Proof step 1 from proof allwritesOK-ind-swap-to-non-pending (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

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

Rule argument

Additional information

Proof step 2 from proof allwritesOK-ind-swap-to-non-pending (previous step) (back to tree)

User applied rule structural induction

And obtained the following new goals

• Proof step 3:

  allWritesOKInd(pending(t := true), persisted, vpe + h0), ¬ allWritesOKInd(pending(t := false), persisted, vpe + h0), ∀ pending, t, persisted. ¬(allWritesOKInd(pending(t := true), persisted, h0) ∧ ¬ allWritesOKInd(pending(t := false), persisted, 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-to-non-pending (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 4:

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

Rule argument

Additional information

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

Proof step 4 from proof allwritesOK-ind-swap-to-non-pending (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, vpe ' + h0), ¬ allWritesOKInd(pending(t := false), persisted, vpe ' + h0), ∀ pending0, t0, persisted0. ¬ allWritesOKInd(pending0(t0 := true), persisted0, h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0, h0) ⊦

Rule argument

Additional information

Proof step 5 from proof allwritesOK-ind-swap-to-non-pending (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 6:

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

Rule argument

Additional information

Proof step 6 from proof allwritesOK-ind-swap-to-non-pending (previous step) (back to tree)

User applied rule constructor cut

And obtained the following new goals

• Proof step 7:

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

• Proof step 9:

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

Rule argument

Additional information

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

Proof step 7 from proof allwritesOK-ind-swap-to-non-pending (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 8:

  allWritesOKInd(pending(t := true), persisted, h0), ¬ allWritesOKInd(pending(t := false), persisted, h0), ∀ pending0, t0, persisted0. ¬ allWritesOKInd(pending0(t0 := true), persisted0, h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0, 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-to-non-pending (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

Proof step 9 from proof allwritesOK-ind-swap-to-non-pending (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 := b), h0), ¬ allWritesOKInd(pending(t := false)(t1 := ¬ b), persisted(l := b), h0), ¬ pending(t1), b → persisted(l), ∀ pending0, t0, persisted0. ¬ allWritesOKInd(pending0(t0 := true), persisted0, h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0, 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-to-non-pending (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 11:

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

• Proof step 12:

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

Rule argument

Additional information

Used terms: ¬ b, pending, true, t1, t

Proof step 11 from proof allwritesOK-ind-swap-to-non-pending (previous step) (back to tree)

Lemma pending-update-swap

Proof step 12 from proof allwritesOK-ind-swap-to-non-pending (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 := b), h0), ¬ allWritesOKInd(pending(t1 := ¬ b)(t := false), persisted(l := b), h0), ¬ pending(t1), b → persisted(l), ∀ pending0, t0, persisted0. ¬ allWritesOKInd(pending0(t0 := true), persisted0, h0) ∨ allWritesOKInd(pending0(t0 := false), persisted0, h0) ⊦

Rule argument

Additional information

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

Proof step 13 from proof allwritesOK-ind-swap-to-non-pending (previous step) (back to tree)

Lemma pending-update-swap

Proof step 14 from proof allwritesOK-ind-swap-to-non-pending (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(l := b)