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

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  ¬(reachable(r, r1, H(r0 := ce)) ↔ false), r0 ≠ null, r0 ∈ H, ¬ reachable(r, r1, H), ¬ reachable(r, r0, H) ⊦

Rule argument

Additional information

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  r0 ≠ null, reachable(r, r1, H(r0 := ce)), r0 ∈ H, ¬ reachable(r, r1, H), ¬ reachable(r, r0, H) ⊦

Rule argument

Additional information

Proof step 3 from proof modnotreachable-reachable-rev-weak (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:

  r0 ≠ null, r0 ∈ H, ¬ reachable(r, r1, H), ¬ reachable(r, r0, H), ∃ x. path(r ' + x, H(r0 := ce)) ∧ (r ' + x).last = r1 ⊦

Rule argument

Additional information

Used terms: H(r0 := ce), r1, r

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

Lemma reachable-def

Proof step 5 from proof modnotreachable-reachable-rev-weak (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 6:

  r0 ≠ null, path(r ' + x, H(r0 := ce)), r0 ∈ H, ¬ reachable(r, (r ' + x).last, H), ¬ reachable(r, r0, H) ⊦

Rule argument

Additional information

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

User applied rule structural induction

And obtained the following new goals

• Proof step 7:

  r0 ≠ null, path(r ' + (r1 + x0), H(r0 := ce)), r0 ∈ H, ¬ reachable(r, (r ' + (r1 + x0)).last, H), ¬ reachable(r, r0, H), ∀ r0, r, H, ce. ¬(r0 ≠ null ∧ path(r ' + x0, H(r0 := ce)) ∧ r0 ∈ H ∧ ¬ reachable(r, (r ' + x0).last, H) ∧ ¬ reachable(r, r0, H)) ⊦

Rule argument

Additional information

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 8:

  r ≠ null, r0 ≠ null, path((H(r0 := ce)(r)).nxt ' + x0, H(r0 := ce)), r ∈ H(r0 := ce), r0 ∈ H, ¬ reachable(r, ((H(r0 := ce)(r)).nxt ' + x0).last, H), ¬ reachable(r, r0, H), ∀ r2, r3, H0, ce0. r2 = null ∨ ¬ path(r3 ' + x0, H0(r2 := ce0)) ∨ ¬ r2 ∈ H0 ∨ reachable(r3, (r3 ' + x0).last, H0) ∨ reachable(r3, r2, H0) ⊦

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 list[reflist] ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) ⊦ a + x = a ' + x; (s)

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 9:

  ⊦ a ≠ b → st(b := d)(a) = st(a)

• Proof step 10:

  r ≠ r0, r ≠ null, r0 ≠ null, path((H(r)).nxt ' + x0, H(r0 := ce)), r0 ∈ H, r ∈ H(r0 := ce), ¬ reachable(r, ((H(r)).nxt ' + x0).last, H), ¬ reachable(r, r0, H), ∀ r2, r3, H0, ce0. r2 = null ∨ ¬ path(r3 ' + x0, H0(r2 := ce0)) ∨ ¬ r2 ∈ H0 ∨ reachable(r3, (r3 ' + x0).last, H0) ∨ reachable(r3, r2, H0) ⊦

Rule argument

Additional information

Used simplifier rules: self-reachable: ⊦ reachable(r, r, h) ↔ r ≠ null ∧ r ∈ h; ⊦ a ∈ st(a := d); (s) ⊦ st(a := d)(a) = d; (s) Used terms: r, ce, r0, H

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

A spec-lemma

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 11:

  r ≠ r0, r ≠ null, r0 ≠ null, path((H(r)).nxt ' + x0, H(r0 := ce)), r0 ∈ H, r ∈ H, ¬ reachable(r, ((H(r)).nxt ' + x0).last, H), ¬ reachable(r, r0, H), ∀ r2, r3, H0, ce0. r2 = null ∨ ¬ path(r3 ' + x0, H0(r2 := ce0)) ∨ ¬ r2 ∈ H0 ∨ reachable(r3, (r3 ' + x0).last, H0) ∨ reachable(r3, r2, H0) ⊦

Rule argument

Additional information

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

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

User applied rule all left

And obtained the following new goals

• Proof step 12:

  r ≠ r0, r ≠ null, r0 ≠ null, path((H(r)).nxt ' + x0, H(r0 := ce)), r ∈ H, r0 ∈ H, ¬ reachable(r, ((H(r)).nxt ' + x0).last, H), ¬ reachable(r, r0, H), r0 = null ∨ ¬ path((H(r)).nxt ' + x0, H(r0 := ce)) ∨ ¬ r0 ∈ H ∨ reachable((H(r)).nxt, ((H(r)).nxt ' + x0).last, H) ∨ reachable((H(r)).nxt, r0, H) ⊦

Rule argument

Additional information

Used terms: r0, (H(r)).nxt, H, ce

Proof step 12 from proof modnotreachable-reachable-rev-weak (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 13:

  r ≠ null, r ≠ r0, r0 ≠ null, path((H(r)).nxt ' + x0, H(r0 := ce)), reachable((H(r)).nxt, ((H(r)).nxt ' + x0).last, H), r ∈ H, r0 ∈ H, ¬ reachable(r, r0, H), ¬ reachable(r, ((H(r)).nxt ' + x0).last, H), ¬ reachable((H(r)).nxt, r0, H) ⊦

Rule argument

Additional information

Used simplifier rules: fromnext-reachable-rev: ⊦ reachable((h(r)).nxt, r0, h) ∧ r ≠ r0 ∧ r ≠ null ∧ r ∈ h → (reachable(r, r0, h) ↔ true);

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

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 14:

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

• Proof step 15:

  r ≠ null, r ≠ r0, r0 ≠ null, path((H(r)).nxt ' + x0, H(r0 := ce)), r0 ∈ H, r ∈ H, ¬ reachable(r, r0, H), ¬ reachable(r, ((H(r)).nxt ' + x0).last, H), ¬ reachable((H(r)).nxt, r0, H), ∃ x. path((H(r)).nxt ' + x, H) ∧ ((H(r)).nxt ' + x).last = ((H(r)).nxt ' + x0).last ⊦

Rule argument

Additional information

Used terms: H, ((H(r)).nxt ' + x0).last, (H(r)).nxt

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

Lemma reachable-def

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 16:

  ((H(r)).nxt ' + x).last = ((H(r)).nxt ' + x0).last, r ≠ null, r ≠ r0, r0 ≠ null, path((H(r)).nxt ' + x, H), path((H(r)).nxt ' + x0, H(r0 := ce)), r ∈ H, r0 ∈ H, ¬ reachable(r, r0, H), ¬ reachable(r, ((H(r)).nxt ' + x0).last, H), ¬ reachable((H(r)).nxt, r0, H) ⊦

Rule argument

Additional information

Proof step 16 from proof modnotreachable-reachable-rev-weak (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 17:

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

• Proof step 18:

  ((H(r)).nxt ' + x).last = ((H(r)).nxt ' + x0).last, r ≠ null, r ≠ r0, r0 ≠ null, path((H(r)).nxt ' + x0, H(r0 := ce)), path((H(r)).nxt ' + x, H), r0 ∈ H, r ∈ H, ¬ reachable(r, r0, H), ¬ reachable((H(r)).nxt, r0, H), ∀ x. ¬(path(r ' + x, H) ∧ (r ' + x).last = ((H(r)).nxt ' + x0).last) ⊦

Rule argument

Additional information

Used terms: H, ((H(r)).nxt ' + x0).last, r

Proof step 17 from proof modnotreachable-reachable-rev-weak (previous step) (back to tree)

Lemma reachable-def

Proof step 18 from proof modnotreachable-reachable-rev-weak (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); ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) a: ⊦ (x + y) + z = x + y + z; from specification list[reflist] Used terms: (H(r)).nxt ' + x