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Proof step 1 from proof abs-persistFO-weak (conclusion) (back to tree)

⊦ allReadsOK(h) ∧ isPersistFO(n, p(l)) ∧ abs(h, rs, p) → abs(h, rs, p(l := p(l) -p n))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof abs-persistFO-weak (previous step) (back to tree)

abs(h, rs, p), allReadsOK(h), isPersistFO(n, p(l)), ¬ abs(h, rs, p(l := p(l) -p n)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:abs-def-reduce, seq:⊦ abs(h, rs, p) → ( abs(h0, rs0, p0) ↔ (∀ l. absloc(events((h | l) | false), rs(l), p(l)) → absloc(events((h0 | l) | false), rs0(l), p0(l)))), subst:[rs0, h0, rs, p, h, p0] → [rs, h, rs, p, h, p(l := p(l) -p n)], <all paths>

Additional information

Used terms: p(l := p(l) -p n), rs, h, rs, p, h

Proof step 3 from proof abs-persistFO-weak (previous step) (back to tree)

⊦ abs(h, rs, p) → (abs(h0, rs0, p0) ↔ (∀ l. absloc(events((h | l) | false), rs(l), p(l)) → absloc(events((h0 | l) | false), rs0(l), p0(l))))

Lemma abs-def-reduce

Proof step 4 from proof abs-persistFO-weak (previous step) (back to tree)

abs(h, rs, p), allReadsOK(h), isPersistFO(n, p(l)), ∃ l0. ¬(absloc(events((h | l0) | false), rs(l0), p(l0)) → absloc(events((h | l0) | false), rs(l0), p(l := p(l) -p n)(l0))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 5 from proof abs-persistFO-weak (previous step) (back to tree)

abs(h, rs, p), absloc(events((h | l0) | false), rs(l0), p(l0)), allReadsOK(h), isPersistFO(n, p(l)), ¬ absloc(events((h | l0) | false), rs(l0), (l = l0 ⊃ p(l0) -p n;p(l0))) ⊦

Heuristic nonsplitting cut applied rule cut formula

And obtained the following new goals

Rule argument

fma: l = l0

Additional information

Used formulas: l = l0

Proof step 6 from proof abs-persistFO-weak (previous step) (back to tree)

l = l0, abs(h, rs, p), absloc(events((h | l0) | false), rs(l0), p(l0)), allReadsOK(h), isPersistFO(n, p(l)), ¬ absloc(events((h | l0) | false), rs(l0), (l = l0 ⊃ p(l0) -p n;p(l0))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 7 from proof abs-persistFO-weak (previous step) (back to tree)

abs(h, rs, p), absloc(events((h | l0) | false), rs(l0), p(l0)), allReadsOK(h), isPersistFO(n, p(l0)), ¬ absloc(events((h | l0) | false), rs(l0), p(l0) -p n) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:absloc-persistFO, seq:⊦ (el = [] ∨ ¬ el.head.R?) ∧ absloc(el, rset, fowl) ∧ isPersistFO(n, fowl) → absloc(el, rset, fowl -p n), subst:[el, rset, fowl, n] → [events((h | l0) | false), rs(l0), p(l0), n], <all paths>

Additional information

Used terms: events((h | l0) | false), rs(l0), p(l0), n

Proof step 8 from proof abs-persistFO-weak (previous step) (back to tree)

⊦ (el = [] ∨ ¬ el.head.R?) ∧ absloc(el, rset, fowl) ∧ isPersistFO(n, fowl) → absloc(el, rset, fowl -p n)

Lemma absloc-persistFO

Proof step 9 from proof abs-persistFO-weak (previous step) (back to tree)

abs(h, rs, p), absloc(events((h | l0) | false), rs(l0), p(l0)), allReadsOK(h), isPersistFO(n, p(l0)), ¬ absloc(events((h | l0) | false), rs(l0), p(l0) -p n) ⊦ (events((h | l0) | false) = [] ∨ ¬ events((h | l0) | false).head.R?) ∧ absloc(events((h | l0) | false), rs(l0), p(l0)) ∧ isPersistFO(n, p(l0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 10 from proof abs-persistFO-weak (previous step) (back to tree)

events((h | l0) | false) ≠ [], events((h | l0) | false).head.R?, abs(h, rs, p), absloc(events((h | l0) | false), rs(l0), p(l0)), allReadsOK(h), isPersistFO(n, p(l0)), ¬ absloc(events((h | l0) | false), rs(l0), p(l0) -p n) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-data, name:elim-add, seq: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail), subst:[x, head, tail] → [events((h | l0) | false), e, el]; ['a] → [RWevent]

Additional information

Used simplifier rules: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail); (elim) Used terms: events((h | l0) | false), e, el

Proof step 11 from proof abs-persistFO-weak (previous step) (back to tree)

events((h | l0) | false) = e + el, events((h | l0) | false) ≠ [], e.R?, abs(h, rs, p), absloc(events((h | l0) | false), rs(l0), p(l0)), allReadsOK(h), isPersistFO(n, p(l0)), ¬ absloc(events((h | l0) | false), rs(l0), p(l0) -p n) ⊦

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

Proof step 12 from proof abs-persistFO-weak (previous step) (back to tree)

events((h | l0) | false) = e ' + el, e.R?, abs(h, rs, p), absloc(e ' + el, rs(l0), p(l0)), allReadsOK(h), isPersistFO(n, p(l0)), ¬ absloc(e ' + el, rs(l0), p(l0) -p n) ⊦

User applied rule constructor cut

And closed the goal!

Rule argument

var: e

Additional information

Used simplifier rules: RWevent freely generated by R, W;; ⊦ ¬ W(t, l, v).R?; (s) ⊦ allReadsOK(h) → events((h | l) | false) ≠ R(t0, l0, v0) ' + el; (as) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof Tree for abs-persistFO-weak (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