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