Proof step 1 from proof allwritesOK-ind-swap-flags-persist-one-write (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  allWritesOKInd(pending(t := true), persisted(l := false), h), ¬ allWritesOKInd(pending(t := false), persisted(l := true), h) ⊦

Rule argument

Additional information

Proof step 2 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

User applied rule structural induction

And obtained the following new goals

• Proof step 3:

  allWritesOKInd(pending(t := true), persisted(l := false), vpe + h0), ¬ allWritesOKInd(pending(t := false), persisted(l := true), vpe + h0), ∀ pending, t, persisted, l. ¬(allWritesOKInd(pending(t := true), persisted(l := false), h0) ∧ ¬ allWritesOKInd(pending(t := false), persisted(l := true), h0)) ⊦

Rule argument

Additional information

Used simplifier rules: allWritesOKInd-base: ⊦ allWritesOKInd(pending, persisted, []); List('a) freely generated by [], +;;

Proof step 3 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 4:

  allWritesOKInd(pending(t := true), persisted(l := false), vpe ' + h0), ¬ allWritesOKInd(pending(t := false), persisted(l := true), vpe ' + h0), ∀ pending0, t0, persisted0, l0. ¬ allWritesOKInd(pending0(t0 := true), persisted0(l0 := false), h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0(l0 := true), h0) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s)

Proof step 4 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

User applied rule constructor cut

And obtained the following new goals

• Proof step 5:

  vpe = (e, b), allWritesOKInd(pending(t := true), persisted(l := false), vpe ' + h0), ¬ allWritesOKInd(pending(t := false), persisted(l := true), vpe ' + h0), ∀ pending0, t0, persisted0, l0. ¬ allWritesOKInd(pending0(t0 := true), persisted0(l0 := false), h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0(l0 := true), h0) ⊦

Rule argument

Additional information

Proof step 5 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 6:

  allWritesOKInd(pending(t := true), persisted(l := false), (e, b) ' + h0), ¬ allWritesOKInd(pending(t := false), persisted(l := true), (e, b) ' + h0), ∀ pending0, t0, persisted0, l0. ¬ allWritesOKInd(pending0(t0 := true), persisted0(l0 := false), h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0(l0 := true), h0) ⊦

Rule argument

Additional information

Proof step 6 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

User applied rule constructor cut

And obtained the following new goals

• Proof step 7:

  e = R(t1, l1, v), allWritesOKInd(pending(t := true), persisted(l := false), (e, b) ' + h0), ¬ allWritesOKInd(pending(t := false), persisted(l := true), (e, b) ' + h0), ∀ pending0, t0, persisted0, l0. ¬ allWritesOKInd(pending0(t0 := true), persisted0(l0 := false), h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0(l0 := true), h0) ⊦

• Proof step 9:

  e = W(t1, l1, v), allWritesOKInd(pending(t := true), persisted(l := false), (e, b) ' + h0), ¬ allWritesOKInd(pending(t := false), persisted(l := true), (e, b) ' + h0), ∀ pending0, t0, persisted0, l0. ¬ allWritesOKInd(pending0(t0 := true), persisted0(l0 := false), h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0(l0 := true), h0) ⊦

Rule argument

Additional information

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

Proof step 7 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 8:

  allWritesOKInd(pending(t := true), persisted(l := false), h0), ¬ allWritesOKInd(pending(t := false), persisted(l := true), h0), ∀ pending0, t0, persisted0, l0. ¬ allWritesOKInd(pending0(t0 := true), persisted0(l0 := false), h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0(l0 := true), h0) ⊦

Rule argument

Additional information

Used simplifier rules: allWritesOKInd-recR: ⊦ allWritesOKInd(pending, persisted, (R(t, l, v), b) ' + h) ↔ allWritesOKInd(pending, persisted, h); a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 8 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: pending, t, persisted, l

Proof step 9 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 10:

  t ≠ t1, allWritesOKInd(pending(t := true)(t1 := ¬ b), persisted(l := false)(l1 := b), h0), ¬ pending(t1), b → l ≠ l1 ∧ persisted(l1), ∀ pending0, t0, persisted0, l0. ¬ allWritesOKInd(pending0(t0 := true), persisted0(l0 := false), h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0(l0 := true), h0) ⊦ (b → l ≠ l1 → persisted(l1)) ∧ allWritesOKInd(pending(t := false)(t1 := ¬ b), persisted(l := true)(l1 := b), h0)

Rule argument

Additional information

Used simplifier rules: allWritesOKInd-recW: ⊦ allWritesOKInd(pending, persisted, (W(t, l, v), b) ' + h) ↔ ¬ pending(t) ∧ (b → persisted(l)) ∧ allWritesOKInd(pending(t := ¬ b), persisted(l := b), h); a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 10 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

Heuristic nonsplitting cut applied rule cut formula

And obtained the following new goals

• Proof step 11:

  t ≠ t1, allWritesOKInd(pending(t := true)(t1 := ¬ b), persisted(l := false)(l1 := b), h0), ¬ allWritesOKInd(pending(t := false)(t1 := ¬ b), persisted(l := true)(l1 := b), h0), ¬ pending(t1), b → l ≠ l1 ∧ persisted(l1), ∀ pending0, t0, persisted0, l0. ¬ allWritesOKInd(pending0(t0 := true), persisted0(l0 := false), h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0(l0 := true), h0) ⊦ (b → l ≠ l1 → persisted(l1)) ∧ allWritesOKInd(pending(t := false)(t1 := ¬ b), persisted(l := true)(l1 := b), h0)

Rule argument

Additional information

Used formulas: allWritesOKInd(pending(t := false)(t1 := ¬ b), persisted(l := true)(l1 := b), h0)

Proof step 11 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 12:

  t ≠ t1, allWritesOKInd(pending(t := true)(t1 := ¬ b), persisted(l := false)(l1 := b), h0), ¬ allWritesOKInd(pending(t := false)(t1 := ¬ b), persisted(l := true)(l1 := b), h0), ¬ pending(t1), b → l ≠ l1 ∧ persisted(l1), ∀ pending0, t0, persisted0, l0. ¬ allWritesOKInd(pending0(t0 := true), persisted0(l0 := false), h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0(l0 := true), h0) ⊦

Rule argument

Additional information

Proof step 12 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 13:

  ⊦ t0 ≠ t1 → pending(t0 := b0)(t1 := b1) = pending(t1 := b1)(t0 := b0)

• Proof step 14:

  t ≠ t1, allWritesOKInd(pending(t1 := ¬ b)(t := true), persisted(l := false)(l1 := b), h0), ¬ allWritesOKInd(pending(t := false)(t1 := ¬ b), persisted(l := true)(l1 := b), h0), ¬ pending(t1), b → l ≠ l1 ∧ persisted(l1), ∀ pending0, t0, persisted0, l0. ¬ allWritesOKInd(pending0(t0 := true), persisted0(l0 := false), h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0(l0 := true), h0) ⊦

Rule argument

Additional information

Used terms: ¬ b, pending, true, t1, t

Proof step 13 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

Lemma pending-update-swap

Proof step 14 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 15:

  ⊦ l0 ≠ l1 → persisted(l0 := b0)(l1 := b1) = persisted(l1 := b1)(l0 := b0)

• Proof step 16:

  l = l1, t ≠ t1, allWritesOKInd(pending(t1 := ¬ b)(t := true), persisted(l := false)(l1 := b), h0), ¬ allWritesOKInd(pending(t := false)(t1 := ¬ b), persisted(l := true)(l1 := b), h0), ¬ pending(t1), b → l ≠ l1 ∧ persisted(l1), ∀ pending0, t0, persisted0, l0. ¬ allWritesOKInd(pending0(t0 := true), persisted0(l0 := false), h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0(l0 := true), h0) ⊦

• Proof step 23:

  l ≠ l1, t ≠ t1, allWritesOKInd(pending(t1 := ¬ b)(t := true), persisted(l1 := b)(l := false), h0), ¬ allWritesOKInd(pending(t := false)(t1 := ¬ b), persisted(l := true)(l1 := b), h0), ¬ pending(t1), b → l ≠ l1 ∧ persisted(l1), ∀ pending0, t0, persisted0, l0. ¬ allWritesOKInd(pending0(t0 := true), persisted0(l0 := false), h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0(l0 := true), h0) ⊦

Rule argument

Additional information

Used terms: persisted, b, l1, false, l

Proof step 15 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

Lemma persisted-update-swap

Proof step 16 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 17:

  t ≠ t1, allWritesOKInd(pending(t1 := true)(t := true), persisted(l1 := false), h0), ¬ allWritesOKInd(pending(t := false)(t1 := true), persisted(l1 := false), h0), ¬ pending(t1), ∀ pending0, t0, persisted0, l0. ¬ allWritesOKInd(pending0(t0 := true), persisted0(l0 := false), h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0(l0 := true), h0) ⊦

Rule argument

Additional information

Used simplifier rules: persisted-update-same: ⊦ persisted(l := b0)(l := b) = persisted(l := b);

Proof step 17 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 18:

  ⊦ t0 ≠ t1 → pending(t0 := b0)(t1 := b1) = pending(t1 := b1)(t0 := b0)

• Proof step 19:

  t ≠ t1, allWritesOKInd(pending(t := true)(t1 := true), persisted(l1 := false), h0), ¬ allWritesOKInd(pending(t := false)(t1 := true), persisted(l1 := false), h0), ¬ pending(t1), ∀ pending0, t0, persisted0, l0. ¬ allWritesOKInd(pending0(t0 := true), persisted0(l0 := false), h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0(l0 := true), h0) ⊦

Rule argument

Additional information

Used terms: true, pending, true, t, t1

Proof step 18 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

Lemma pending-update-swap

Proof step 19 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 20:

  ⊦ t0 ≠ t1 → pending(t0 := b0)(t1 := b1) = pending(t1 := b1)(t0 := b0)

• Proof step 21:

  t ≠ t1, allWritesOKInd(pending(t1 := true)(t := true), persisted(l1 := false), h0), ¬ allWritesOKInd(pending(t := false)(t1 := true), persisted(l1 := false), h0), ¬ pending(t1), ∀ pending0, t0, persisted0, l0. ¬ allWritesOKInd(pending0(t0 := true), persisted0(l0 := false), h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0(l0 := true), h0) ⊦

Rule argument

Additional information

Used terms: true, pending, true, t1, t

Proof step 20 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

Lemma pending-update-swap

Proof step 21 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 22:

  ⊦ t0 ≠ t1 → pending(t0 := b0)(t1 := b1) = pending(t1 := b1)(t0 := b0)

Rule argument

Additional information

Used simplifier rules: allwritesOK-ind-swap-to-non-pending: ⊦ allWritesOKInd(pending(t := true), persisted, h) → allWritesOKInd(pending(t := false), persisted, h); Used terms: true, pending, false, t1, t

Proof step 22 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

Lemma pending-update-swap

Proof step 23 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 24:

  l ≠ l1, t ≠ t1, allWritesOKInd(pending(t1 := ¬ b)(t := true), persisted(l1 := b)(l := false), h0), ¬ allWritesOKInd(pending(t := false)(t1 := ¬ b), persisted(l := true)(l1 := b), h0), ¬ pending(t1), b → persisted(l1), ∀ pending0, t0, persisted0, l0. ¬ allWritesOKInd(pending0(t0 := true), persisted0(l0 := false), h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0(l0 := true), h0) ⊦

Rule argument

Additional information

Proof step 24 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 25:

  ⊦ t0 ≠ t1 → pending(t0 := b0)(t1 := b1) = pending(t1 := b1)(t0 := b0)

• Proof step 26:

  l ≠ l1, t ≠ t1, allWritesOKInd(pending(t1 := ¬ b)(t := true), persisted(l1 := b)(l := false), h0), ¬ allWritesOKInd(pending(t1 := ¬ b)(t := false), persisted(l := true)(l1 := b), h0), ¬ pending(t1), b → persisted(l1), ∀ pending0, t0, persisted0, l0. ¬ allWritesOKInd(pending0(t0 := true), persisted0(l0 := false), h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0(l0 := true), h0) ⊦

Rule argument

Additional information

Used terms: ¬ b, pending, false, t1, t

Proof step 25 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

Lemma pending-update-swap

Proof step 26 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 27:

  ⊦ l0 ≠ l1 → persisted(l0 := b0)(l1 := b1) = persisted(l1 := b1)(l0 := b0)

• Proof step 28:

  l ≠ l1, t ≠ t1, allWritesOKInd(pending(t1 := ¬ b)(t := true), persisted(l1 := b)(l := false), h0), ¬ allWritesOKInd(pending(t1 := ¬ b)(t := false), persisted(l1 := b)(l := true), h0), ¬ pending(t1), b → persisted(l1), ∀ pending0, t0, persisted0, l0. ¬ allWritesOKInd(pending0(t0 := true), persisted0(l0 := false), h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0(l0 := true), h0) ⊦

Rule argument

Additional information

Used terms: persisted, b, l1, true, l

Proof step 27 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

Lemma persisted-update-swap

Proof step 28 from proof allwritesOK-ind-swap-flags-persist-one-write (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: pending(t1 := ¬ b), t, persisted(l1 := b), l