Proof step 1 from proof persistDependentReads-restrict-other-loc (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  l ≠ l0, persistDependentReads(l, h) | l0 ≠ h | l0 ⊦

Rule argument

Additional information

Proof step 2 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  persistDependentReads(l, h).filter(λ vpe. vpe._1.l = l0) ≠ h.filter(λ vpe. vpe._1.l = l0), l ≠ l0 ⊦

Rule argument

Additional information

Used simplifier rules: restrict-loc: ⊦ h | l = h.filter(λ vpe. vpe._1.l = l);

Proof step 3 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

User applied rule structural induction

And obtained the following new goals

• Proof step 4:

  persistDependentReads(l, vpe + h0).filter(λ vpe. vpe._1.l = l0) ≠ (vpe + h0).filter(λ vpe. vpe._1.l = l0), l ≠ l0, ∀ l, l0. ¬(persistDependentReads(l, h0).filter(λ vpe. vpe._1.l = l0) ≠ h0.filter(λ vpe. vpe._1.l = l0) ∧ l ≠ l0) ⊦

Rule argument

Additional information

Used simplifier rules: persistDependentReads-base: ⊦ persistDependentReads(l, []) = []; List('a) freely generated by [], +;; ⊦ [].filter(p) = []; (s)

Proof step 4 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 5:

  persistDependentReads(l, vpe ' + h0).filter(λ vpe. vpe._1.l = l0) ≠ (vpe._1.l = l0 ⊃ vpe ';[]) + h0.filter(λ vpe. vpe._1.l = l0), l ≠ l0, ∀ l1, l2. persistDependentReads(l1, h0).filter(λ vpe. vpe._1.l = l2) = h0.filter(λ vpe. vpe._1.l = l2) ∨ l1 = l2 ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s) ⊦ a + x = a ' + x; (s)

Proof step 5 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 6:

  vpe = (e, boolvar), persistDependentReads(l, vpe ' + h0).filter(λ vpe. vpe._1.l = l0) ≠ (e.l = l0 ⊃ vpe ';[]) + h0.filter(λ vpe. vpe._1.l = l0), l ≠ l0, ∀ l1, l2. persistDependentReads(l1, h0).filter(λ vpe. vpe._1.l = l2) = h0.filter(λ vpe. vpe._1.l = l2) ∨ l1 = l2 ⊦

Rule argument

Additional information

Used terms: e, boolvar

Proof step 6 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 7:

  persistDependentReads(l, (e, boolvar) ' + h0).filter(λ vpe. vpe._1.l = l0) ≠ (e.l = l0 ⊃ (e, boolvar) ';[]) + h0.filter(λ vpe. vpe._1.l = l0), l ≠ l0, ∀ l1, l2. persistDependentReads(l1, h0).filter(λ vpe. vpe._1.l = l2) = h0.filter(λ vpe. vpe._1.l = l2) ∨ l1 = l2 ⊦

Rule argument

Additional information

Proof step 7 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

• Proof step 8:

  e.l = l0, persistDependentReads(l, (e, boolvar) ' + h0).filter(λ vpe. vpe._1.l = l0) ≠ (e, boolvar) ' + h0.filter(λ vpe. vpe._1.l = l0), l ≠ l0, ∀ l1, l2. persistDependentReads(l1, h0).filter(λ vpe. vpe._1.l = l2) = h0.filter(λ vpe. vpe._1.l = l2) ∨ l1 = l2 ⊦

• Proof step 14:

  persistDependentReads(l, (e, boolvar) ' + h0).filter(λ vpe. vpe._1.l = l0) ≠ [] + h0.filter(λ vpe. vpe._1.l = l0), e.l ≠ l0, l ≠ l0, ∀ l1, l2. persistDependentReads(l1, h0).filter(λ vpe. vpe._1.l = l2) = h0.filter(λ vpe. vpe._1.l = l2) ∨ l1 = l2 ⊦

Rule argument

Additional information

Proof step 8 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 9:

  persistDependentReads(l, (e, boolvar) ' + h0).filter(λ vpe. vpe._1.l = e.l) ≠ (e, boolvar) ' + h0.filter(λ vpe. vpe._1.l = e.l), l ≠ e.l, ∀ l1, l2. persistDependentReads(l1, h0).filter(λ vpe. vpe._1.l = l2) = h0.filter(λ vpe. vpe._1.l = l2) ∨ l1 = l2 ⊦

Rule argument

Additional information

Proof step 9 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

User applied rule constructor cut

And obtained the following new goals

• Proof step 10:

  e = R(t, l0, v), persistDependentReads(l, (e, boolvar) ' + h0).filter(λ vpe. vpe._1.l = e.l) ≠ (e, boolvar) ' + h0.filter(λ vpe. vpe._1.l = e.l), l ≠ e.l, ∀ l1, l2. persistDependentReads(l1, h0).filter(λ vpe. vpe._1.l = l2) = h0.filter(λ vpe. vpe._1.l = l2) ∨ l1 = l2 ⊦

• Proof step 12:

  e = W(t, l0, v), persistDependentReads(l, (e, boolvar) ' + h0).filter(λ vpe. vpe._1.l = e.l) ≠ (e, boolvar) ' + h0.filter(λ vpe. vpe._1.l = e.l), l ≠ e.l, ∀ l1, l2. persistDependentReads(l1, h0).filter(λ vpe. vpe._1.l = l2) = h0.filter(λ vpe. vpe._1.l = l2) ∨ l1 = l2 ⊦

Rule argument

Additional information

Used simplifier rules: RWevent freely generated by R, W;;

Proof step 10 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 11:

  persistDependentReads(l, h0).filter(λ vpe. vpe._1.l = l0) ≠ h0.filter(λ vpe. vpe._1.l = l0), l ≠ l0, ∀ l1, l2. persistDependentReads(l1, h0).filter(λ vpe. vpe._1.l = l2) = h0.filter(λ vpe. vpe._1.l = l2) ∨ l1 = l2 ⊦

Rule argument

Additional information

Used simplifier rules: persistDependentReads-otherread: ⊦ l ≠ l0 → persistDependentReads(l, (R(t, l0, v), b) ' + h) = (R(t, l0, v), b) + persistDependentReads(l, h); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ # x = # z → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws) ⊦ R(t, l, v).l = l; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s) ⊦ a + x = a ' + x; (s)

Proof step 11 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: l, l0

Proof step 12 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 13:

  persistDependentReads(l, h0).filter(λ vpe. vpe._1.l = l0) ≠ h0.filter(λ vpe. vpe._1.l = l0), l ≠ l0, ∀ l1, l2. persistDependentReads(l1, h0).filter(λ vpe. vpe._1.l = l2) = h0.filter(λ vpe. vpe._1.l = l2) ∨ l1 = l2 ⊦

Rule argument

Additional information

Used simplifier rules: persistDependentReads-otherwrite: ⊦ l ≠ l0 → persistDependentReads(l, (W(t, l0, v), b) ' + h) = (W(t, l0, v), b) ' + persistDependentReads(l, h); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ # x = # z → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws) ⊦ W(t, l, v).l = l; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s)

Proof step 13 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: l, l0

Proof step 14 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 15:

  persistDependentReads(l, (e, boolvar) ' + h0).filter(λ vpe. vpe._1.l = l0) ≠ h0.filter(λ vpe. vpe._1.l = l0), e.l ≠ l0, l ≠ l0, ∀ l1, l2. persistDependentReads(l1, h0).filter(λ vpe. vpe._1.l = l2) = h0.filter(λ vpe. vpe._1.l = l2) ∨ l1 = l2 ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ [] + x = x; (s)

Proof step 15 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

User applied rule constructor cut

And obtained the following new goals

• Proof step 16:

  e = R(t, l3, v), persistDependentReads(l, (e, boolvar) ' + h0).filter(λ vpe. vpe._1.l = l0) ≠ h0.filter(λ vpe. vpe._1.l = l0), e.l ≠ l0, l ≠ l0, ∀ l1, l2. persistDependentReads(l1, h0).filter(λ vpe. vpe._1.l = l2) = h0.filter(λ vpe. vpe._1.l = l2) ∨ l1 = l2 ⊦

• Proof step 23:

  e = W(t, l3, v), persistDependentReads(l, (e, boolvar) ' + h0).filter(λ vpe. vpe._1.l = l0) ≠ h0.filter(λ vpe. vpe._1.l = l0), e.l ≠ l0, l ≠ l0, ∀ l1, l2. persistDependentReads(l1, h0).filter(λ vpe. vpe._1.l = l2) = h0.filter(λ vpe. vpe._1.l = l2) ∨ l1 = l2 ⊦

Rule argument

Additional information

Used simplifier rules: RWevent freely generated by R, W;;

Proof step 16 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 17:

  persistDependentReads(l, (R(t, l3, v), boolvar) ' + h0).filter(λ vpe. vpe._1.l = l0) ≠ h0.filter(λ vpe. vpe._1.l = l0), l ≠ l0, l3 ≠ l0, ∀ l1, l2. persistDependentReads(l1, h0).filter(λ vpe. vpe._1.l = l2) = h0.filter(λ vpe. vpe._1.l = l2) ∨ l1 = l2 ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ R(t, l, v).l = l; (s)

Proof step 17 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 18:

  ⊦ l ≠ l0 → persistDependentReads(l, (R(t, l0, v), b) ' + h) = (R(t, l0, v), b) + persistDependentReads(l, h)

• Proof step 19:

  l = l3, persistDependentReads(l, (R(t, l3, v), boolvar) ' + h0).filter(λ vpe. vpe._1.l = l0) ≠ h0.filter(λ vpe. vpe._1.l = l0), l ≠ l0, l3 ≠ l0, ∀ l1, l2. persistDependentReads(l1, h0).filter(λ vpe. vpe._1.l = l2) = h0.filter(λ vpe. vpe._1.l = l2) ∨ l1 = l2 ⊦

• Proof step 21:

  ((R(t, l3, v), boolvar) + persistDependentReads(l, h0)).filter(λ vpe. vpe._1.l = l0) ≠ h0.filter(λ vpe. vpe._1.l = l0), l ≠ l3, l ≠ l0, l3 ≠ l0, ∀ l1, l2. persistDependentReads(l1, h0).filter(λ vpe. vpe._1.l = l2) = h0.filter(λ vpe. vpe._1.l = l2) ∨ l1 = l2 ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: v, boolvar, l, l3, t, h0

Proof step 18 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

Lemma persistDependentReads-otherread

Proof step 19 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 20:

  persistDependentReads(l3, h0).filter(λ vpe. vpe._1.l = l0) ≠ h0.filter(λ vpe. vpe._1.l = l0), l3 ≠ l0, ∀ l1, l2. persistDependentReads(l1, h0).filter(λ vpe. vpe._1.l = l2) = h0.filter(λ vpe. vpe._1.l = l2) ∨ l1 = l2 ⊦

Rule argument

Additional information

Used simplifier rules: persistDependentReads-read: ⊦ persistDependentReads(l, (R(t, l, v), b) ' + h) = (R(t, l, v), true) ' + persistDependentReads(l, h); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ [] + x = x; (s) ⊦ R(t, l, v).l = l; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s)

Proof step 20 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: l3, l0

Proof step 21 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 22:

  persistDependentReads(l, h0).filter(λ vpe. vpe._1.l = l0) ≠ h0.filter(λ vpe. vpe._1.l = l0), l ≠ l3, l ≠ l0, l3 ≠ l0, ∀ l1, l2. persistDependentReads(l1, h0).filter(λ vpe. vpe._1.l = l2) = h0.filter(λ vpe. vpe._1.l = l2) ∨ l1 = l2 ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ [] + x = x; (s) ⊦ R(t, l, v).l = l; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s) ⊦ a + x = a ' + x; (s)

Proof step 22 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: l, l0

Proof step 23 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 24:

  persistDependentReads(l, (W(t, l3, v), boolvar) ' + h0).filter(λ vpe. vpe._1.l = l0) ≠ h0.filter(λ vpe. vpe._1.l = l0), l ≠ l0, l3 ≠ l0, ∀ l1, l2. persistDependentReads(l1, h0).filter(λ vpe. vpe._1.l = l2) = h0.filter(λ vpe. vpe._1.l = l2) ∨ l1 = l2 ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ W(t, l, v).l = l; (s)

Proof step 24 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 25:

  ⊦ l ≠ l0 → persistDependentReads(l, (W(t, l0, v), b) ' + h) = (W(t, l0, v), b) ' + persistDependentReads(l, h)

• Proof step 26:

  ((W(t, l3, v), boolvar) ' + persistDependentReads(l, h0)).filter(λ vpe. vpe._1.l = l0) ≠ h0.filter(λ vpe. vpe._1.l = l0), l ≠ l3, l ≠ l0, l3 ≠ l0, ∀ l1, l2. persistDependentReads(l1, h0).filter(λ vpe. vpe._1.l = l2) = h0.filter(λ vpe. vpe._1.l = l2) ∨ l1 = l2 ⊦

Rule argument

Additional information

Used simplifier rules: persistDependentReads-nextwrite: ⊦ persistDependentReads(l, (W(t, l, v), b) ' + h) = (W(t, l, v), b) ' + h; ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s) ⊦ W(t, l, v).l = l; (s) ⊦ [] + x = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: v, boolvar, l, l3, t, h0

Proof step 25 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

Lemma persistDependentReads-otherwrite

Proof step 26 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 27:

  persistDependentReads(l, h0).filter(λ vpe. vpe._1.l = l0) ≠ h0.filter(λ vpe. vpe._1.l = l0), l ≠ l3, l ≠ l0, l3 ≠ l0, ∀ l1, l2. persistDependentReads(l1, h0).filter(λ vpe. vpe._1.l = l2) = h0.filter(λ vpe. vpe._1.l = l2) ∨ l1 = l2 ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ [] + x = x; (s) ⊦ W(t, l, v).l = l; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s)

Proof step 27 from proof persistDependentReads-restrict-other-loc (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: l, l0