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Proof step 1 from proof abs-loc-no-w-last-fo-then-no-pending (conclusion) (back to tree)

⊦ absloc(events((h | l) | false), rset, fowl) ∧ allReadsOKInd(mem, persisted, h) ∧ persisted(l) ∧ isPersistFO(n, fowl) ∧ (fowl -p n)|W | t = [] ∧ ¬ FO(t) ∈ fowl -p n ∧ FO(t) ∈ fowl ∧ ¬ t ∈ rset → ((h | l) | t) | false = []

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof abs-loc-no-w-last-fo-then-no-pending (previous step) (back to tree)

(fowl -p n)|W | t = [], ((h | l) | t) | false ≠ [], absloc(events((h | l) | false), rset, fowl), allReadsOKInd(mem, persisted, h), isPersistFO(n, fowl), persisted(l), FO(t) ∈ fowl, ¬ t ∈ rset, ¬ FO(t) ∈ fowl -p n ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: delpos-get-fw: ⊦ ¬ a ∈ x -p n ∧ a ∈ x → a = x[n]; (forward) from specification list-del ⊦ isPersistFO(n, fowl) → (fowl -p n)|W = fowl |W; (s)

Proof step 3 from proof abs-loc-no-w-last-fo-then-no-pending (previous step) (back to tree)

FO(t) = fowl[n], fowl |W | t = [], ((h | l) | t) | false ≠ [], absloc(events((h | l) | false), rset, fowl), allReadsOKInd(mem, persisted, h), isPersistFO(n, fowl), persisted(l), FO(t) ∈ fowl, ¬ t ∈ rset, ¬ FO(t) ∈ fowl -p n ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: fowl

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ [] |W = []; (s) ⊦ [] | t = []; (s) ⊦ ¬ a ∈ []; (s)

Proof step 4 from proof abs-loc-no-w-last-fo-then-no-pending (previous step) (back to tree)

FO(t) = (fow + fowl0)[n], (fow + fowl0)|W | t = [], ((h | l) | t) | false ≠ [], absloc(events((h | l) | false), rset, fow + fowl0), allReadsOKInd(mem, persisted, h), isPersistFO(n, fow + fowl0), persisted(l), FO(t) ∈ fow + fowl0, ¬ t ∈ rset, ¬ FO(t) ∈ (fow + fowl0) -p n, ∀ t, n, h, l, rset, mem, persisted. ¬ ( FO(t) = fowl0[n] ∧ fowl0 |W | t = [] ∧ ((h | l) | t) | false ≠ [] ∧ absloc(events((h | l) | false), rset, fowl0) ∧ allReadsOKInd(mem, persisted, h) ∧ isPersistFO(n, fowl0) ∧ persisted(l) ∧ FO(t) ∈ fowl0 ∧ ¬ t ∈ rset ∧ ¬ FO(t) ∈ fowl0 -p n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ fowl + fowl0 | t = (fowl | t) + (fowl0 | t); (s) ⊦ (fowl + fowl0)|W = fowl |W + fowl0 |W; (s)

Proof step 5 from proof abs-loc-no-w-last-fo-then-no-pending (previous step) (back to tree)

FO(t) = (fow ' + fowl0)[n], fow ' |W | t = [], fowl0 |W | t = [], ((h | l) | t) | false ≠ [], absloc(events((h | l) | false), rset, fow ' + fowl0), allReadsOKInd(mem, persisted, h), isPersistFO(n, fow ' + fowl0), persisted(l), ¬ t ∈ rset, ¬ FO(t) ∈ (fow ' + fowl0) -p n, ∀ t0, n0, h0, l0, rset0, mem0, persisted0. FO(t0) ≠ fowl0[n0] ∨ fowl0 |W | t0 ≠ [] ∨ ((h0 | l0) | t0) | false = [] ∨ ¬ absloc(events((h0 | l0) | false), rset0, fowl0) ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ isPersistFO(n0, fowl0) ∨ ¬ persisted0(l0) ∨ ¬ FO(t0) ∈ fowl0 ∨ t0 ∈ rset0 ∨ FO(t0) ∈ fowl0 -p n0, FO(t) = fow ∨ FO(t) ∈ fowl0 ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: n

Additional information

Used simplifier rules: abs-loc-no-w-last-firstfo-then-no-pending: ⊦ absloc(events((h | l) | false), rset, FO(t) ' + fowl) ∧ allReadsOKInd(mem, persisted, h) ∧ persisted(l) ∧ fowl |W | t = [] ∧ ¬ FO(t) ∈ fowl ∧ ¬ t ∈ rset → ((h | l) | t) | false = []; Nat freely generated by 0, +1;; ⊦ (a ' + x)[0] = a; (s) ⊦ (a ' + x) -p 0 = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ FO(t) ' |W = []; (s) ⊦ [] | t = []; (s)

Proof step 6 from proof abs-loc-no-w-last-fo-then-no-pending (previous step) (back to tree)

FO(t) = (fow ' + fowl0)[n], n = n1 +1, fowl0 |W | t = [], fow ' |W | t = [], ((h | l) | t) | false ≠ [], absloc(events((h | l) | false), rset, fow ' + fowl0), allReadsOKInd(mem, persisted, h), isPersistFO(n, fow ' + fowl0), persisted(l), ¬ t ∈ rset, ¬ FO(t) ∈ (fow ' + fowl0) -p n, ∀ t0, n0, h0, l0, rset0, mem0, persisted0. FO(t0) ≠ fowl0[n0] ∨ fowl0 |W | t0 ≠ [] ∨ ((h0 | l0) | t0) | false = [] ∨ ¬ absloc(events((h0 | l0) | false), rset0, fowl0) ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ isPersistFO(n0, fowl0) ∨ ¬ persisted0(l0) ∨ ¬ FO(t0) ∈ fowl0 ∨ t0 ∈ rset0 ∨ FO(t0) ∈ fowl0 -p n0, FO(t) = fow ∨ FO(t) ∈ fowl0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ FO(tid).tid = tid; (s) delpos-get-fw: ⊦ ¬ a ∈ x -p n ∧ a ∈ x → a = x[n]; (forward) from specification list-del a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ∈ a0 ' + x ↔ ¬(a ≠ a0 ∧ ¬ a ∈ x); (ss) 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) -p (n + 1) = a + (x -p n); (s) ⊦ m +1 = m + 1; (s) ⊦ isPersistFO(n + 1, fow ' + fowl) ↔ isPersistFO(n, fowl) ∧ fow.FO? ∧ fowl[n].tid ≠ fow.tid; (s) ⊦ (a ' + x)[n + 1] = x[n]; (s)

Proof step 7 from proof abs-loc-no-w-last-fo-then-no-pending (previous step) (back to tree)

FO(t) = fowl0[n1], fowl0 |W | t = [], fow ' |W | t = [], FO(t) ≠ fow, t ≠ fow.tid, ((h | l) | t) | false ≠ [], fow.FO?, absloc(events((h | l) | false), rset, fow ' + fowl0), allReadsOKInd(mem, persisted, h), isPersistFO(n1, fowl0), persisted(l), FO(t) ∈ fowl0, ¬ FO(t) ∈ fowl0 -p n1, ¬ t ∈ rset, ∀ t0, n0, h0, l0, rset0, mem0, persisted0. FO(t0) ≠ fowl0[n0] ∨ fowl0 |W | t0 ≠ [] ∨ ((h0 | l0) | t0) | false = [] ∨ ¬ absloc(events((h0 | l0) | false), rset0, fowl0) ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ isPersistFO(n0, fowl0) ∨ ¬ persisted0(l0) ∨ ¬ FO(t0) ∈ fowl0 ∨ t0 ∈ rset0 ∨ FO(t0) ∈ fowl0 -p n0 ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: FOW, name:elim-FO, seq: ⊦ fow.FO? → (tid = fow.tid ↔ fow = FO(tid)), subst:[fow, tid] → [fow, tid]

Additional information

Used simplifier rules: ⊦ fow.FO? → (tid = fow.tid ↔ fow = FO(tid)); (elim) Used terms: fow, tid

Proof step 8 from proof abs-loc-no-w-last-fo-then-no-pending (previous step) (back to tree)

FO(t) = fowl0[n1], fow = FO(tid), fow ' |W | t = [], fowl0 |W | t = [], FO(t) ≠ fow, t ≠ tid, ((h | l) | t) | false ≠ [], fow.FO?, absloc(events((h | l) | false), rset, fow ' + fowl0), allReadsOKInd(mem, persisted, h), isPersistFO(n1, fowl0), persisted(l), FO(t) ∈ fowl0, ¬ FO(t) ∈ fowl0 -p n1, ¬ t ∈ rset, ∀ t0, n0, h0, l0, rset0, mem0, persisted0. FO(t0) ≠ fowl0[n0] ∨ fowl0 |W | t0 ≠ [] ∨ ((h0 | l0) | t0) | false = [] ∨ ¬ absloc(events((h0 | l0) | false), rset0, fowl0) ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ isPersistFO(n0, fowl0) ∨ ¬ persisted0(l0) ∨ ¬ FO(t0) ∈ fowl0 ∨ t0 ∈ rset0 ∨ FO(t0) ∈ fowl0 -p n0 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ FO(tid).FO?; (s) ⊦ FO(tid) = FO(tid0) ↔ tid = tid0; (s) ⊦ [] | t = []; (s) ⊦ FO(t) ' |W = []; (s)

Proof step 9 from proof abs-loc-no-w-last-fo-then-no-pending (previous step) (back to tree)

FO(t) = fowl0[n1], fowl0 |W | t = [], t ≠ tid, ((h | l) | t) | false ≠ [], absloc(events((h | l) | false), rset, FO(tid) ' + fowl0), allReadsOKInd(mem, persisted, h), isPersistFO(n1, fowl0), persisted(l), FO(t) ∈ fowl0, ¬ FO(t) ∈ fowl0 -p n1, ¬ t ∈ rset, ∀ t0, n0, h0, l0, rset0, mem0, persisted0. FO(t0) ≠ fowl0[n0] ∨ fowl0 |W | t0 ≠ [] ∨ ((h0 | l0) | t0) | false = [] ∨ ¬ absloc(events((h0 | l0) | false), rset0, fowl0) ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ isPersistFO(n0, fowl0) ∨ ¬ persisted0(l0) ∨ ¬ FO(t0) ∈ fowl0 ∨ t0 ∈ rset0 ∨ FO(t0) ∈ fowl0 -p n0 ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:absloc-persistfirstFO, seq:⊦ el = [] ∨ ¬ el.head.R? → (absloc(el, rset, FO(t) ' + fowl) ↔ absloc(el, rset, fowl)), subst:[el, rset, t, fowl] → [events((h | l) | false), rset, tid, fowl0], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: events((h | l) | false), rset, tid, fowl0

Proof step 10 from proof abs-loc-no-w-last-fo-then-no-pending (previous step) (back to tree)

⊦ el = [] ∨ ¬ el.head.R? → (absloc(el, rset, FO(t) ' + fowl) ↔ absloc(el, rset, fowl))

Lemma absloc-persistfirstFO

Proof step 11 from proof abs-loc-no-w-last-fo-then-no-pending (previous step) (back to tree)

FO(t) = fowl0[n1], fowl0 |W | t = [], events((h | l) | false) ≠ [], t ≠ tid, ((h | l) | t) | false ≠ [], events((h | l) | false).head.R?, absloc(events((h | l) | false), rset, FO(tid) ' + fowl0), allReadsOKInd(mem, persisted, h), isPersistFO(n1, fowl0), persisted(l), FO(t) ∈ fowl0, ¬ FO(t) ∈ fowl0 -p n1, ¬ t ∈ rset, ∀ t0, n0, h0, l0, rset0, mem0, persisted0. FO(t0) ≠ fowl0[n0] ∨ fowl0 |W | t0 ≠ [] ∨ ((h0 | l0) | t0) | false = [] ∨ ¬ absloc(events((h0 | l0) | false), rset0, fowl0) ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ isPersistFO(n0, fowl0) ∨ ¬ persisted0(l0) ∨ ¬ FO(t0) ∈ fowl0 ∨ t0 ∈ rset0 ∨ FO(t0) ∈ fowl0 -p n0 ⊦

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 | l) | 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 | l) | false), e, el

Proof step 12 from proof abs-loc-no-w-last-fo-then-no-pending (previous step) (back to tree)

FO(t) = fowl0[n1], events((h | l) | false) = e + el, fowl0 |W | t = [], events((h | l) | false) ≠ [], t ≠ tid, ((h | l) | t) | false ≠ [], e.R?, absloc(events((h | l) | false), rset, FO(tid) ' + fowl0), allReadsOKInd(mem, persisted, h), isPersistFO(n1, fowl0), persisted(l), FO(t) ∈ fowl0, ¬ FO(t) ∈ fowl0 -p n1, ¬ t ∈ rset, ∀ t0, n0, h0, l0, rset0, mem0, persisted0. FO(t0) ≠ fowl0[n0] ∨ fowl0 |W | t0 ≠ [] ∨ ((h0 | l0) | t0) | false = [] ∨ ¬ absloc(events((h0 | l0) | false), rset0, fowl0) ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ isPersistFO(n0, fowl0) ∨ ¬ persisted0(l0) ∨ ¬ FO(t0) ∈ fowl0 ∨ t0 ∈ rset0 ∨ FO(t0) ∈ fowl0 -p n0 ⊦

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 13 from proof abs-loc-no-w-last-fo-then-no-pending (previous step) (back to tree)

FO(t) = fowl0[n1], events((h | l) | false) = e ' + el, fowl0 |W | t = [], t ≠ tid, ((h | l) | t) | false ≠ [], e.R?, absloc(e ' + el, rset, FO(tid) ' + fowl0), allReadsOKInd(mem, persisted, h), isPersistFO(n1, fowl0), persisted(l), FO(t) ∈ fowl0, ¬ FO(t) ∈ fowl0 -p n1, ¬ t ∈ rset, ∀ t0, n0, h0, l0, rset0, mem0, persisted0. FO(t0) ≠ fowl0[n0] ∨ fowl0 |W | t0 ≠ [] ∨ ((h0 | l0) | t0) | false = [] ∨ ¬ absloc(events((h0 | l0) | false), rset0, fowl0) ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ isPersistFO(n0, fowl0) ∨ ¬ persisted0(l0) ∨ ¬ FO(t0) ∈ fowl0 ∨ t0 ∈ rset0 ∨ FO(t0) ∈ fowl0 -p n0 ⊦

User applied rule constructor cut

And closed the goal!

Rule argument

var: e

Additional information

Used simplifier rules: absloc-recW-FO: ⊦ absloc(W(t, l, v) ' + el, rset, FO(t0) ' + fowl) ↔ absloc(W(t, l, v) ' + el, rset, fowl); RWevent freely generated by R, W;; a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ¬ W(t, l, v).R?; (s) ⊦ allReadsOKInd(mem, persisted, h) ∧ persisted(l) → events((h | l) | false) ≠ R(t0, l0, v0) ' + el; (as)

Proof step 14 from proof abs-loc-no-w-last-fo-then-no-pending (previous step) (back to tree)

FO(t) = fowl0[n1], fowl0 |W | t = [], t ≠ tid, ((h | l) | t) | false ≠ [], absloc(events((h | l) | false), rset, fowl0), allReadsOKInd(mem, persisted, h), isPersistFO(n1, fowl0), persisted(l), FO(t) ∈ fowl0, ¬ FO(t) ∈ fowl0 -p n1, ¬ t ∈ rset, ∀ t0, n0, h0, l0, rset0, mem0, persisted0. FO(t0) ≠ fowl0[n0] ∨ fowl0 |W | t0 ≠ [] ∨ ((h0 | l0) | t0) | false = [] ∨ ¬ absloc(events((h0 | l0) | false), rset0, fowl0) ∨ ¬ allReadsOKInd(mem0, persisted0, h0) ∨ ¬ isPersistFO(n0, fowl0) ∨ ¬ persisted0(l0) ∨ ¬ FO(t0) ∈ fowl0 ∨ t0 ∈ rset0 ∨ FO(t0) ∈ fowl0 -p n0, events((h | l) | false) = [] ∨ ¬ events((h | l) | false).head.R? ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 12 terms: t, n1, h, l, rset, mem, persisted abort: F discard: T computed: T

Additional information

Used terms: t, n1, h, l, rset, mem, persisted

Proof Tree for abs-loc-no-w-last-fo-then-no-pending (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