[Imprint][Data Privacy Statement][E-Mail]



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Proof step 1 from proof allReadsOK-ind-no-initial-unpersisted-read (conclusion) (back to tree)

⊦ allReadsOKInd(mem, persisted, h) ∧ persisted(l) → events((h | l) | false) ≠ R(t0, l0, v0) ' + el

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof allReadsOK-ind-no-initial-unpersisted-read (previous step) (back to tree)

events((h | l) | false) = R(t0, l0, v0) ' + el, allReadsOKInd(mem, persisted, h), persisted(l) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: restrict-loc: ⊦ h | l = h.filter(λ vpe. vpe._1.l = l); restrict-vp: ⊦ h | b = h.filter(λ vpe. vpe._2 ↔ b); restrict-events: ⊦ events(h) = h.map(λ vpe. vpe._1); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list

Proof step 3 from proof allReadsOK-ind-no-initial-unpersisted-read (previous step) (back to tree)

h.filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2).map(._1) = R(t0, l0, v0) ' + el, allReadsOKInd(mem, persisted, h), persisted(l) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: h

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ [].filter(p) = []; (s) ⊦ [].map(f) = []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a ' ≠ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 4 from proof allReadsOK-ind-no-initial-unpersisted-read (previous step) (back to tree)

(vpe + h0).filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2).map(._1) = R(t0, l0, v0) ' + el, allReadsOKInd(mem, persisted, vpe + h0), persisted(l), ∀ l, t0, l0, v0, el, mem, persisted. ¬(h0.filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2).map(._1) = R(t0, l0, v0) ' + el ∧ allReadsOKInd(mem, persisted, h0) ∧ persisted(l)) ⊦

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 ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ a + x = a ' + x; (s) ⊦ (x + y).map(f) = x.map(f) + y.map(f); (s) ⊦ (x + y).filter(p) = x.filter(p) + y.filter(p); (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s)

Proof step 5 from proof allReadsOK-ind-no-initial-unpersisted-read (previous step) (back to tree)

(vpe._1.l = l ⊃ vpe ';[]).filter(λ vpe. ¬ vpe._2).map(._1) + h0.filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2).map(._1) = R(t0, l0, v0) ' + el, allReadsOKInd(mem, persisted, vpe ' + h0), persisted(l), ∀ l1, t, l2, v, el0, mem0, persisted0. h0.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2).map(._1) ≠ R(t, l2, v) ' + el0 ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ persisted0(l1), R(t0, l0, v0) ∈ (vpe._1.l = l ⊃ vpe ';[]).filter(λ vpe. ¬ vpe._2).map(._1) ∨ R(t0, l0, v0) ∈ h0.filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2).map(._1) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

term: vpe._1

Additional information

Used terms: e, boolvar

Proof step 6 from proof allReadsOK-ind-no-initial-unpersisted-read (previous step) (back to tree)

(e.l = l ⊃ vpe ';[]).filter(λ vpe. ¬ vpe._2).map(._1) + h0.filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2).map(._1) = R(t0, l0, v0) ' + el, vpe = (e, boolvar), allReadsOKInd(mem, persisted, vpe ' + h0), persisted(l), ∀ l1, t, l2, v, el0, mem0, persisted0. h0.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2).map(._1) ≠ R(t, l2, v) ' + el0 ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ persisted0(l1), R(t0, l0, v0) ∈ (e.l = l ⊃ vpe ';[]).filter(λ vpe. ¬ vpe._2).map(._1) ∨ R(t0, l0, v0) ∈ h0.filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2).map(._1) ⊦

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 ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list

Proof step 7 from proof allReadsOK-ind-no-initial-unpersisted-read (previous step) (back to tree)

(e.l = l ⊃ (e, boolvar) ';[]).filter(λ vpe. ¬ vpe._2).map(._1) + h0.filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2).map(._1) = R(t0, l0, v0) ' + el, allReadsOKInd(mem, persisted, (e, boolvar) ' + h0), persisted(l), ∀ l1, t, l2, v, el0, mem0, persisted0. h0.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2).map(._1) ≠ R(t, l2, v) ' + el0 ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ persisted0(l1), R(t0, l0, v0) ∈ (e.l = l ⊃ (e, boolvar) ';[]).filter(λ vpe. ¬ vpe._2).map(._1) ∨ R(t0, l0, v0) ∈ h0.filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2).map(._1) ⊦

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 1

Additional information

Proof step 8 from proof allReadsOK-ind-no-initial-unpersisted-read (previous step) (back to tree)

(e, boolvar) '.filter(λ vpe. ¬ vpe._2).map(._1) + h0.filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2).map(._1) = R(t0, l0, v0) ' + el, e.l = l, allReadsOKInd(mem, persisted, (e, boolvar) ' + h0), persisted(l), ∀ l1, t, l2, v, el0, mem0, persisted0. h0.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2).map(._1) ≠ R(t, l2, v) ' + el0 ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ persisted0(l1), R(t0, l0, v0) ∈ (e.l = l ⊃ (e, boolvar) ';[]).filter(λ vpe. ¬ vpe._2).map(._1) ∨ R(t0, l0, v0) ∈ h0.filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2).map(._1) ⊦

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 ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ a '.filter(p) = (p(a) ⊃ a ';[]); (s)

Proof step 9 from proof allReadsOK-ind-no-initial-unpersisted-read (previous step) (back to tree)

(¬ boolvar ⊃ (e, false) ';[]).map(._1) + h0.filter(λ vpe. vpe._1.l = e.l).filter(λ vpe. ¬ vpe._2).map(._1) = R(t0, l0, v0) ' + el, allReadsOKInd(mem, persisted, (e, boolvar) ' + h0), persisted(e.l), ∀ l1, t, l2, v, el0, mem0, persisted0. h0.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2).map(._1) ≠ R(t, l2, v) ' + el0 ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ persisted0(l1), R(t0, l0, v0) ∈ (¬ boolvar ⊃ (e, false) ';[]).map(._1) ∨ R(t0, l0, v0) ∈ h0.filter(λ vpe. vpe._1.l = e.l).filter(λ vpe. ¬ vpe._2).map(._1) ⊦

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 1

Additional information

Used simplifier rules: allReadsOKInd-recR: ⊦ allReadsOKInd(mem, persisted, (R(t, l, v), b) ' + h) ↔ v = mem(l) ∧ (b ↔ persisted(l)) ∧ allReadsOKInd(mem, persisted, h); ⊦ a ' + x = a0 ' + y ↔ a = a0 ∧ x = y; (ws) ⊦ a '.map(f) = f(a) '; (s) ⊦ a ∈ a0 ' ↔ a = a0; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ R(t, l, v).l = l; (s)

Proof step 10 from proof allReadsOK-ind-no-initial-unpersisted-read (previous step) (back to tree)

[].map(._1) + h0.filter(λ vpe. vpe._1.l = e.l).filter(λ vpe. ¬ vpe._2).map(._1) = R(t0, l0, v0) ' + el, boolvar, allReadsOKInd(mem, persisted, (e, boolvar) ' + h0), persisted(e.l), ∀ l1, t, l2, v, el0, mem0, persisted0. h0.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2).map(._1) ≠ R(t, l2, v) ' + el0 ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ persisted0(l1), R(t0, l0, v0) ∈ (¬ boolvar ⊃ (e, false) ';[]).map(._1) ∨ R(t0, l0, v0) ∈ h0.filter(λ vpe. vpe._1.l = e.l).filter(λ vpe. ¬ vpe._2).map(._1) ⊦

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 ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ ¬ a ∈ []; (s) ⊦ [].map(f) = []; (s) ⊦ [] + x = x; (s)

Proof step 11 from proof allReadsOK-ind-no-initial-unpersisted-read (previous step) (back to tree)

h0.filter(λ vpe. vpe._1.l = e.l).filter(λ vpe. ¬ vpe._2).map(._1) = R(t0, l0, v0) ' + el, allReadsOKInd(mem, persisted, (e, true) ' + h0), persisted(e.l), ∀ l1, t, l2, v, el0, mem0, persisted0. h0.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2).map(._1) ≠ R(t, l2, v) ' + el0 ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ persisted0(l1) ⊦

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 12 from proof allReadsOK-ind-no-initial-unpersisted-read (previous step) (back to tree)

h0.filter(λ vpe. vpe._1.l = e.l).filter(λ vpe. ¬ vpe._2).map(._1) = R(t0, l0, v0) ' + el, e = R(t1, l, v1), allReadsOKInd(mem, persisted, (e, true) ' + h0), persisted(e.l), ∀ l1, t, l2, v, el0, mem0, persisted0. h0.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2).map(._1) ≠ R(t, l2, v) ' + el0 ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ persisted0(l1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: allReadsOKInd-recR: ⊦ allReadsOKInd(mem, persisted, (R(t, l, v), b) ' + h) ↔ v = mem(l) ∧ (b ↔ persisted(l)) ∧ allReadsOKInd(mem, persisted, h); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ R(t, l, v).l = l; (s)

Proof step 13 from proof allReadsOK-ind-no-initial-unpersisted-read (previous step) (back to tree)

h0.filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2).map(._1) = R(t0, l0, v0) ' + el, allReadsOKInd(mem, persisted, h0), persisted(l), ∀ l1, t, l2, v, el0, mem0, persisted0. h0.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2).map(._1) ≠ R(t, l2, v) ' + el0 ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ persisted0(l1) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 4 terms: l, t0, l0, v0, el, mem, persisted

Additional information

Used terms: l, t0, l0, v0, el, mem, persisted

Proof step 14 from proof allReadsOK-ind-no-initial-unpersisted-read (previous step) (back to tree)

h0.filter(λ vpe. vpe._1.l = e.l).filter(λ vpe. ¬ vpe._2).map(._1) = R(t0, l0, v0) ' + el, e = W(t1, l, v1), allReadsOKInd(mem, persisted, (e, true) ' + h0), persisted(e.l), ∀ l1, t, l2, v, el0, mem0, persisted0. h0.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2).map(._1) ≠ R(t, l2, v) ' + el0 ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ persisted0(l1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: allReadsOKInd-recW: ⊦ allReadsOKInd(mem, persisted, (W(t, l, v), b) ' + h) ↔ allReadsOKInd(mem(l := v), persisted(l := b), h); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ W(t, l, v).l = l; (s)

Proof step 15 from proof allReadsOK-ind-no-initial-unpersisted-read (previous step) (back to tree)

h0.filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2).map(._1) = R(t0, l0, v0) ' + el, allReadsOKInd(mem(l := v1), persisted(l := true), h0), persisted(l), ∀ l1, t, l2, v, el0, mem0, persisted0. h0.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2).map(._1) ≠ R(t, l2, v) ' + el0 ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ persisted0(l1) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 4 terms: l, t0, l0, v0, el, mem(l := v1), persisted(l := true)

Additional information

Used terms: l, t0, l0, v0, el, mem(l := v1), persisted(l := true)

Proof step 16 from proof allReadsOK-ind-no-initial-unpersisted-read (previous step) (back to tree)

[].filter(λ vpe. ¬ vpe._2).map(._1) + h0.filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2).map(._1) = R(t0, l0, v0) ' + el, e.l ≠ l, allReadsOKInd(mem, persisted, (e, boolvar) ' + h0), persisted(l), ∀ l1, t, l2, v, el0, mem0, persisted0. h0.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2).map(._1) ≠ R(t, l2, v) ' + el0 ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ persisted0(l1), R(t0, l0, v0) ∈ (e.l = l ⊃ (e, boolvar) ';[]).filter(λ vpe. ¬ vpe._2).map(._1) ∨ R(t0, l0, v0) ∈ h0.filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2).map(._1) ⊦

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 ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ ¬ a ∈ []; (s) ⊦ [].map(f) = []; (s) ⊦ [].filter(p) = []; (s) ⊦ [] + x = x; (s)

Proof step 17 from proof allReadsOK-ind-no-initial-unpersisted-read (previous step) (back to tree)

h0.filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2).map(._1) = R(t0, l0, v0) ' + el, e.l ≠ l, allReadsOKInd(mem, persisted, (e, boolvar) ' + h0), persisted(l), ∀ l1, t, l2, v, el0, mem0, persisted0. h0.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2).map(._1) ≠ R(t, l2, v) ' + el0 ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ persisted0(l1) ⊦

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 18 from proof allReadsOK-ind-no-initial-unpersisted-read (previous step) (back to tree)

h0.filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2).map(._1) = R(t0, l0, v0) ' + el, e = R(t1, l3, v1), e.l ≠ l, allReadsOKInd(mem, persisted, (e, boolvar) ' + h0), persisted(l), ∀ l1, t, l2, v, el0, mem0, persisted0. h0.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2).map(._1) ≠ R(t, l2, v) ' + el0 ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ persisted0(l1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 19 from proof allReadsOK-ind-no-initial-unpersisted-read (previous step) (back to tree)

h0.filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2).map(._1) = R(t0, l0, v0) ' + el, l3 ≠ l, allReadsOKInd(mem, persisted, h0), persisted(l), ∀ l1, t, l2, v, el0, mem0, persisted0. h0.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2).map(._1) ≠ R(t, l2, v) ' + el0 ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ persisted0(l1) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 5 terms: l, t0, l0, v0, el, mem, persisted

Additional information

Used terms: l, t0, l0, v0, el, mem, persisted

Proof step 20 from proof allReadsOK-ind-no-initial-unpersisted-read (previous step) (back to tree)

h0.filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2).map(._1) = R(t0, l0, v0) ' + el, e = W(t1, l3, v1), e.l ≠ l, allReadsOKInd(mem, persisted, (e, boolvar) ' + h0), persisted(l), ∀ l1, t, l2, v, el0, mem0, persisted0. h0.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2).map(._1) ≠ R(t, l2, v) ' + el0 ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ persisted0(l1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: allReadsOKInd-recW: ⊦ allReadsOKInd(mem, persisted, (W(t, l, v), b) ' + h) ↔ allReadsOKInd(mem(l := v), persisted(l := b), h); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ W(t, l, v).l = l; (s)

Proof step 21 from proof allReadsOK-ind-no-initial-unpersisted-read (previous step) (back to tree)

h0.filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2).map(._1) = R(t0, l0, v0) ' + el, l3 ≠ l, allReadsOKInd(mem(l3 := v1), persisted(l3 := boolvar), h0), persisted(l), ∀ l1, t, l2, v, el0, mem0, persisted0. h0.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2).map(._1) ≠ R(t, l2, v) ' + el0 ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ persisted0(l1) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 5 terms: l, t0, l0, v0, el, mem(l3 := v1), persisted(l3 := boolvar)

Additional information

Used terms: l, t0, l0, v0, el, mem(l3 := v1), persisted(l3 := boolvar)

Proof Tree for allReadsOK-ind-no-initial-unpersisted-read (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