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

System applied rule normalize

And obtained the following new goals

• Proof step 2:

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

Rule argument

Additional information

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

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

Rule argument

Additional information

Proof step 3 from proof modified-stays-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:

  reachable(r, r0, h), ∀ x. ¬(path(r ' + x, h(r0 := ce)) ∧ (r ' + x).last = r0) ⊦

Rule argument

Additional information

Used terms: h(r0 := ce), r0, r

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

Lemma reachable-def

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 6:

  reachable(r, r0, h), ∀ x. ¬ path(r ' + x, h((r ' + x).last := ce)) ∨ (r ' + x).last ≠ r0 ⊦

Rule argument

Additional information

Proof step 6 from proof modified-stays-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:

  ∀ x. ¬ path(r ' + x, h((r ' + x).last := ce)) ∨ (r ' + x).last ≠ r0, ∃ x. path(r ' + x, h) ∧ (r ' + x).last = r0 ⊦

Rule argument

Additional information

Used terms: h, r0, r

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

Lemma reachable-def

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 9:

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

Rule argument

Additional information

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

User applied rule structural induction

And obtained the following new goals

• Proof step 10:

  path(r ' + [], h), ∀ x0. ¬ path(r ' + x0, h((r ' + x0).last := ce)) ∨ (r ' + x0).last ≠ (r ' + []).last ⊦

• Proof step 12:

  path(r ' + (r0 + x0), h), ∀ x1. ¬ path(r ' + x1, h((r ' + x1).last := ce)) ∨ (r ' + x1).last ≠ (r ' + (r0 + x0)).last, ∀ r, h, ce. ¬(path(r ' + x0, h) ∧ (∀ x1. ¬ path(r ' + x1, h((r ' + x1).last := ce)) ∨ (r ' + x1).last ≠ (r ' + x0).last)) ⊦

Rule argument

Additional information

Used simplifier rules: List[A] freely generated by [], +;

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 11:

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

Rule argument

Additional information

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

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

User applied rule all left

And closed the goal!

Rule argument

Additional information

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 13:

  r ≠ null, path((h(r)).nxt ' + x0, h), r ∈ h, ∀ x1. ¬ path(r ' + x1, h((r ' + x1).last := ce)) ∨ (r ' + x1).last ≠ ((h(r)).nxt ' + x0).last, ∀ r1, h0, ce0. ¬ path(r1 ' + x0, h0) ∨ ¬(∀ x1. ¬ path(r1 ' + x1, h0((r1 ' + x1).last := ce0)) ∨ (r1 ' + x1).last ≠ (r1 ' + x0).last) ⊦

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 13 from proof modified-stays-reachable (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 14:

  r ≠ null, path((h(r)).nxt ' + x0, h), r ∈ h, ∀ x1. ¬ path(r ' + x1, h((r ' + x1).last := ce)) ∨ (r ' + x1).last ≠ ((h(r)).nxt ' + x0).last, ¬ path((h(r)).nxt ' + x0, h) ∨ ¬(∀ x1. ¬ path((h(r)).nxt ' + x1, h(((h(r)).nxt ' + x1).last := ce)) ∨ ((h(r)).nxt ' + x1).last ≠ ((h(r)).nxt ' + x0).last) ⊦

Rule argument

Additional information

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 15:

  ((h(r)).nxt ' + x1).last = ((h(r)).nxt ' + x0).last, r ≠ null, path((h(r)).nxt ' + x0, h), path((h(r)).nxt ' + x1, h(((h(r)).nxt ' + x0).last := ce)), r ∈ h, ∀ x1. ¬ path(r ' + x1, h((r ' + x1).last := ce)) ∨ (r ' + x1).last ≠ ((h(r)).nxt ' + x0).last ⊦

Rule argument

Additional information

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

User applied rule all left

And obtained the following new goals

• Proof step 16:

  ((h(r)).nxt ' + x1).last = ((h(r)).nxt ' + x0).last, r ≠ null, path((h(r)).nxt ' + x1, h(((h(r)).nxt ' + x0).last := ce)), path((h(r)).nxt ' + x0, h), r ∈ h, ∀ x1. ¬ path(r ' + x1, h((r ' + x1).last := ce)) ∨ (r ' + x1).last ≠ ((h(r)).nxt ' + x0).last, ¬ path(r ' + ((h(r)).nxt + x1), h((r ' + ((h(r)).nxt + x1)).last := ce)) ∨ (r ' + ((h(r)).nxt + x1)).last ≠ ((h(r)).nxt ' + x0).last ⊦

Rule argument

Additional information

Used terms: (h(r)).nxt + x1

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 17:

  ((h(r)).nxt ' + x1).last = ((h(r)).nxt ' + x0).last, (h(((h(r)).nxt ' + x0).last := ce)(r)).nxt ≠ (h(r)).nxt, r ≠ null, path((h(r)).nxt ' + x1, h(((h(r)).nxt ' + x0).last := ce)), path((h(r)).nxt ' + x0, h), r ∈ h, r ∈ h(((h(r)).nxt ' + x0).last := ce), ∀ x1. ¬ path(r ' + x1, h((r ' + x1).last := ce)) ∨ (r ' + x1).last ≠ ((h(r)).nxt ' + x0).last ⊦

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 → st(a) = ∅(a); (s) ⊦ 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 17 from proof modified-stays-reachable (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: path-one: ⊦ path(r ', h) ↔ r ∈ h ∧ r ≠ null; ⊦ a ∈ st(a := d); (s) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification polylist ⊦ a '.last = a; (s) ⊦ a ≠ b → st(b := d)(a) = st(a); (s) Used terms: []