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

⊦ allWritesOKInd(pending, persisted, h1 + (W(t, l, v), false) ' + h2) → (allWritesOKInd(pending, persisted, h1 + (W(t, l, v), true) ' + h2) ↔ persisted(l) ∧ (h1 |W | l) | false = [])

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

¬(allWritesOKInd(pending, persisted, h1 + (W(t, l, v), true) ' + h2) ↔ persisted(l) ∧ (h1 |W | l) | false = []), allWritesOKInd(pending, persisted, h1 + (W(t, l, v), false) ' + h2) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: h1

Additional information

Used simplifier rules: restrict-W-empty: ⊦ [] |W = []; restrict-l-empty: ⊦ [] | l = []; restrict-b-empty: ⊦ [] | b = []; allWritesOKInd-recW: ⊦ allWritesOKInd(pending, persisted, (W(t, l, v), b) ' + h) ↔ ¬ pending(t) ∧ (b → persisted(l)) ∧ allWritesOKInd(pending(t := ¬ b), persisted(l := b), h); allwritesOK-ind-swap-flags-persist-one-write: ⊦ allWritesOKInd(pending(t := true), persisted(l := false), h) → allWritesOKInd(pending(t := false), persisted(l := true), h); List('a) freely generated by [], +;; ⊦ [] + x = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

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

¬(allWritesOKInd(pending, persisted, (vpe + h) + (W(t, l, v), true) ' + h2) ↔ persisted(l) ∧ ((vpe + h)|W | l) | false = []), allWritesOKInd(pending, persisted, (vpe + h) + (W(t, l, v), false) ' + h2), ∀ pending, persisted, t, l, v, h2. ¬ ( ¬(allWritesOKInd(pending, persisted, h + (W(t, l, v), true) ' + h2) ↔ persisted(l) ∧ (h |W | l) | false = []) ∧ allWritesOKInd(pending, persisted, h + (W(t, l, v), false) ' + h2)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: restrict-W-append: ⊦ (h + h0)|W = h |W + h0 |W; restrict-write-one: ⊦ vpe ' |W = (vpe._1.W? ⊃ vpe ';[]); restrict-l-append: ⊦ h + h0 | l0 = (h | l0) + (h0 | l0); restrict-b-append: ⊦ h + h0 | b = (h | b) + (h0 | b); restrict-W: ⊦ h |W = h.filter(λ vpe. vpe._1.W?); restrict-loc: ⊦ h | l = h.filter(λ vpe. vpe._1.l = l); restrict-vp: ⊦ h | b = h.filter(λ vpe. vpe._2 ↔ b); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a + x = a ' + x; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

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

¬ ( allWritesOKInd(pending, persisted, (vpe ' + h) + (W(t, l, v), true) ' + h2) ↔ persisted(l) ∧ (vpe._1.W? ⊃ vpe ';[]).filter(λ vpe. vpe._1.l = l).filter(λ vpe0. ¬ vpe0._2) = [] ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe0. ¬ vpe0._2) = []), allWritesOKInd(pending, persisted, (vpe ' + h) + (W(t, l, v), false) ' + h2), ∀ pending0, persisted0, t0, l0, v0, h0. ( allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), true) ' + h0) ↔ persisted0(l0) ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe0. ¬ vpe0._2) = []) ∨ ¬ allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), false) ' + h0) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: vpe

Additional information

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

vpe = (e, b), ¬ ( allWritesOKInd(pending, persisted, (vpe ' + h) + (W(t, l, v), true) ' + h2) ↔ persisted(l) ∧ (vpe._1.W? ⊃ vpe ';[]).filter(λ vpe. vpe._1.l = l).filter(λ vpe0. ¬ vpe0._2) = [] ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe0. ¬ vpe0._2) = []), allWritesOKInd(pending, persisted, (vpe ' + h) + (W(t, l, v), false) ' + h2), ∀ pending0, persisted0, t0, l0, v0, h0. ( allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), true) ' + h0) ↔ persisted0(l0) ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe0. ¬ vpe0._2) = []) ∨ ¬ allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), false) ' + h0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

¬ ( allWritesOKInd(pending, persisted, ((e, b) ' + h) + (W(t, l, v), true) ' + h2) ↔ persisted(l) ∧ (e.W? ⊃ (e, b) ';[]).filter(λ vpe. vpe._1.l = l).filter(λ vpe0. ¬ vpe0._2) = [] ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe0. ¬ vpe0._2) = []), allWritesOKInd(pending, persisted, ((e, b) ' + h) + (W(t, l, v), false) ' + h2), ∀ pending0, persisted0, t0, l0, v0, h0. ( allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), true) ' + h0) ↔ persisted0(l0) ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe0. ¬ vpe0._2) = []) ∨ ¬ allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), false) ' + h0) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: e

Additional information

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

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

e = R(t1, l1, v1), ¬ ( allWritesOKInd(pending, persisted, ((e, b) ' + h) + (W(t, l, v), true) ' + h2) ↔ persisted(l) ∧ (e.W? ⊃ (e, b) ';[]).filter(λ vpe. vpe._1.l = l).filter(λ vpe0. ¬ vpe0._2) = [] ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe0. ¬ vpe0._2) = []), allWritesOKInd(pending, persisted, ((e, b) ' + h) + (W(t, l, v), false) ' + h2), ∀ pending0, persisted0, t0, l0, v0, h0. ( allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), true) ' + h0) ↔ persisted0(l0) ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe0. ¬ vpe0._2) = []) ∨ ¬ allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), false) ' + h0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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 ⊦ [].filter(p) = []; (s) ⊦ ¬ R(t, l, v).W?; (s)

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

¬ ( allWritesOKInd(pending, persisted, h + (W(t, l, v), true) ' + h2) ↔ persisted(l) ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe0. ¬ vpe0._2) = []), allWritesOKInd(pending, persisted, h + (W(t, l, v), false) ' + h2), ∀ pending0, persisted0, t0, l0, v0, h0. ( allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), true) ' + h0) ↔ persisted0(l0) ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe0. ¬ vpe0._2) = []) ∨ ¬ allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), false) ' + h0) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 3 terms: pending, persisted, t, l, v, h2

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: pending, persisted, t, l, v, h2

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

e = W(t1, l1, v1), ¬ ( allWritesOKInd(pending, persisted, ((e, b) ' + h) + (W(t, l, v), true) ' + h2) ↔ persisted(l) ∧ (e.W? ⊃ (e, b) ';[]).filter(λ vpe. vpe._1.l = l).filter(λ vpe0. ¬ vpe0._2) = [] ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe0. ¬ vpe0._2) = []), allWritesOKInd(pending, persisted, ((e, b) ' + h) + (W(t, l, v), false) ' + h2), ∀ pending0, persisted0, t0, l0, v0, h0. ( allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), true) ' + h0) ↔ persisted0(l0) ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe0. ¬ vpe0._2) = []) ∨ ¬ allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), false) ' + h0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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 ⊦ W(t, l, v).l = l; (s) ⊦ a '.filter(p) = (p(a) ⊃ a ';[]); (s) ⊦ W(t, l, v).W?; (s)

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

¬ ( allWritesOKInd(pending(t1 := ¬ b), persisted(l1 := b), h + (W(t, l, v), true) ' + h2) ↔ persisted(l) ∧ (l1 = l ⊃ (W(t1, l, v1), b) ';[]).filter(λ vpe0. ¬ vpe0._2) = [] ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe0. ¬ vpe0._2) = []), allWritesOKInd(pending(t1 := ¬ b), persisted(l1 := b), h + (W(t, l, v), false) ' + h2), ¬ pending(t1), b → persisted(l1), ∀ pending0, persisted0, t0, l0, v0, h0. ( allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), true) ' + h0) ↔ persisted0(l0) ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe0. ¬ vpe0._2) = []) ∨ ¬ allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), false) ' + h0) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 5 terms: pending(t1 := ¬ b), persisted(l1 := b), t, l, v, h2 abort: F discard: F computed: T

Additional information

Used terms: pending(t1 := ¬ b), persisted(l1 := b), t, l, v, h2

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

¬ ( allWritesOKInd(pending(t1 := ¬ b), persisted(l1 := b), h + (W(t, l, v), true) ' + h2) ↔ persisted(l) ∧ (l1 = l ⊃ (W(t1, l, v1), b) ';[]).filter(λ vpe0. ¬ vpe0._2) = [] ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe0. ¬ vpe0._2) = []), allWritesOKInd(pending(t1 := ¬ b), persisted(l1 := b), h + (W(t, l, v), false) ' + h2), ¬ pending(t1), b → persisted(l1), ∀ pending0, persisted0, t0, l0, v0, h0. ( allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), true) ' + h0) ↔ persisted0(l0) ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe0. ¬ vpe0._2) = []) ∨ ¬ allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), false) ' + h0), ( allWritesOKInd(pending(t1 := ¬ b), persisted(l1 := b), h + (W(t, l, v), true) ' + h2) ↔ persisted(l1 := b)(l) ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe0. ¬ vpe0._2) = []) ∨ ¬ allWritesOKInd(pending(t1 := ¬ b), persisted(l1 := b), h + (W(t, l, v), false) ' + h2) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

allWritesOKInd(pending(t1 := ¬ b), persisted(l1 := b), h + (W(t, l, v), true) ' + h2) ↔ (l1 = l ⊃ b;persisted(l)) ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe0. ¬ vpe0._2) = [], ¬ ( allWritesOKInd(pending(t1 := ¬ b), persisted(l1 := b), h + (W(t, l, v), true) ' + h2) ↔ persisted(l) ∧ (l1 = l ⊃ (W(t1, l, v1), b) ';[]).filter(λ vpe0. ¬ vpe0._2) = [] ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe0. ¬ vpe0._2) = []), allWritesOKInd(pending(t1 := ¬ b), persisted(l1 := b), h + (W(t, l, v), false) ' + h2), ¬ pending(t1), b → persisted(l1), ∀ pending0, persisted0, t0, l0, v0, h0. ( allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), true) ' + h0) ↔ persisted0(l0) ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe0. ¬ vpe0._2) = []) ∨ ¬ allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), false) ' + h0) ⊦

Heuristic nonsplitting cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: allWritesOKInd(pending(t1 := ¬ b), persisted(l1 := b), h + (W(t, l, v), true) ' + h2)

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a '.filter(p) = (p(a) ⊃ a ';[]); (s) ⊦ a ' ≠ []; (s) Used formulas: allWritesOKInd(pending(t1 := ¬ b), persisted(l1 := b), h + (W(t, l, v), true) ' + h2)

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

allWritesOKInd(pending(t1 := ¬ b), persisted(l1 := b), h + (W(t, l, v), true) ' + h2) ↔ (l1 = l ⊃ b;persisted(l)) ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe0. ¬ vpe0._2) = [], ¬ ( allWritesOKInd(pending(t1 := ¬ b), persisted(l1 := b), h + (W(t, l, v), true) ' + h2) ↔ persisted(l) ∧ (l1 = l ⊃ (W(t1, l, v1), b) ';[]).filter(λ vpe0. ¬ vpe0._2) = [] ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe0. ¬ vpe0._2) = []), allWritesOKInd(pending(t1 := ¬ b), persisted(l1 := b), h + (W(t, l, v), false) ' + h2), allWritesOKInd(pending(t1 := ¬ b), persisted(l1 := b), h + (W(t, l, v), true) ' + h2), ¬ pending(t1), b → persisted(l1), ∀ pending0, persisted0, t0, l0, v0, h0. ( allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), true) ' + h0) ↔ persisted0(l0) ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe0. ¬ vpe0._2) = []) ∨ ¬ allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), false) ' + h0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe0. ¬ vpe0._2) = [], allWritesOKInd(pending(t1 := ¬ b), persisted(l1 := b), h + (W(t, l, v), true) ' + h2), allWritesOKInd(pending(t1 := ¬ b), persisted(l1 := b), h + (W(t, l, v), false) ' + h2), ¬ pending(t1), b → persisted(l1), (l1 = l ⊃ b;persisted(l)), ∀ pending0, persisted0, t0, l0, v0, h0. ( allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), true) ' + h0) ↔ persisted0(l0) ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe0. ¬ vpe0._2) = []) ∨ ¬ allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), false) ' + h0) ⊦ persisted(l) ∧ (l1 = l ⊃ (W(t1, l, v1), b) ';[]).filter(λ vpe0. ¬ vpe0._2) = []

Heuristic nonsplitting cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: ¬ b

Additional information

Used simplifier rules: ⊦ [].filter(p) = []; (s) Used formulas: ¬ b

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

h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe0. ¬ vpe0._2) = [], b, allWritesOKInd(pending(t1 := ¬ b), persisted(l1 := b), h + (W(t, l, v), false) ' + h2), allWritesOKInd(pending(t1 := ¬ b), persisted(l1 := b), h + (W(t, l, v), true) ' + h2), ¬ pending(t1), b → persisted(l1), (l1 = l ⊃ b;persisted(l)), ∀ pending0, persisted0, t0, l0, v0, h0. ( allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), true) ' + h0) ↔ persisted0(l0) ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe0. ¬ vpe0._2) = []) ∨ ¬ allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), false) ' + h0) ⊦ persisted(l) ∧ (l1 = l ⊃ (W(t1, l, v1), b) ';[]).filter(λ vpe0. ¬ vpe0._2) = []

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe0. ¬ vpe0._2) = [], (l1 = l ⊃ (W(t1, l, v1), true) ';[]).filter(λ vpe0. ¬ vpe0._2) ≠ [], allWritesOKInd(pending(t1 := false), persisted(l1 := true), h + (W(t, l, v), true) ' + h2), allWritesOKInd(pending(t1 := false), persisted(l1 := true), h + (W(t, l, v), false) ' + h2), persisted(l1), persisted(l), ¬ pending(t1), ∀ pending0, persisted0, t0, l0, v0, h0. ( allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), true) ' + h0) ↔ persisted0(l0) ∧ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe0. ¬ vpe0._2) = []) ∨ ¬ allWritesOKInd(pending0, persisted0, h + (W(t0, l0, v0), false) ' + h0) ⊦

Heuristic nonsplitting cut applied rule cut formula

And closed the goal!

Rule argument

fma: l1 = l

Additional information

Used simplifier rules: ⊦ [].filter(p) = []; (s) ⊦ a '.filter(p) = (p(a) ⊃ a ';[]); (s) Used formulas: l1 = l

Proof Tree for allwritesOK-ind-persist-one-write (View conclusion)

The following image shows the proof tree of sequent calculus. The proved goal is at the bottom. You can click on the nodes to view additional rules. See here for details on colors in a proof tree.

the proof tree