Proof step 1 from proof compact-local (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  !(me, hO, hP0) * !(me, hO, hP1) ≠ !(me, hO, hP0 * hP1) ⊦

Rule argument

Additional information

Proof step 2 from proof compact-local (previous step) (back to tree)

User applied rule expand left

And obtained the following new goals

• Proof step 3:

  ¬(((!(me, hO, hP0) * !(me, hO, hP1)))(h) ↔ !(me, hO, hP0 * hP1)(h)) ⊦

Rule argument

Additional information

Proof step 3 from proof compact-local (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 4:

  ((!(me, hO, hP0) * !(me, hO, hP1)))(h), ¬ !(me, hO, hP0 * hP1)(h) ⊦

• Proof step 12:

  !(me, hO, hP0 * hP1)(h), ¬ ((!(me, hO, hP0) * !(me, hO, hP1)))(h) ⊦

Rule argument

Additional information

Proof step 4 from proof compact-local (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 5:

  ⊦ ((hP0 * hP1))(h) ↔ (∃ h0, h1. h0 ⊥ h1 ∧ h = h0 ∪ h1 ∧ hP0(h0) ∧ hP1(h1))

• Proof step 6:

  ¬ !(me, hO, hP0 * hP1)(h), ∃ h0, h1. h0 ⊥ h1 ∧ h = h0 ∪ h1 ∧ !(me, hO, hP0)(h0) ∧ !(me, hO, hP1)(h1) ⊦

Rule argument

Additional information

Used terms: h, !(me, hO, hP1), !(me, hO, hP0)

Proof step 5 from proof compact-local (previous step) (back to tree)

A spec-lemma

Proof step 6 from proof compact-local (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 7:

  !(me, hO, hP0)(h0), !(me, hO, hP1)(h1), h0 ⊥ h1, ¬ !(me, hO, hP0 * hP1)(h0 ∪ h1) ⊦

Rule argument

Additional information

Proof step 7 from proof compact-local (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 8:

  ⊦ !(id, of, hP)(h) ↔ hP(h) ∧ (∀ r. r ∈ h → of(r) = id)

• Proof step 9:

  !(me, hO, hP1)(h1), !(me, hO, hP0)(h0), h0 ⊥ h1, ¬ ((hP0 * hP1))(h0 ∪ h1) ∨ (∃ r. ¬(r ∈ h0 ∪ h1 → hO(r) = me)) ⊦

Rule argument

Additional information

Used terms: h0 ∪ h1, hP0 * hP1, hO, me

Proof step 8 from proof compact-local (previous step) (back to tree)

Lemma owned

Proof step 9 from proof compact-local (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 10:

  !(me, hO, hP0)(h0), !(me, hO, hP1)(h1), h0 ⊥ h1, ¬ ((hP0 * hP1))(h0 ∪ h1) ⊦

Rule argument

Additional information

Used simplifier rules: owned-r: ⊦ !(me, hO, hP)(h) ∧ hO(r) ≠ me → ¬ r ∈ h; ⊦ h0 ⊥ h1 → (r ∈ h0 ∪ h1 ↔ r ∈ h0 ∨ r ∈ h1); (s)

Proof step 10 from proof compact-local (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 11:

  ⊦ hP0(h0) ∧ hP1(h1) ∧ h0 ⊥ h1 → ((hP0 * hP1))(h0 ∪ h1)

Rule argument

Additional information

Used simplifier rules: owned-P: ⊦ !(me, hO, hP)(h) ∧ ¬ hP(h) → false; Used terms: h1, h0, hP1, hP0

Proof step 11 from proof compact-local (previous step) (back to tree)

A spec-lemma

Proof step 12 from proof compact-local (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 13:

  ⊦ !(id, of, hP)(h) ↔ hP(h) ∧ (∀ r. r ∈ h → of(r) = id)

• Proof step 14:

  ((hP0 * hP1))(h), ¬ ((!(me, hO, hP0) * !(me, hO, hP1)))(h), ∀ r. r ∈ h → hO(r) = me ⊦

Rule argument

Additional information

Used terms: h, hP0 * hP1, hO, me

Proof step 13 from proof compact-local (previous step) (back to tree)

Lemma owned

Proof step 14 from proof compact-local (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 15:

  ((hP0 * hP1))(h), ¬ ((!(me, hO, hP0) * !(me, hO, hP1)))(h), ∀ r. r ∈ h → hO(r) = me ⊦

Rule argument

Additional information

Proof step 15 from proof compact-local (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 16:

  ⊦ ((hP0 * hP1))(h) ↔ (∃ h0, h1. h0 ⊥ h1 ∧ h = h0 ∪ h1 ∧ hP0(h0) ∧ hP1(h1))

• Proof step 17:

  ¬ ((!(me, hO, hP0) * !(me, hO, hP1)))(h), ∀ r. r ∈ h → hO(r) = me, ∃ h0, h1. h0 ⊥ h1 ∧ h = h0 ∪ h1 ∧ hP0(h0) ∧ hP1(h1) ⊦

Rule argument

Additional information

Used terms: h, hP1, hP0

Proof step 16 from proof compact-local (previous step) (back to tree)

A spec-lemma

Proof step 17 from proof compact-local (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 18:

  hP0(h0), hP1(h1), h0 ⊥ h1, ¬ ((!(me, hO, hP0) * !(me, hO, hP1)))(h0 ∪ h1), ∀ r. r ∈ h0 ∨ r ∈ h1 → hO(r) = me ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ h0 ⊥ h1 → (r ∈ h0 ∪ h1 ↔ r ∈ h0 ∨ r ∈ h1); (s)

Proof step 18 from proof compact-local (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 19:

  ⊦ hP0(h0) ∧ hP1(h1) ∧ h0 ⊥ h1 → ((hP0 * hP1))(h0 ∪ h1)

• Proof step 20:

  hP0(h0), hP1(h1), h0 ⊥ h1, ¬ ((!(me, hO, hP0) * !(me, hO, hP1)))(h0 ∪ h1), ∀ r. r ∈ h0 ∨ r ∈ h1 → hO(r) = me ⊦ !(me, hO, hP0)(h0) ∧ !(me, hO, hP1)(h1) ∧ h0 ⊥ h1

Rule argument

Additional information

Used terms: h1, h0, !(me, hO, hP1), !(me, hO, hP0)

Proof step 19 from proof compact-local (previous step) (back to tree)

A spec-lemma

Proof step 20 from proof compact-local (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 21:

  hP0(h0), hP1(h1), h0 ⊥ h1, ¬ ((!(me, hO, hP0) * !(me, hO, hP1)))(h0 ∪ h1), ∀ r. r ∈ h0 ∨ r ∈ h1 → hO(r) = me ⊦ !(me, hO, hP0)(h0) ∧ !(me, hO, hP1)(h1)

Rule argument

Additional information

Proof step 21 from proof compact-local (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 22:

  hP0(h0), hP1(h1), h0 ⊥ h1, ¬ !(me, hO, hP0)(h0), ¬ ((!(me, hO, hP0) * !(me, hO, hP1)))(h0 ∪ h1), ∀ r. r ∈ h0 ∨ r ∈ h1 → hO(r) = me ⊦

• Proof step 26:

  hP0(h0), hP1(h1), h0 ⊥ h1, ¬ !(me, hO, hP1)(h1), ¬ ((!(me, hO, hP0) * !(me, hO, hP1)))(h0 ∪ h1), ∀ r. r ∈ h0 ∨ r ∈ h1 → hO(r) = me ⊦

Rule argument

Additional information

Proof step 22 from proof compact-local (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 23:

  ⊦ !(id, of, hP)(h) ↔ hP(h) ∧ (∀ r. r ∈ h → of(r) = id)

• Proof step 24:

  hP0(h0), hP1(h1), h0 ⊥ h1, ¬ ((!(me, hO, hP0) * !(me, hO, hP1)))(h0 ∪ h1), ∀ r. r ∈ h0 ∨ r ∈ h1 → hO(r) = me, ¬ hP0(h0) ∨ (∃ r. ¬(r ∈ h0 → hO(r) = me)) ⊦

Rule argument

Additional information

Used terms: h0, hP0, hO, me

Proof step 23 from proof compact-local (previous step) (back to tree)

Lemma owned

Proof step 24 from proof compact-local (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 25:

  hO(r) ≠ me, hP0(h0), hP1(h1), r ∈ h0, h0 ⊥ h1, ¬ ((!(me, hO, hP0) * !(me, hO, hP1)))(h0 ∪ h1), ∀ r. r ∈ h0 ∨ r ∈ h1 → hO(r) = me ⊦

Rule argument

Additional information

Proof step 25 from proof compact-local (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: r

Proof step 26 from proof compact-local (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 27:

  ⊦ !(id, of, hP)(h) ↔ hP(h) ∧ (∀ r. r ∈ h → of(r) = id)

• Proof step 28:

  hP0(h0), hP1(h1), h0 ⊥ h1, ¬ ((!(me, hO, hP0) * !(me, hO, hP1)))(h0 ∪ h1), ∀ r. r ∈ h0 ∨ r ∈ h1 → hO(r) = me, ¬ hP1(h1) ∨ (∃ r. ¬(r ∈ h1 → hO(r) = me)) ⊦

Rule argument

Additional information

Used terms: h1, hP1, hO, me

Proof step 27 from proof compact-local (previous step) (back to tree)

Lemma owned

Proof step 28 from proof compact-local (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 29:

  hO(r) ≠ me, hP0(h0), hP1(h1), r ∈ h1, h0 ⊥ h1, ¬ ((!(me, hO, hP0) * !(me, hO, hP1)))(h0 ∪ h1), ∀ r. r ∈ h0 ∨ r ∈ h1 → hO(r) = me ⊦

Rule argument

Additional information

Proof step 29 from proof compact-local (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ r ∈ h1 ∧ h0 ⊥ h1 → (r ∈ h0 ↔ false); (s) Used terms: r