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

⊦ firstVolWrite(l, h) < # h → (∃ t, v. h = h.take(firstVolWrite(l, h)) + ((W(t, l, v), false) + h.drop(firstVolWrite(l, h) + 1)))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

firstVolWrite(l, h) < # h ⊦ ∃ t, v. h = h.take(firstVolWrite(l, h)) + ((W(t, l, v), false) + h.drop(firstVolWrite(l, h) + 1))

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 a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a + x = a ' + x; (s)

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

h ≠ [], firstVolWrite(l, h) < # h ⊦ ∃ t, v. h = h.take(firstVolWrite(l, h)) + (W(t, l, v), false) ' + h.drop(firstVolWrite(l, h) + 1)

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 [], +;;

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

vpe + h0 ≠ [], firstVolWrite(l, vpe + h0) < #(vpe + h0), ∀ l. h0 ≠ [] ∧ firstVolWrite(l, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1)) ⊦ ∃ t, v. vpe + h0 = (vpe + h0).take(firstVolWrite(l, vpe + h0)) + (W(t, l, v), false) ' + (vpe + h0).drop(firstVolWrite(l, vpe + h0) + 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 c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (a ' + x).drop(n + 1) = x.drop(n); (s) ⊦ a + x = a ' + x; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

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

firstVolWrite(l, vpe ' + h0) ≤ # h0, ∀ l. h0 ≠ [] ∧ firstVolWrite(l, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1)) ⊦ ∃ t, v. vpe ' + h0 = (vpe ' + h0).take(firstVolWrite(l, vpe ' + h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, vpe ' + h0))

User applied rule all left

And obtained the following new goals

Rule argument

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

Additional information

Used terms: l

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

firstVolWrite(l, vpe ' + h0) ≤ # h0, h0 ≠ [] ∧ firstVolWrite(l, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1)), ∀ l. h0 ≠ [] ∧ firstVolWrite(l, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1)) ⊦ ∃ t, v. vpe ' + h0 = (vpe ' + h0).take(firstVolWrite(l, vpe ' + h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, vpe ' + h0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

firstVolWrite(l, vpe ' + h0) ≤ # h0, h0 ≠ [] ∧ firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l. h0 ≠ [] ∧ firstVolWrite(l, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1)) ⊦ ∃ t, v. vpe ' + h0 = (vpe ' + h0).take(firstVolWrite(l, vpe ' + h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, vpe ' + h0))

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: h0 = []

Additional information

Used formulas: h0 = []

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

h0 = [], firstVolWrite(l, vpe ' + h0) ≤ # h0, h0 ≠ [] ∧ firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l. h0 ≠ [] ∧ firstVolWrite(l, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1)) ⊦ ∃ t, v. vpe ' + h0 = (vpe ' + h0).take(firstVolWrite(l, vpe ' + h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, vpe ' + h0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a ' = a0 ' ↔ a = a0; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x + [] = x; (s) ⊦ [] + x = x; (s) ⊦ x.drop(0) = x; (s) ⊦ x.take(0) = []; (s) ⊦ m ≤ 0 ↔ m = 0; (s) ⊦ # [] = 0; (s)

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

firstVolWrite(l, vpe ') = 0 ⊦ ∃ t, v. vpe = (W(t, l, v), false)

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, vpe '], <all paths>

Additional information

Used terms: l, vpe '

Proof step 10 from proof first-vol-write-split (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 11 from proof first-vol-write-split (previous step) (back to tree)

vpe '.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) = 0 ⊦ ∃ t, v. vpe = (W(t, l, v), false)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a '.indexWhere(p) = (p(a) ⊃ 0;1); (s)

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

vpe._1.W? ⊦ vpe._2, ∃ t, v. vpe = (W(t, vpe._1.l, v), false)

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

¬ boolvar, vpe = (e, boolvar), e.W? ⊦ ∃ t, v. vpe = (W(t, e.l, v), false)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

e.W? ⊦ ∃ t, v. e = W(t, e.l, v)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: RWevent, name:ex-dotWque, seq: ⊦ e.W? ↔ (∃ t, l, v. e = W(t, l, v)), subst:[e] → [e], <all paths>

Additional information

Used terms: e

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

⊦ e.W? ↔ (∃ t, l, v. e = W(t, l, v))

A spec-lemma

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

∃ t, l, v. e = W(t, l, v) ⊦ ∃ t, v. e = W(t, e.l, v)

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)

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

⊦ ∃ t0, v0. W(t, l, v) = W(t0, l, v0)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: t, v

Additional information

Used terms: t, v

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

h0 ≠ [], firstVolWrite(l, vpe ' + h0) ≤ # h0, h0 ≠ [] ∧ firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l. h0 ≠ [] ∧ firstVolWrite(l, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1)) ⊦ ∃ t, v. vpe ' + h0 = (vpe ' + h0).take(firstVolWrite(l, vpe ' + h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, vpe ' + h0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

h0 ≠ [], firstVolWrite(l, vpe ' + h0) ≤ # h0, firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t, v. vpe ' + h0 = (vpe ' + h0).take(firstVolWrite(l, vpe ' + h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, vpe ' + h0))

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: vpe

Additional information

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

vpe = (e, b), h0 ≠ [], firstVolWrite(l, vpe ' + h0) ≤ # h0, firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t, v. vpe ' + h0 = (vpe ' + h0).take(firstVolWrite(l, vpe ' + h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, vpe ' + h0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

h0 ≠ [], firstVolWrite(l, (e, b) ' + h0) ≤ # h0, firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t, v. (e, b) ' + h0 = ((e, b) ' + h0).take(firstVolWrite(l, (e, b) ' + h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, (e, b) ' + 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 22 from proof first-vol-write-split (previous step) (back to tree)

e = R(t0, l1, v0), h0 ≠ [], firstVolWrite(l, (e, b) ' + h0) ≤ # h0, firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t, v. (e, b) ' + h0 = ((e, b) ' + h0).take(firstVolWrite(l, (e, b) ' + h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, (e, b) ' + h0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

h0 ≠ [], firstVolWrite(l, (R(t0, l1, v0), b) ' + h0) ≤ # h0, firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t, v. (R(t0, l1, v0), b) ' + h0 = ((R(t0, l1, v0), b) ' + h0).take(firstVolWrite(l, (R(t0, l1, v0), b) ' + h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, (R(t0, l1, v0), b) ' + h0))

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, (R(t0, l1, v0), b) ' + h0], <all paths>

Additional information

Used terms: l, (R(t0, l1, v0), b) ' + h0

Proof step 24 from proof first-vol-write-split (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 25 from proof first-vol-write-split (previous step) (back to tree)

h0 ≠ [], ((R(t0, l1, v0), b) ' + h0).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) ≤ # h0, firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t, v. (R(t0, l1, v0), b) ' + h0 = ((R(t0, l1, v0), b) ' + h0).take(((R(t0, l1, v0), b) ' + h0).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)) + (W(t, l, v), false) ' + h0.drop(((R(t0, l1, v0), b) ' + h0).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2))

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: 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 = l ∧ ¬ vpe._2, (R(t0, l1, v0), b), h0]; ['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 = l ∧ ¬ vpe._2, (R(t0, l1, v0), b), h0

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

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

A spec-lemma

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

h0 ≠ [], 1 + h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) ≤ # h0, firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t, v. (R(t0, l1, v0), b) ' + h0 = ((R(t0, l1, v0), b) ' + h0).take(1 + h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)) + (W(t, l, v), false) ' + h0.drop(1 + h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)), (R(t0, l1, v0), b)._1.W? ∧ (R(t0, l1, v0), b)._1.l = l ∧ ¬ (R(t0, l1, v0), b)._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 a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ # x = # z → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ a + x = a ' + x; (s) ⊦ (a ' + x).take(n + 1) = a + x.take(n); (s) ⊦ ¬ R(t, l, v).W?; (s) ⊦ m + 1 ≤ n ↔ m < n; (s)

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

h0 ≠ [], h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) < # h0, firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t1, v1. h0 = h0.take(h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)) + (W(t1, l, v1), false) ' + h0.drop(1 + h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ 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, h0], <all paths>

Additional information

Used terms: l, h0

Proof step 29 from proof first-vol-write-split (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 30 from proof first-vol-write-split (previous step) (back to tree)

h0 ≠ [], h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) < # h0, h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) < # h0 → h0 = h0.take(h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)) + (W(t, l, v), false) ' + h0.drop(h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) + 1), ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t1, v1. h0 = h0.take(h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)) + (W(t1, l, v1), false) ' + h0.drop(1 + h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

h0 = h0.take(h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)) + (W(t, l, v), false) ' + h0.drop(h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) + 1), h0 ≠ [], h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) < # h0, ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t1, v1. h0 = h0.take(h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)) + (W(t1, l, v1), false) ' + h0.drop(1 + h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2))

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: t, v

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: t, v

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

e = W(t0, l1, v0), h0 ≠ [], firstVolWrite(l, (e, b) ' + h0) ≤ # h0, firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t, v. (e, b) ' + h0 = ((e, b) ' + h0).take(firstVolWrite(l, (e, b) ' + h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, (e, b) ' + h0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

h0 ≠ [], firstVolWrite(l, (W(t0, l1, v0), b) ' + h0) ≤ # h0, firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t, v. (W(t0, l1, v0), b) ' + h0 = ((W(t0, l1, v0), b) ' + h0).take(firstVolWrite(l, (W(t0, l1, v0), b) ' + h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, (W(t0, l1, v0), b) ' + h0))

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: ¬ b ∧ l1 = l

Additional information

Used formulas: ¬ b ∧ l1 = l

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

¬ b, l1 = l, h0 ≠ [], firstVolWrite(l, (W(t0, l1, v0), b) ' + h0) ≤ # h0, firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t, v. (W(t0, l1, v0), b) ' + h0 = ((W(t0, l1, v0), b) ' + h0).take(firstVolWrite(l, (W(t0, l1, v0), b) ' + h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, (W(t0, l1, v0), b) ' + h0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

h0 ≠ [], firstVolWrite(l, (W(t0, l, v0), false) ' + h0) ≤ # h0, firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t, v. (W(t0, l, v0), false) ' + h0 = ((W(t0, l, v0), false) ' + h0).take(firstVolWrite(l, (W(t0, l, v0), false) ' + h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, (W(t0, l, v0), false) ' + h0))

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, (W(t0, l, v0), false) ' + h0], <all paths>

Additional information

Used terms: l, (W(t0, l, v0), false) ' + h0

Proof step 36 from proof first-vol-write-split (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 37 from proof first-vol-write-split (previous step) (back to tree)

h0 ≠ [], ((W(t0, l, v0), false) ' + h0).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) ≤ # h0, firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t, v. (W(t0, l, v0), false) ' + h0 = ((W(t0, l, v0), false) ' + h0).take(((W(t0, l, v0), false) ' + h0).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)) + (W(t, l, v), false) ' + h0.drop(((W(t0, l, v0), false) ' + h0).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2))

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: indexWhere, name:indexWhere-rec1, seq: ⊦ p(a) → (a ' + x).indexWhere(p) = 0, subst:[p, a, x] → [λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2, (W(t0, l, v0), false), h0]; ['a] → [(RWevent, Bool)], <all paths>

Additional information

Used simplifier rules: ⊦ W(t, l, v).W?; (s) ⊦ 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 = l ∧ ¬ vpe._2, (W(t0, l, v0), false), h0

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

⊦ p(a) → (a ' + x).indexWhere(p) = 0

A spec-lemma

Proof step 39 from proof first-vol-write-split (previous step) (back to tree)

(W(t0, l, v0), false)._1.l = l, h0 ≠ [], (W(t0, l, v0), false)._1.W?, 0 ≤ # h0, firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ (W(t0, l, v0), false)._2, ∃ t, v. (W(t0, l, v0), false) ' + h0 = ((W(t0, l, v0), false) ' + h0).take(0) + (W(t, l, v), false) ' + h0.drop(0)

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 ' ↔ a = a0; (s) ⊦ # y = # y0 → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws) ⊦ [] + x = x; (s) ⊦ x.drop(0) = x; (s) ⊦ x.take(0) = []; (s) ⊦ W(t, l, v).W?; (s) ⊦ 0 ≤ m; (s) ⊦ W(t, l, v).l = l; (s)

Proof step 40 from proof first-vol-write-split (previous step) (back to tree)

h0 ≠ [], firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t1, v1. W(t0, l, v0) = W(t1, l, v1)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: t0, v0

Additional information

Used terms: t0, v0

Proof step 41 from proof first-vol-write-split (previous step) (back to tree)

h0 ≠ [], firstVolWrite(l, (W(t0, l1, v0), b) ' + h0) ≤ # h0, firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t, v. (W(t0, l1, v0), b) ' + h0 = ((W(t0, l1, v0), b) ' + h0).take(firstVolWrite(l, (W(t0, l1, v0), b) ' + h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, (W(t0, l1, v0), b) ' + h0)), ¬ b ∧ l1 = l

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, (W(t0, l1, v0), b) ' + h0], <all paths>

Additional information

Used terms: l, (W(t0, l1, v0), b) ' + h0

Proof step 42 from proof first-vol-write-split (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 43 from proof first-vol-write-split (previous step) (back to tree)

h0 ≠ [], ((W(t0, l1, v0), b) ' + h0).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) ≤ # h0, firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t, v. (W(t0, l1, v0), b) ' + h0 = ((W(t0, l1, v0), b) ' + h0).take(((W(t0, l1, v0), b) ' + h0).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)) + (W(t, l, v), false) ' + h0.drop(((W(t0, l1, v0), b) ' + h0).indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)), ¬ b ∧ l1 = l

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: 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 = l ∧ ¬ vpe._2, (W(t0, l1, v0), b), h0]; ['a] → [(RWevent, Bool)], <all paths>

Additional information

Used simplifier rules: ⊦ W(t, l, v).W?; (s) ⊦ 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 = l ∧ ¬ vpe._2, (W(t0, l1, v0), b), h0

Proof step 44 from proof first-vol-write-split (previous step) (back to tree)

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

A spec-lemma

Proof step 45 from proof first-vol-write-split (previous step) (back to tree)

h0 ≠ [], 1 + h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) ≤ # h0, firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t, v. (W(t0, l1, v0), b) ' + h0 = ((W(t0, l1, v0), b) ' + h0).take(1 + h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)) + (W(t, l, v), false) ' + h0.drop(1 + h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)), ¬ b ∧ l1 = l, (W(t0, l1, v0), b)._1.W? ∧ (W(t0, l1, v0), b)._1.l = l ∧ ¬ (W(t0, l1, v0), b)._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 a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ # x = # z → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ a + x = a ' + x; (s) ⊦ (a ' + x).take(n + 1) = a + x.take(n); (s) ⊦ W(t, l, v).l = l; (s) ⊦ W(t, l, v).W?; (s) ⊦ m + 1 ≤ n ↔ m < n; (s)

Proof step 46 from proof first-vol-write-split (previous step) (back to tree)

h0 ≠ [], h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) < # h0, firstVolWrite(l, h0) < # h0 → h0 = h0.take(firstVolWrite(l, h0)) + (W(t, l, v), false) ' + h0.drop(firstVolWrite(l, h0) + 1), ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t1, v1. h0 = h0.take(h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)) + (W(t1, l, v1), false) ' + h0.drop(1 + h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)), l1 = l ∧ ¬ b, ¬ b ∧ l1 = l

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, h0], <all paths>

Additional information

Used terms: l, h0

Proof step 47 from proof first-vol-write-split (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 48 from proof first-vol-write-split (previous step) (back to tree)

h0 ≠ [], h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) < # h0, h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) < # h0 → h0 = h0.take(h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)) + (W(t, l, v), false) ' + h0.drop(h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) + 1), ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t1, v1. h0 = h0.take(h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)) + (W(t1, l, v1), false) ' + h0.drop(1 + h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)), ¬ b ∧ l1 = l, l1 = l ∧ ¬ b

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 49 from proof first-vol-write-split (previous step) (back to tree)

h0 = h0.take(h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)) + (W(t, l, v), false) ' + h0.drop(h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) + 1), h0 ≠ [], h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2) < # h0, ∀ l0. firstVolWrite(l0, h0) < # h0 → (∃ t, v. h0 = h0.take(firstVolWrite(l0, h0)) + (W(t, l0, v), false) ' + h0.drop(firstVolWrite(l0, h0) + 1)) ⊦ ∃ t1, v1. h0 = h0.take(h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)) + (W(t1, l, v1), false) ' + h0.drop(1 + h0.indexWhere(λ vpe. vpe._1.W? ∧ vpe._1.l = l ∧ ¬ vpe._2)), l1 = l ∧ ¬ b, ¬ b ∧ l1 = l

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: t, v

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: t, v

Proof Tree for first-vol-write-split (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