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



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Proof step 1 from proof first-vol-write-size (conclusion) (back to tree)

⊦ firstVolWrite(l, h) < # h ↔ (h |W | l) | false ≠ []

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof first-vol-write-size (previous step) (back to tree)

¬(firstVolWrite(l, h) < # h ↔ (h |W | l) | false ≠ []) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: 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);

Proof step 3 from proof first-vol-write-size (previous step) (back to tree)

¬(firstVolWrite(l, h) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2) ≠ []) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:firstVolWrite-def, seq: ⊦ firstVolWrite(l, h) = h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2), subst:[l, h] → [l, h], <all paths>

Additional information

Used terms: l, h

Proof step 4 from proof first-vol-write-size (previous step) (back to tree)

⊦ firstVolWrite(l, h) = h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)

Lemma firstVolWrite-def

Proof step 5 from proof first-vol-write-size (previous step) (back to tree)

¬ ( h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).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. ¬ ( h0 = h ∧ ¬ ( h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l) .filter (λ vpe. ¬ vpe._2 ) ≠ []))]

Additional information

Used terms: h, λ h0. ∀ h, l. ¬ ( h0 = h ∧ ¬ ( h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2) ≠ []))

Proof step 6 from proof first-vol-write-size (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 7 from proof first-vol-write-size (previous step) (back to tree)

[] = h, ¬ ( h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2) ≠ []) ⊦

Heuristic simplifier applied rule simplifier

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ [].filter(p) = []; (s) ⊦ # [] = 0; (s) ⊦ [].indexWhere(p) = 0; (s)

Proof step 8 from proof first-vol-write-size (previous step) (back to tree)

∀ h1, l. ¬ ( h = h1 ∧ ¬ ( h1.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) < # h1 ↔ h1.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2) ≠ [])) ⊦ ∀ h1, l0. ¬ ( (W(t, l, v), true) ' + h = h1 ∧ ¬ ( h1.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) < # h1 ↔ h1.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).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) ⊦ W(t, l, v).W?; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s)

Proof step 9 from proof first-vol-write-size (previous step) (back to tree)

¬ ( ¬ # h < ((W(t, l, v), true) ' + h).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) ↔ (l = l0 ⊃ (W(t, l0, v), true) ';[]).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) ≠ []), ∀ l1. h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l1 ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2) ≠ [] ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: forall-indexWhere, name:indexWhere-rec2, seq: ⊦ ¬ p(a) → (a ' + x).indexWhere(p) = 1 + x.indexWhere(p), subst:[p, a, x] → [λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2, (W(t, l, v), true), h]; ['a] → [(RWevent, Bool)], <all paths>

Additional information

Used simplifier rules: ⊦ a '.filter(p) = (p(a) ⊃ a ';[]); (s) ⊦ W(t, l, v).l = l; (s) ⊦ W(t, l, v).W?; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2, (W(t, l, v), true), h

Proof step 10 from proof first-vol-write-size (previous step) (back to tree)

⊦ ¬ p(a) → (a ' + x).indexWhere(p) = 1 + x.indexWhere(p)

A spec-lemma

Proof step 11 from proof first-vol-write-size (previous step) (back to tree)

¬ ( ¬ # h < 1 + h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) ↔ (l = l0 ⊃ (W(t, l0, v), true) ';[]).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) ≠ []), ∀ l1. h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l1 ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2) ≠ [] ⊦ (W(t, l, v), true)._1.W? ∧ (W(t, l, v), true)._1.l = l0 ∧ ¬ (W(t, l, v), true)._2

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ W(t, l, v).l = l; (s) ⊦ W(t, l, v).W?; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n + 1 ↔ ¬ n < m; (s)

Proof step 12 from proof first-vol-write-size (previous step) (back to tree)

¬ ( h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) < # h ↔ (l = l0 ⊃ (W(t, l0, v), true) ';[]).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) ≠ []), ∀ l1. h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l1 ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2) ≠ [] ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 2 terms: l0 abort: F discard: T computed: T

Additional information

Used terms: l0

Proof step 13 from proof first-vol-write-size (previous step) (back to tree)

h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) ≠ [], ¬ ( h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) < # h ↔ (l = l0 ⊃ (W(t, l0, v), true) ';[]).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) ≠ []) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 14 from proof first-vol-write-size (previous step) (back to tree)

h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) = [], (l = l0 ⊃ (W(t, l0, v), true) ';[]).filter(λ vpe. ¬ vpe._2) ≠ [], # h ≤ h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) ⊦

User applied rule case distinction

And closed the goal!

Rule argument

pos: left 2

Additional information

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

Proof step 15 from proof first-vol-write-size (previous step) (back to tree)

∀ h1, l. ¬ ( h = h1 ∧ ¬ ( h1.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) < # h1 ↔ h1.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2) ≠ [])) ⊦ ∀ h1, l0. ¬ ( (W(t, l, v), false) ' + h = h1 ∧ ¬ ( h1.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) < # h1 ↔ h1.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).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) ⊦ W(t, l, v).W?; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s)

Proof step 16 from proof first-vol-write-size (previous step) (back to tree)

¬ ( ¬ # h < ((W(t, l, v), false) ' + h).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) ↔ (l = l0 ⊃ (W(t, l0, v), false) ';[]).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) ≠ []), ∀ l1. h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l1 ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2) ≠ [] ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 1

Additional information

Proof step 17 from proof first-vol-write-size (previous step) (back to tree)

(l = l0 ⊃ (W(t, l0, v), false) ';[]).filter(λ vpe. ¬ vpe._2) = [], h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) = [], ¬ # h < ((W(t, l, v), false) ' + h).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2), ∀ l1. h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l1 ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2) ≠ [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 18 from proof first-vol-write-size (previous step) (back to tree)

h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) = [], (l = l0 ⊃ (W(t, l0, v), false) ';[]).filter(λ vpe. ¬ vpe._2) = [], ((W(t, l, v), false) ' + h).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) ≤ # h, ∀ l1. h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l1 ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2) ≠ [] ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: forall-indexWhere, name:indexWhere-rec2, seq: ⊦ ¬ p(a) → (a ' + x).indexWhere(p) = 1 + x.indexWhere(p), subst:[p, a, x] → [λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2, (W(t, l, v), false), h]; ['a] → [(RWevent, Bool)], <all paths>

Additional information

Used simplifier rules: ⊦ W(t, l, v).l = l; (s) ⊦ a '.filter(p) = (p(a) ⊃ a ';[]); (s) ⊦ a ' ≠ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2, (W(t, l, v), false), h

Proof step 19 from proof first-vol-write-size (previous step) (back to tree)

⊦ ¬ p(a) → (a ' + x).indexWhere(p) = 1 + x.indexWhere(p)

A spec-lemma

Proof step 20 from proof first-vol-write-size (previous step) (back to tree)

(l = l0 ⊃ (W(t, l0, v), false) ';[]).filter(λ vpe. ¬ vpe._2) = [], h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) = [], 1 + h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) ≤ # h, ∀ l1. h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l1 ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2) ≠ [] ⊦ (W(t, l, v), false)._1.W? ∧ (W(t, l, v), false)._1.l = l0 ∧ ¬ (W(t, l, v), false)._2

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ [].filter(p) = []; (s) ⊦ W(t, l, v).l = l; (s) ⊦ W(t, l, v).W?; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + 1 ≤ n ↔ m < n; (s)

Proof step 21 from proof first-vol-write-size (previous step) (back to tree)

h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) = [], h ≠ [], l ≠ l0, h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) < # h, ∀ l1. h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l1 ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2) ≠ [] ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 5 terms: l0

Additional information

Used terms: l0

Proof step 22 from proof first-vol-write-size (previous step) (back to tree)

# h < ((W(t, l, v), false) ' + h).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2), ∀ l1. h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l1 ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2) ≠ [], (l = l0 ⊃ (W(t, l0, v), false) ';[]).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) ≠ [] ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: forall-indexWhere, name:indexWhere-rec1, seq: ⊦ p(a) → (a ' + x).indexWhere(p) = 0, subst:[p, a, x] → [λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2, (W(t, l, v), false), h]; ['a] → [(RWevent, Bool)], <all paths>

Additional information

Used simplifier rules: ⊦ ¬ n < 0; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2, (W(t, l, v), false), h

Proof step 23 from proof first-vol-write-size (previous step) (back to tree)

⊦ p(a) → (a ' + x).indexWhere(p) = 0

A spec-lemma

Proof step 24 from proof first-vol-write-size (previous step) (back to tree)

# h < ((W(t, l, v), false) ' + h).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2), ∀ l1. h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l1 ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2) ≠ [], (l = l0 ⊃ (W(t, l0, v), false) ';[]).filter(λ vpe. ¬ vpe._2) ≠ [] ∨ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) ≠ [] ⊦ (W(t, l, v), false)._1.W? ∧ (W(t, l, v), false)._1.l = l0 ∧ ¬ (W(t, l, v), false)._2

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ [].filter(p) = []; (s) ⊦ W(t, l, v).l = l; (s) ⊦ W(t, l, v).W?; (s)

Proof step 25 from proof first-vol-write-size (previous step) (back to tree)

h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) ≠ [], l ≠ l0, # h < ((W(t, l, v), false) ' + h).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2), ∀ l1. h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l1 ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2) ≠ [] ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 4 terms: l0 abort: F discard: T computed: T

Additional information

Used terms: l0

Proof step 26 from proof first-vol-write-size (previous step) (back to tree)

h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) ≠ [], h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) ≠ [], l ≠ l0, # h < ((W(t, l, v), false) ' + h).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s) ⊦ # x = 0 ↔ x = []; (s) trans-Nat: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 27 from proof first-vol-write-size (previous step) (back to tree)

h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) ≠ [], h ≠ [], l ≠ l0, h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) < ((W(t, l, v), false) ' + h).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2), h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) < # h, # h < ((W(t, l, v), false) ' + h).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2), h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) + 1 < ((W(t, l, v), false) ' + h).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: forall-indexWhere, name:indexWhere-rec2, seq: ⊦ ¬ p(a) → (a ' + x).indexWhere(p) = 1 + x.indexWhere(p), subst:[p, a, x] → [λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2, (W(t, l, v), false), h]; ['a] → [(RWevent, Bool)], <all paths>

Additional information

Used simplifier rules: ⊦ ¬ n < n; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ W(t, l, v).l = l; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2, (W(t, l, v), false), h

Proof step 28 from proof first-vol-write-size (previous step) (back to tree)

⊦ ¬ p(a) → (a ' + x).indexWhere(p) = 1 + x.indexWhere(p)

A spec-lemma

Proof step 29 from proof first-vol-write-size (previous step) (back to tree)

∀ h1, l. ¬ ( h = h1 ∧ ¬ ( h1.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) < # h1 ↔ h1.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2) ≠ [])) ⊦ ∀ h1, l0. ¬ ( (R(t, l, v), true) ' + h = h1 ∧ ¬ ( h1.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) < # h1 ↔ h1.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).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 = x; (s) ⊦ ¬ R(t, l, v).W?; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s)

Proof step 30 from proof first-vol-write-size (previous step) (back to tree)

¬ ( ¬ # h < ((R(t, l, v), true) ' + h).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) ≠ []), ∀ l1. h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l1 ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2) ≠ [] ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: forall-indexWhere, name:indexWhere-rec2, seq: ⊦ ¬ p(a) → (a ' + x).indexWhere(p) = 1 + x.indexWhere(p), subst:[p, a, x] → [λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2, (R(t, l, v), true), h]; ['a] → [(RWevent, Bool)], <all paths>

Additional information

Used simplifier rules: ⊦ R(t, l, v).l = l; (s) ⊦ ¬ R(t, l, v).W?; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2, (R(t, l, v), true), h

Proof step 31 from proof first-vol-write-size (previous step) (back to tree)

⊦ ¬ p(a) → (a ' + x).indexWhere(p) = 1 + x.indexWhere(p)

A spec-lemma

Proof step 32 from proof first-vol-write-size (previous step) (back to tree)

¬ ( ¬ # h < 1 + h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) ≠ []), ∀ l1. h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l1 ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2) ≠ [] ⊦ (R(t, l, v), true)._1.W? ∧ (R(t, l, v), true)._1.l = l0 ∧ ¬ (R(t, l, v), true)._2

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ R(t, l, v).W?; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n + 1 ↔ ¬ n < m; (s)

Proof step 33 from proof first-vol-write-size (previous step) (back to tree)

¬ ( h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) ≠ []), ∀ l1. h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l1 ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2) ≠ [] ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 2 terms: l0 abort: F discard: T computed: T

Additional information

Used terms: l0

Proof step 34 from proof first-vol-write-size (previous step) (back to tree)

∀ h1, l. ¬ ( h = h1 ∧ ¬ ( h1.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) < # h1 ↔ h1.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l).filter(λ vpe. ¬ vpe._2) ≠ [])) ⊦ ∀ h1, l0. ¬ ( (R(t, l, v), false) ' + h = h1 ∧ ¬ ( h1.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) < # h1 ↔ h1.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).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 = x; (s) ⊦ ¬ R(t, l, v).W?; (s) ⊦ (a ' + x).filter(p) = (p(a) ⊃ a ';[]) + x.filter(p); (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s)

Proof step 35 from proof first-vol-write-size (previous step) (back to tree)

¬ ( ¬ # h < ((R(t, l, v), false) ' + h).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) ≠ []), ∀ l1. h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l1 ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2) ≠ [] ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: forall-indexWhere, name:indexWhere-rec2, seq: ⊦ ¬ p(a) → (a ' + x).indexWhere(p) = 1 + x.indexWhere(p), subst:[p, a, x] → [λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2, (R(t, l, v), false), h]; ['a] → [(RWevent, Bool)], <all paths>

Additional information

Used simplifier rules: ⊦ R(t, l, v).l = l; (s) ⊦ ¬ R(t, l, v).W?; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2, (R(t, l, v), false), h

Proof step 36 from proof first-vol-write-size (previous step) (back to tree)

⊦ ¬ p(a) → (a ' + x).indexWhere(p) = 1 + x.indexWhere(p)

A spec-lemma

Proof step 37 from proof first-vol-write-size (previous step) (back to tree)

¬ ( ¬ # h < 1 + h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) ≠ []), ∀ l1. h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l1 ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2) ≠ [] ⊦ (R(t, l, v), false)._1.W? ∧ (R(t, l, v), false)._1.l = l0 ∧ ¬ (R(t, l, v), false)._2

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ R(t, l, v).W?; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n + 1 ↔ ¬ n < m; (s)

Proof step 38 from proof first-vol-write-size (previous step) (back to tree)

¬ ( h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l0 ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l0).filter(λ vpe. ¬ vpe._2) ≠ []), ∀ l1. h.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l1 ∧ ¬ vpe._2) < # h ↔ h.filter(λ vpe. vpe._1.W?).filter(λ vpe. vpe._1.l = l1).filter(λ vpe. ¬ vpe._2) ≠ [] ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 2 terms: l0 abort: F discard: T computed: T

Additional information

Used terms: l0

Proof Tree for first-vol-write-size (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