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



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

⊦ ((h | l) | t) | false = [] → ((persistDependentReads(l, h) | l) | t) | false = []

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

((h | l) | t) | false = [], ((persistDependentReads(l, h) | l) | t) | false ≠ [] ⊦

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-thread: ⊦ h | t = h.filter(λ vpe. vpe._1.t = t); restrict-vp: ⊦ h | b = h.filter(λ vpe. vpe._2 ↔ b);

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

h.filter(λ vpe. vpe._1.l = l).filter(λ vpe. vpe._1.t = t).filter(λ vpe. ¬ vpe._2) = [], persistDependentReads(l, h).filter(λ vpe. vpe._1.l = l).filter(λ vpe. vpe._1.t = t).filter(λ vpe. ¬ vpe._2) ≠ [] ⊦

User applied rule induction

And obtained the following new goals

Rule argument

spec: <toplevel>, name:hist-ind, seq:hPred([]), ∀ t, l, v, h. hPred(h) → hPred((W(t, l, v), true) ' + h), ∀ t, l, v, h. hPred(h) → hPred((W(t, l, v), false) ' + h), ∀ t, l, v, h. hPred(h) → hPred((R(t, l, v), true) ' + h), ∀ t, l, v, h. hPred(h) → hPred((R(t, l, v), false) ' + h) ⊦ hPred(h), subst:[h, hPred] → [h, λ h0. ∀ h, l, t. ¬ ( h0 = h ∧ h.filter(λ vpe. vpe._1.l = l).filter(λ vpe. vpe._1.t = t) .filter (λ vpe. ¬ vpe._2 ) = [] ∧ persistDependentReads(l, h).filter(λ vpe. vpe._1.l = l) .filter (λ vpe. vpe._1.t = t ) .filter (λ vpe. ¬ vpe._2 ) ≠ [])]

Additional information

Used terms: h, λ h0. ∀ h, l, t. ¬ ( h0 = h ∧ h.filter(λ vpe. vpe._1.l = l).filter(λ vpe. vpe._1.t = t).filter(λ vpe. ¬ vpe._2) = [] ∧ persistDependentReads(l, h).filter(λ vpe. vpe._1.l = l) .filter (λ vpe. vpe._1.t = t ) .filter (λ vpe. ¬ vpe._2 ) ≠ [])

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

hPred([]), ∀ t, l, v, h. hPred(h) → hPred((W(t, l, v), true) ' + h), ∀ t, l, v, h. hPred(h) → hPred((W(t, l, v), false) ' + h), ∀ t, l, v, h. hPred(h) → hPred((R(t, l, v), true) ' + h), ∀ t, l, v, h. hPred(h) → hPred((R(t, l, v), false) ' + h) ⊦ hPred(h)

Lemma hist-ind

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

h.filter(λ vpe. vpe._1.l = l).filter(λ vpe. vpe._1.t = t).filter(λ vpe. ¬ vpe._2) = [], [] = h, persistDependentReads(l, h).filter(λ vpe. vpe._1.l = l).filter(λ vpe. vpe._1.t = t).filter(λ vpe. ¬ vpe._2) ≠ [] ⊦

Heuristic simplifier applied rule simplifier

And closed the goal!

Rule argument

Additional information

Used simplifier rules: persistDependentReads-base: ⊦ persistDependentReads(l, []) = []; ⊦ [].filter(p) = []; (s)

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

∀ h1, l, t. ¬ ( h = h1 ∧ h1.filter(λ vpe. vpe._1.l = l).filter(λ vpe. vpe._1.t = t).filter(λ vpe. ¬ vpe._2) = [] ∧ persistDependentReads(l, h1).filter(λ vpe. vpe._1.l = l).filter(λ vpe. vpe._1.t = t).filter(λ vpe. ¬ vpe._2) ≠ []) ⊦ ∀ h1, l0, t0. ¬ ( (W(t, l, v), true) ' + h = h1 ∧ h1.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [] ∧ persistDependentReads(l0, h1).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) ≠ [])

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 ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ (x + y).filter(p) = x.filter(p) + y.filter(p); (s) ⊦ W(t, l, v).l = l; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s)

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

(l = l0 ⊃ (W(t, l0, v), true) ';[]).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], persistDependentReads(l0, (W(t, l, v), true) ' + h).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) ≠ [], ∀ l1, t1. h.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ persistDependentReads(l1, h).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) = [] ⊦

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: persistDependentReads-nextwrite: ⊦ persistDependentReads(l, (W(t, l, v), b) ' + h) = (W(t, l, v), b) ' + h; ⊦ a '.filter(p) = (p(a) ⊃ a ';[]); (s) ⊦ W(t, l, v).l = l; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s) ⊦ W(t, l, v).t = t; (s) ⊦ (x + y).filter(p) = x.filter(p) + y.filter(p); (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

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

h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], [].filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], persistDependentReads(l0, (W(t, l, v), true) ' + h).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) ≠ [], l ≠ l0, ∀ l1, t1. h.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ persistDependentReads(l1, h).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) = [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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

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

h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], persistDependentReads(l0, h).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) ≠ [], l ≠ l0, ∀ l1, t1. h.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ persistDependentReads(l1, h).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) = [] ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 4 terms: l0, t0

Additional information

Used terms: l0, t0

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

∀ h1, l, t. ¬ ( h = h1 ∧ h1.filter(λ vpe. vpe._1.l = l).filter(λ vpe. vpe._1.t = t).filter(λ vpe. ¬ vpe._2) = [] ∧ persistDependentReads(l, h1).filter(λ vpe. vpe._1.l = l).filter(λ vpe. vpe._1.t = t).filter(λ vpe. ¬ vpe._2) ≠ []) ⊦ ∀ h1, l0, t0. ¬ ( (W(t, l, v), false) ' + h = h1 ∧ h1.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [] ∧ persistDependentReads(l0, h1).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) ≠ [])

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 ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ (x + y).filter(p) = x.filter(p) + y.filter(p); (s) ⊦ W(t, l, v).l = l; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s)

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

(l = l0 ⊃ (W(t, l0, v), false) ';[]).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], persistDependentReads(l0, (W(t, l, v), false) ' + h).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) ≠ [], ∀ l1, t1. h.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ persistDependentReads(l1, h).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) = [] ⊦

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: persistDependentReads-nextwrite: ⊦ persistDependentReads(l, (W(t, l, v), b) ' + h) = (W(t, l, v), b) ' + h; ⊦ a '.filter(p) = (p(a) ⊃ a ';[]); (s) ⊦ W(t, l, v).l = l; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s) ⊦ W(t, l, v).t = t; (s) ⊦ (x + y).filter(p) = x.filter(p) + y.filter(p); (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

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

h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], [].filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], persistDependentReads(l0, (W(t, l, v), false) ' + h).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) ≠ [], l ≠ l0, ∀ l1, t1. h.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ persistDependentReads(l1, h).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) = [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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

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

h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], persistDependentReads(l0, h).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) ≠ [], l ≠ l0, ∀ l1, t1. h.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ persistDependentReads(l1, h).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) = [] ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 4 terms: l0, t0

Additional information

Used terms: l0, t0

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

∀ h1, l, t. ¬ ( h = h1 ∧ h1.filter(λ vpe. vpe._1.l = l).filter(λ vpe. vpe._1.t = t).filter(λ vpe. ¬ vpe._2) = [] ∧ persistDependentReads(l, h1).filter(λ vpe. vpe._1.l = l).filter(λ vpe. vpe._1.t = t).filter(λ vpe. ¬ vpe._2) ≠ []) ⊦ ∀ h1, l0, t0. ¬ ( (R(t, l, v), true) ' + h = h1 ∧ h1.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [] ∧ persistDependentReads(l0, h1).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) ≠ [])

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 ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ (x + y).filter(p) = x.filter(p) + y.filter(p); (s) ⊦ R(t, l, v).l = l; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s)

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

(l = l0 ⊃ (R(t, l0, v), true) ';[]).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], persistDependentReads(l0, (R(t, l, v), true) ' + h).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) ≠ [], ∀ l1, t1. h.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ persistDependentReads(l1, h).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) = [] ⊦

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 1

Additional information

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

h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], (R(t, l0, v), true) '.filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], l = l0, persistDependentReads(l0, (R(t, l, v), true) ' + h).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) ≠ [], ∀ l1, t1. h.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ persistDependentReads(l1, h).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) = [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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 + y = [] ↔ x = [] ∧ y = []; (s) ⊦ (x + y).filter(p) = x.filter(p) + y.filter(p); (s) ⊦ R(t, l, v).t = t; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s) ⊦ R(t, l, v).l = l; (s) ⊦ a '.filter(p) = (p(a) ⊃ a ';[]); (s)

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

h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], (t = t0 ⊃ (R(t0, l0, v), true) ';[]).filter(λ vpe. ¬ vpe._2) = [], persistDependentReads(l0, h).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) ≠ [], ∀ l1, t1. h.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ persistDependentReads(l1, h).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) = [] ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 4 terms: l0, t0

Additional information

Used terms: l0, t0

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

h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], [].filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], persistDependentReads(l0, (R(t, l, v), true) ' + h).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) ≠ [], l ≠ l0, ∀ l1, t1. h.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ persistDependentReads(l1, h).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) = [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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 = 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) ⊦ [].filter(p) = []; (s)

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

h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], persistDependentReads(l0, h).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) ≠ [], l ≠ l0, ∀ l1, t1. h.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ persistDependentReads(l1, h).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) = [] ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 4 terms: l0, t0

Additional information

Used terms: l0, t0

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

∀ h1, l, t. ¬ ( h = h1 ∧ h1.filter(λ vpe. vpe._1.l = l).filter(λ vpe. vpe._1.t = t).filter(λ vpe. ¬ vpe._2) = [] ∧ persistDependentReads(l, h1).filter(λ vpe. vpe._1.l = l).filter(λ vpe. vpe._1.t = t).filter(λ vpe. ¬ vpe._2) ≠ []) ⊦ ∀ h1, l0, t0. ¬ ( (R(t, l, v), false) ' + h = h1 ∧ h1.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [] ∧ persistDependentReads(l0, h1).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) ≠ [])

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 ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ (x + y).filter(p) = x.filter(p) + y.filter(p); (s) ⊦ R(t, l, v).l = l; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s)

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

(l = l0 ⊃ (R(t, l0, v), false) ';[]).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], persistDependentReads(l0, (R(t, l, v), false) ' + h).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) ≠ [], ∀ l1, t1. h.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ persistDependentReads(l1, h).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) = [] ⊦

Heuristic if-then-else split applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 1

Additional information

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

h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], (R(t, l0, v), false) '.filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], l = l0, persistDependentReads(l0, (R(t, l, v), false) ' + h).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) ≠ [], ∀ l1, t1. h.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ persistDependentReads(l1, h).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) = [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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 + y = [] ↔ x = [] ∧ y = []; (s) ⊦ (x + y).filter(p) = x.filter(p) + y.filter(p); (s) ⊦ R(t, l, v).t = t; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s) ⊦ R(t, l, v).l = l; (s) ⊦ a '.filter(p) = (p(a) ⊃ a ';[]); (s)

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

h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], (t = t0 ⊃ (R(t0, l0, v), false) ';[]).filter(λ vpe. ¬ vpe._2) = [], ∀ l1, t1. h.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ persistDependentReads(l1, h).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) = [] ⊦ (t = t0 ⊃ (R(t0, l0, v), true) ';[]).filter(λ vpe. ¬ vpe._2) = [] ∧ persistDependentReads(l0, h).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = []

Heuristic nonsplitting cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: t = t0

Additional information

Used simplifier rules: ⊦ a '.filter(p) = (p(a) ⊃ a ';[]); (s) ⊦ a ' ≠ []; (s) Used formulas: t = t0

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

(t = t0 ⊃ (R(t0, l0, v), false) ';[]).filter(λ vpe. ¬ vpe._2) = [], h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], t ≠ t0, ∀ l1, t1. h.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ persistDependentReads(l1, h).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) = [] ⊦ (t = t0 ⊃ (R(t0, l0, v), true) ';[]).filter(λ vpe. ¬ vpe._2) = [] ∧ persistDependentReads(l0, h).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = []

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], persistDependentReads(l0, h).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) ≠ [], t ≠ t0, ∀ l1, t1. h.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ persistDependentReads(l1, h).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) = [] ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 4 terms: l0, t0

Additional information

Used terms: l0, t0

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

h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], [].filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], persistDependentReads(l0, (R(t, l, v), false) ' + h).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) ≠ [], l ≠ l0, ∀ l1, t1. h.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ persistDependentReads(l1, h).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) = [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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 = 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) ⊦ [].filter(p) = []; (s)

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

h.filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) = [], persistDependentReads(l0, h).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. vpe._1.t = t0).filter(λ vpe. ¬ vpe._2) ≠ [], l ≠ l0, ∀ l1, t1. h.filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ persistDependentReads(l1, h).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. vpe._1.t = t1).filter(λ vpe. ¬ vpe._2) = [] ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 4 terms: l0, t0

Additional information

Used terms: l0, t0

Proof Tree for persistDependentReads-restrict-empty (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