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Proof step 1 from proof abs-single-FO-no-pending (conclusion) (back to tree)

⊦ abs(h, rs, p) ∧ allReadsOK(h) ∧ noPendingW(t, p) ∧ isPersistFO(n, p(l)) ∧ ¬ FO(t) ∈ p(l) -p n ∧ FO(t) ∈ fowl ∧ ¬ t ∈ rs(l) → ((h | l) | t) | false = []

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof abs-single-FO-no-pending (previous step) (back to tree)

((h | l) | t) | false ≠ [], abs(h, rs, p), allReadsOK(h), isPersistFO(n, p(l)), noPendingW(t, p), FO(t) ∈ fowl, ¬ t ∈ rs(l), ¬ FO(t) ∈ p(l) -p n ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 3 from proof abs-single-FO-no-pending (previous step) (back to tree)

((h | l) | t) | false ≠ [], abs(h, rs, p), allReadsOK(h), isPersistFO(n, p(l)), noPendingW(t, p), FO(t) ∈ fowl, ¬ t ∈ rs(l), ¬ FO(t) ∈ p(l) -p n ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

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

Additional information

Used terms: h, rs, p

Proof step 4 from proof abs-single-FO-no-pending (previous step) (back to tree)

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

Lemma abs-def

Proof step 5 from proof abs-single-FO-no-pending (previous step) (back to tree)

((h | l) | t) | false ≠ [], allReadsOK(h), isPersistFO(n, p(l)), noPendingW(t, p), FO(t) ∈ fowl, ¬ t ∈ rs(l), ¬ FO(t) ∈ p(l) -p n, ∀ l. absloc(events((h | l) | false), rs(l), p(l)) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 8 terms: l abort: F discard: T computed: T

Additional information

Used terms: l

Proof step 6 from proof abs-single-FO-no-pending (previous step) (back to tree)

((h | l) | t) | false ≠ [], absloc(events((h | l) | false), rs(l), p(l)), allReadsOK(h), isPersistFO(n, p(l)), noPendingW(t, p), FO(t) ∈ fowl, ¬ t ∈ rs(l), ¬ FO(t) ∈ p(l) -p n ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: PLIB-preds, name:allReadsOK-def, seq: ⊦ allReadsOK(h) ↔ allReadsOKInd(∅, λ l. true, h), subst:[h] → [h], <all paths>

Additional information

Used terms: h

Proof step 7 from proof abs-single-FO-no-pending (previous step) (back to tree)

⊦ allReadsOK(h) ↔ allReadsOKInd(∅, λ l. true, h)

A spec-lemma

Proof step 8 from proof abs-single-FO-no-pending (previous step) (back to tree)

((h | l) | t) | false ≠ [], absloc(events((h | l) | false), rs(l), p(l)), allReadsOKInd(∅, λ l. true, h), isPersistFO(n, p(l)), noPendingW(t, p), FO(t) ∈ fowl, ¬ t ∈ rs(l), ¬ FO(t) ∈ p(l) -p n ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: FLITPSC-preds, name:noPendingW, seq: ⊦ noPendingW(t, p) ↔ (∀ l. p(l) |W | t = []), subst:[t, p] → [t, p], <all paths>

Additional information

Used terms: t, p

Proof step 9 from proof abs-single-FO-no-pending (previous step) (back to tree)

⊦ noPendingW(t, p) ↔ (∀ l. p(l) |W | t = [])

A spec-lemma

Proof step 10 from proof abs-single-FO-no-pending (previous step) (back to tree)

((h | l) | t) | false ≠ [], absloc(events((h | l) | false), rs(l), p(l)), allReadsOKInd(∅, λ l. true, h), isPersistFO(n, p(l)), FO(t) ∈ fowl, ¬ t ∈ rs(l), ¬ FO(t) ∈ p(l) -p n, ∀ l. p(l) |W | t = [] ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 8 terms: l abort: F discard: T computed: T

Additional information

Used terms: l

Proof step 11 from proof abs-single-FO-no-pending (previous step) (back to tree)

p(l) |W | t = [], ((h | l) | t) | false ≠ [], absloc(events((h | l) | false), rs(l), p(l)), allReadsOKInd(∅, λ l. true, h), isPersistFO(n, p(l)), FO(t) ∈ fowl, ¬ t ∈ rs(l), ¬ FO(t) ∈ p(l) -p n ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:abs-loc-no-w-last-fo-then-no-pending, seq:⊦ 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 = [], subst:[t, n, persisted, mem, fowl, h, rset, l] → [t, n, λ l. true, ∅, p(l), h, rs(l), l], <all paths>

Additional information

Used simplifier rules: abs-loc-no-w-no-fo-then-no-pending: ⊦ absloc(events((h | l) | false), rset, fowl) ∧ fowl |W | t = [] ∧ ¬ FO(t) ∈ fowl ∧ ¬ t ∈ rset → ((h | l) | t) | false = []; ⊦ isPersistFO(n, fowl) → (fowl -p n)|W = fowl |W; (s) Used terms: λ l. true, p(l), ∅, l, t, n, rs(l), h

Proof step 12 from proof abs-single-FO-no-pending (previous step) (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 = []

Lemma abs-loc-no-w-last-fo-then-no-pending

Proof Tree for abs-single-FO-no-pending (View conclusion)

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the proof tree