Proof step 1 from proof stability (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  ((hP1 * {h} ))(h1), ((hP0 * {h} ))(h0), hP(h0), (hP0 -⊗ hP) * hP1 ⊆ hP, ¬ hP(h1) ⊦

Rule argument

Additional information

Proof step 2 from proof stability (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  ((hP0 * {h} ))(h0), ((hP1 * {h} ))(h1), hP(h0), ¬ hP(h1), ∀ h. (((hP0 -⊗ hP) * hP1))(h) → hP(h) ⊦

Rule argument

Additional information

Used simplifier rules: Imp: ⊦ hP0 ⊆ hP1 ↔ (∀ h. hP0(h) → hP1(h));

Proof step 3 from proof stability (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 4:

  ((hP1 * {h} ))(h1), ((hP0 * {h} ))(h0), hP(h0), ¬ hP(h1), (((hP0 -⊗ hP) * hP1))(h1) → hP(h1) ⊦

Rule argument

Additional information

Used terms: h1

Proof step 4 from proof stability (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 5:

  ((hP0 * {h} ))(h0), ((hP1 * {h} ))(h1), hP(h0), ¬ (((hP0 -⊗ hP) * hP1))(h1), ¬ hP(h1) ⊦

Rule argument

Additional information

Proof step 5 from proof stability (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 6:

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

• Proof step 7:

  ((hP0 * {h} ))(h0), hP(h0), ¬ (((hP0 -⊗ hP) * hP1))(h1), ¬ hP(h1), ∃ h0, h2. h0 ⊥ h2 ∧ h1 = h0 ∪ h2 ∧ hP1(h0) ∧ {h} (h2) ⊦

Rule argument

Additional information

Used terms: h1, {h} , hP1

Proof step 6 from proof stability (previous step) (back to tree)

Lemma sep

Proof step 7 from proof stability (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 8:

  ((hP0 * {h3} ))(h0), hP(h0), hP1(h2), h2 ⊥ h3, ¬ (((hP0 -⊗ hP) * hP1))(h2 ∪ h3), ¬ hP(h2 ∪ h3) ⊦

Rule argument

Additional information

Used simplifier rules: this: ⊦ {h} (h0) ↔ h = h0;

Proof step 8 from proof stability (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 9:

  ⊦ hP0 * hP1 = hP1 * hP0

• Proof step 10:

  ((hP0 * {h3} ))(h0), hP(h0), hP1(h2), h2 ⊥ h3, ¬ ((hP1 * (hP0 -⊗ hP)))(h2 ∪ h3), ¬ hP(h2 ∪ h3) ⊦

Rule argument

Additional information

Used terms: hP1, hP0 -⊗ hP

Proof step 9 from proof stability (previous step) (back to tree)

Lemma sep-com

Proof step 10 from proof stability (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)

• Proof step 12:

  ((hP0 * {h3} ))(h0), hP(h0), hP1(h2), h2 ⊥ h3, ¬ ((hP1 * (hP0 -⊗ hP)))(h2 ∪ h3), ¬ hP(h2 ∪ h3) ⊦ hP1(h2) ∧ ((hP0 -⊗ hP))(h3) ∧ h2 ⊥ h3

Rule argument

Additional information

Used terms: h3, h2, hP0 -⊗ hP, hP1

Proof step 11 from proof stability (previous step) (back to tree)

Lemma split-union

Proof step 12 from proof stability (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 13:

  ((hP0 * {h3} ))(h0), hP(h0), hP1(h2), h2 ⊥ h3, ¬ ((hP1 * (hP0 -⊗ hP)))(h2 ∪ h3), ¬ ((hP0 -⊗ hP))(h3), ¬ hP(h2 ∪ h3) ⊦

Rule argument

Additional information

Proof step 13 from proof stability (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 14:

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

• Proof step 15:

  ((hP0 * {h3} ))(h0), hP(h0), hP1(h2), h2 ⊥ h3, ¬ ((hP1 * (hP0 -⊗ hP)))(h2 ∪ h3), ¬ hP(h2 ∪ h3), ∀ h0, h1. ¬(h0 ⊥ h3 ∧ h1 = h0 ∪ h3 ∧ hP0(h0) ∧ hP(h1)) ⊦

Rule argument

Additional information

Used terms: h3, hP, hP0

Proof step 14 from proof stability (previous step) (back to tree)

Lemma septract

Proof step 15 from proof stability (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 16:

  ((hP0 * {h3} ))(h0), hP(h0), hP1(h2), h2 ⊥ h3, ¬ ((hP1 * (hP0 -⊗ hP)))(h2 ∪ h3), ¬ hP(h2 ∪ h3), ∀ h. ¬ h ⊥ h3 ∨ ¬ hP0(h) ∨ ¬ hP(h ∪ h3) ⊦

Rule argument

Additional information

Proof step 16 from proof stability (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 17:

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

• Proof step 18:

  hP(h0), hP1(h2), h2 ⊥ h3, ¬ ((hP1 * (hP0 -⊗ hP)))(h2 ∪ h3), ¬ hP(h2 ∪ h3), ∀ h. ¬ h ⊥ h3 ∨ ¬ hP0(h) ∨ ¬ hP(h ∪ h3), ∃ h1, h2. h1 ⊥ h2 ∧ h0 = h1 ∪ h2 ∧ hP0(h1) ∧ {h3} (h2) ⊦

Rule argument

Additional information

Used terms: h0, {h3} , hP0

Proof step 17 from proof stability (previous step) (back to tree)

Lemma sep

Proof step 18 from proof stability (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 19:

  hP(h1 ∪ h), hP0(h1), hP1(h2), h1 ⊥ h, h2 ⊥ h, ¬ ((hP1 * (hP0 -⊗ hP)))(h2 ∪ h), ¬ hP(h2 ∪ h), ∀ h0. ¬ h0 ⊥ h ∨ ¬ hP0(h0) ∨ ¬ hP(h0 ∪ h) ⊦

Rule argument

Additional information

Used simplifier rules: this: ⊦ {h} (h0) ↔ h = h0;

Proof step 19 from proof stability (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: h1