma reachable-def

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 8:

  ce.nxt = null, path(r ' + x, h(r2 := ce)), (r ' + x).last ∈ h, ¬ r2 ∈ h, ∀ x0. ¬ path(r ' + x0, h) ∨ (r ' + x0).last ≠ (r ' + x).last ⊦

Rule argument

Additional information

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

User applied rule all left

And obtained the following new goals

• Proof step 9:

  ce.nxt = null, path(r ' + x, h(r2 := ce)), (r ' + x).last ∈ h, ¬ r2 ∈ h, ¬ path(r ' + x, h) ∨ (r ' + x).last ≠ (r ' + x).last ⊦

Rule argument

Additional information

Used terms: x

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 10:

  ce.nxt = null, path(r ' + x, h(r2 := ce)), (r ' + x).last ∈ h, ¬ path(r ' + x, h), ¬ r2 ∈ h ⊦

Rule argument

Additional information

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

User applied rule structural induction

And obtained the following new goals

• Proof step 11:

  ce.nxt = null, path(r ' + (r0 + x0), h(r2 := ce)), (r ' + (r0 + x0)).last ∈ h, ¬ path(r ' + (r0 + x0), h), ¬ r2 ∈ h, ∀ ce, r, h, r2. ¬(ce.nxt = null ∧ path(r ' + x0, h(r2 := ce)) ∧ (r ' + x0).last ∈ h ∧ ¬ path(r ' + x0, h) ∧ ¬ r2 ∈ h) ⊦

Rule argument

Additional information

Used simplifier rules: path-one: ⊦ path(r ', h) ↔ r ∈ h ∧ r ≠ null; List[A] freely generated by [], +; ⊦ a '.last = a; (s) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 12:

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

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 ∈ st ∧ ¬ a ∈ st → st(a := d)(b) = st(b); (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a + x = a ' + x; (s) ⊦ (b ' + a ' + x).last = (a ' + x).last; (s)

Proof step 12 from proof reachable-addnewtonull-rev (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 = null, r ≠ r2, r ≠ null, path((h(r)).nxt ' + x0, h(r2 := ce)), ((h(r)).nxt ' + x0).last ∈ h, r ∈ h(r2 := ce), ¬ r2 ∈ h, ∀ ce0, r1, h0, r3. ce0.nxt ≠ null ∨ ¬ path(r1 ' + x0, h0(r3 := ce0)) ∨ ¬ (r1 ' + x0).last ∈ h0 ∨ path(r1 ' + x0, h0) ∨ r3 ∈ h0 ⊦ r ∈ h ∧ path((h(r)).nxt ' + x0, h)

Rule argument

Additional information

Used simplifier rules: null-not-start-of-path: ⊦ ¬ path(null ' + x, h); ⊦ a ∈ st(b := d) ↔ a = b ∨ a ∈ st; (ws) ⊦ st(a := d)(a) = d; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist Used terms: r, ce, r2, h

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

A spec-lemma

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 15:

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

Rule argument

Additional information

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

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

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: ce, (h(r)).nxt, h, r2