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



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Proof step 1 from proof step-APersist (conclusion) (back to tree)

⊦ (allReadsOK(gs.h) ∧ allWritesOK(gs.h) ∧ gs.vm = ∅ ⊕ gs.h ∧ gs.pm = ∅ ⊕ (gs.h | true)) ∧ PLIBLInv(gs, lsf(t), pcf(t)) ∧ firstVolWrite(l, gs.h) < # gs.h ∧ gs0 = (gs.pm:= gs.pm(l := gs.h[firstVolWrite(l, gs.h)]._1.v)) .h:= ( gs.h.take(firstVolWrite(l, gs.h)) + ((gs.h[firstVolWrite(l, gs.h)]._1, true) + persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1)))) → allReadsOK(gs0.h) ∧ allWritesOK(gs0.h) ∧ gs0.vm = ∅ ⊕ gs0.h ∧ gs0.pm = ∅ ⊕ (gs0.h | true)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof step-APersist (previous step) (back to tree)

gs.pm = ∅ ⊕ (gs.h | true), gs.vm = ∅ ⊕ gs.h, gs0 = (gs.pm:= gs.pm(l := gs.h[firstVolWrite(l, gs.h)]._1.v)) .h:= ( gs.h.take(firstVolWrite(l, gs.h)) + ((gs.h[firstVolWrite(l, gs.h)]._1, true) + persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1)))), firstVolWrite(l, gs.h) < # gs.h, PLIBLInv(gs, lsf(t), pcf(t)), allReadsOK(gs.h), allWritesOK(gs.h) ⊦ allReadsOK(gs0.h) ∧ allWritesOK(gs0.h) ∧ gs0.vm = ∅ ⊕ gs0.h ∧ gs0.pm = ∅ ⊕ (gs0.h | true)

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 ⊦ mem ⊕ (h1 + (persistDependentReads(l, h2) | true)) = mem ⊕ (h1 + (h2 | true)); (s) ⊦ (e, true) ' + h | true = (e, true) ' + (h | true); (s) ⊦ h + h0 | b = (h | b) + (h0 | b); (s) ⊦ (gs.h:= h).h = h; (s) ⊦ a + x = a ' + x; (s) ⊦ (gs.pm:= vm0).pm = vm0; (s) ⊦ (gs.h:= h).pm = gs.pm; (s) ⊦ mem ⊕ (h1 + persistDependentReads(l, h2)) = mem ⊕ (h1 + h2); (s) ⊦ (gs.pm:= vm0).vm = gs.vm; (s) ⊦ (gs.h:= h).vm = gs.vm; (s) ⊦ firstVolWrite(l, h) < # h ↔ (h |W | l) | false ≠ []; (s)

Proof step 3 from proof step-APersist (previous step) (back to tree)

gs.pm = ∅ ⊕ (gs.h | true), gs.vm = ∅ ⊕ gs.h, (gs.h |W | l) | false ≠ [], PLIBLInv(gs, lsf(t), pcf(t)), allReadsOK(gs.h), allWritesOK(gs.h) ⊦ allReadsOK (gs.h.take(firstVolWrite(l, gs.h)) + (gs.h[firstVolWrite(l, gs.h)]._1, true) ' + persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1))) ∧ allWritesOK (gs.h.take(firstVolWrite(l, gs.h)) + (gs.h[firstVolWrite(l, gs.h)]._1, true) ' + persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1))) ∧ gs.vm = ∅ ⊕ ((gs.h.take(firstVolWrite(l, gs.h)) + (gs.h[firstVolWrite(l, gs.h)]._1, true) ') + gs.h.drop(firstVolWrite(l, gs.h) + 1)) ∧ gs.pm(l := gs.h[firstVolWrite(l, gs.h)]._1.v) = ∅ ⊕ (((gs.h.take(firstVolWrite(l, gs.h)) | true) + (gs.h[firstVolWrite(l, gs.h)]._1, true) ') + (gs.h.drop(firstVolWrite(l, gs.h) + 1) | true))

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

term: gs.h[firstVolWrite(l, gs.h)]._1

Additional information

Used terms: e, boolvar

Proof step 4 from proof step-APersist (previous step) (back to tree)

gs.pm = ∅ ⊕ (gs.h | true), gs.vm = ∅ ⊕ gs.h, gs.h[firstVolWrite(l, gs.h)] = (e, boolvar), (gs.h |W | l) | false ≠ [], PLIBLInv(gs, lsf(t), pcf(t)), allReadsOK(gs.h), allWritesOK(gs.h) ⊦ allReadsOK(gs.h.take(firstVolWrite(l, gs.h)) + (e, true) ' + persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1))) ∧ allWritesOK(gs.h.take(firstVolWrite(l, gs.h)) + (e, true) ' + persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1))) ∧ gs.vm = ∅ ⊕ ((gs.h.take(firstVolWrite(l, gs.h)) + (e, true) ') + gs.h.drop(firstVolWrite(l, gs.h) + 1)) ∧ gs.pm(l := e.v) = ∅ ⊕ (((gs.h.take(firstVolWrite(l, gs.h)) | true) + (e, true) ') + (gs.h.drop(firstVolWrite(l, gs.h) + 1) | true))

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: PLIB-preds, name:first-vol-write-split, seq:⊦ firstVolWrite(l, h) < # h → (∃ t, v. h = h.take(firstVolWrite(l, h)) + ((W(t, l, v), false) + h.drop(firstVolWrite(l, h) + 1))), subst:[l, h] → [l, gs.h]

Additional information

Used terms: l, gs.h

Proof step 5 from proof step-APersist (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)))

A spec-lemma

Proof step 6 from proof step-APersist (previous step) (back to tree)

gs.pm = ∅ ⊕ (gs.h | true), gs.vm = ∅ ⊕ gs.h, gs.h[firstVolWrite(l, gs.h)] = (e, boolvar), (gs.h |W | l) | false ≠ [], PLIBLInv(gs, lsf(t), pcf(t)), allReadsOK(gs.h), allWritesOK(gs.h), firstVolWrite(l, gs.h) < # gs.h → (∃ t, v. gs.h = gs.h.take(firstVolWrite(l, gs.h)) + ((W(t, l, v), false) + gs.h.drop(firstVolWrite(l, gs.h) + 1))) ⊦ allReadsOK(gs.h.take(firstVolWrite(l, gs.h)) + (e, true) ' + persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1))) ∧ allWritesOK(gs.h.take(firstVolWrite(l, gs.h)) + (e, true) ' + persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1))) ∧ gs.vm = ∅ ⊕ ((gs.h.take(firstVolWrite(l, gs.h)) + (e, true) ') + gs.h.drop(firstVolWrite(l, gs.h) + 1)) ∧ gs.pm(l := e.v) = ∅ ⊕ (((gs.h.take(firstVolWrite(l, gs.h)) | true) + (e, true) ') + (gs.h.drop(firstVolWrite(l, gs.h) + 1) | true))

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 + x = a ' + x; (s) ⊦ firstVolWrite(l, h) < # h ↔ (h |W | l) | false ≠ []; (s)

Proof step 7 from proof step-APersist (previous step) (back to tree)

gs.h = gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1), gs.pm = ∅ ⊕ (gs.h | true), gs.vm = ∅ ⊕ gs.h, gs.h[firstVolWrite(l, gs.h)] = (e, boolvar), (gs.h |W | l) | false ≠ [], PLIBLInv(gs, lsf(t), pcf(t)), allReadsOK(gs.h), allWritesOK(gs.h) ⊦ allReadsOK(gs.h.take(firstVolWrite(l, gs.h)) + (e, true) ' + persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1))) ∧ allWritesOK(gs.h.take(firstVolWrite(l, gs.h)) + (e, true) ' + persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1))) ∧ gs.vm = ∅ ⊕ ((gs.h.take(firstVolWrite(l, gs.h)) + (e, true) ') + gs.h.drop(firstVolWrite(l, gs.h) + 1)) ∧ gs.pm(l := e.v) = ∅ ⊕ (((gs.h.take(firstVolWrite(l, gs.h)) | true) + (e, true) ') + (gs.h.drop(firstVolWrite(l, gs.h) + 1) | true))

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 1, flags:

Additional information

Proof step 8 from proof step-APersist (previous step) (back to tree)

gs.h = gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1), gs.pm = ∅ ⊕ (gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1) | true), gs.vm = ∅ ⊕ (gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)), (gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1) )[firstVolWrite(l, gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)) ] = (e, boolvar), ((gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1))|W | l) | false ≠ [], PLIBLInv(gs, lsf(t), pcf(t)), allReadsOK(gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)), allWritesOK(gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)) ⊦ allReadsOK ( (gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1) ).take (firstVolWrite(l, gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)) ) + (e, true) ' + persistDependentReads (l, (gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1) ).drop (firstVolWrite(l, gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)) + 1 ))) ∧ allWritesOK ( (gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1) ).take (firstVolWrite(l, gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)) ) + (e, true) ' + persistDependentReads (l, (gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1) ).drop (firstVolWrite(l, gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)) + 1 ))) ∧ gs.vm = ∅ ⊕ ( ( (gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1) ).take (firstVolWrite(l, gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)) ) + (e, true) ') + (gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1) ).drop (firstVolWrite(l, gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)) + 1 )) ∧ gs.pm(l := e.v) = ∅ ⊕ ( ( ( (gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1) ).take (firstVolWrite(l, gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)) ) | true) + (e, true) ') + ( (gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1) ).drop (firstVolWrite(l, gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)) + 1 ) | true))

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-del, name:get-append-grt, seq: ⊦ ¬ n < # x → (x + y)[n] = y[n - # x], subst:[n, x, y] → [firstVolWrite (l, gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)), gs.h.take(firstVolWrite(l, gs.h)), (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)]; ['a] → [(RWevent, Bool)], <all paths>

Additional information

Used simplifier rules: ⊦ n < # x → (x + y)[n] = x[n]; (s) ⊦ (e, false) ' + h | true = h | true; (s) ⊦ (h + h0)|W = h |W + h0 |W; (s) ⊦ ((W(t, l, v), b) ' + h)|W = (W(t, l, v), b) ' + h |W; (s) ⊦ h + h0 | l0 = (h | l0) + (h0 | l0); (s) ⊦ (W(t, l, v), b) ' + h | l = (W(t, l, v), b) ' + (h | l); (s) ⊦ h + h0 | b = (h | b) + (h0 | b); (s) ⊦ ((h.take(firstVolWrite(l, h)))|W | l) | false = []; (s) ⊦ (e, false) ' + h | false = (e, false) ' + (h | false); (s) ⊦ [] + x = x; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a ' ≠ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ # x.take(n) = n; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m < n → n ≠ 0; (s) ⊦ ¬ n < n; (s) Used terms: firstVolWrite (l, gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)), gs.h.take(firstVolWrite(l, gs.h)), (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)

Proof step 9 from proof step-APersist (previous step) (back to tree)

⊦ ¬ n < # x → (x + y)[n] = y[n - # x]

A spec-lemma

Proof step 10 from proof step-APersist (previous step) (back to tree)

gs.h = gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1), gs.pm = ∅ ⊕ (gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1) | true), gs.vm = ∅ ⊕ (gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)), ((W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1) )[ firstVolWrite(l, gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)) - # gs.h.take(firstVolWrite(l, gs.h)) ] = (e, boolvar), ((gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1))|W | l) | false ≠ [], PLIBLInv(gs, lsf(t), pcf(t)), allReadsOK(gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)), allWritesOK(gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)), ¬ firstVolWrite(l, gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)) < # gs.h.take(firstVolWrite(l, gs.h)) ⊦ allReadsOK ( (gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1) ).take (firstVolWrite(l, gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)) ) + (e, true) ' + persistDependentReads (l, (gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1) ).drop (firstVolWrite(l, gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)) + 1 ))) ∧ allWritesOK ( (gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1) ).take (firstVolWrite(l, gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)) ) + (e, true) ' + persistDependentReads (l, (gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1) ).drop (firstVolWrite(l, gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)) + 1 ))) ∧ gs.vm = ∅ ⊕ ( ( (gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1) ).take (firstVolWrite(l, gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)) ) + (e, true) ') + (gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1) ).drop (firstVolWrite(l, gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)) + 1 )) ∧ gs.pm(l := e.v) = ∅ ⊕ ( ( ( (gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1) ).take (firstVolWrite(l, gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)) ) | true) + (e, true) ') + ( (gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1) ).drop (firstVolWrite(l, gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)) + 1 ) | true))

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 ⊦ mem ⊕ (h1 + (W(t, l, v), true) ' + h2) = mem ⊕ (h1 + (W(t, l, v), false) ' + h2); (s) ⊦ W(t, l, v).v = v; (s) ⊦ m ≤ m; (s) ⊦ (a ' + x)[0] = a; (s) ⊦ m - m = 0; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ # x.take(n) = n; (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ [] + x = x; (s) ⊦ (e, false) ' + h | false = (e, false) ' + (h | false); (s) ⊦ ((h.take(firstVolWrite(l, h)))|W | l) | false = []; (s) ⊦ h + h0 | b = (h | b) + (h0 | b); (s) ⊦ (W(t, l, v), b) ' + h | l = (W(t, l, v), b) ' + (h | l); (s) ⊦ h + h0 | l0 = (h | l0) + (h0 | l0); (s) ⊦ ((W(t, l, v), b) ' + h)|W = (W(t, l, v), b) ' + h |W; (s) ⊦ (h + h0)|W = h |W + h0 |W; (s) ⊦ (e, false) ' + h | true = h | true; (s)

Proof step 11 from proof step-APersist (previous step) (back to tree)

gs.h = gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1), gs.pm = ∅ ⊕ ((gs.h.take(firstVolWrite(l, gs.h)) | true) + (gs.h.drop(firstVolWrite(l, gs.h) + 1) | true)), gs.vm = ∅ ⊕ gs.h, PLIBLInv(gs, lsf(t), pcf(t)), allReadsOK(gs.h), allWritesOK(gs.h) ⊦ allReadsOK(gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), true) ' + persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1))) ∧ allWritesOK(gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), true) ' + persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1))) ∧ gs.pm(l := v) = ∅ ⊕ ((gs.h.take(firstVolWrite(l, gs.h)) | true) + (W(t0, l, v), false) ' + (gs.h.drop(firstVolWrite(l, gs.h) + 1) | true))

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: PLIB-preds, name:allreadsok-persist, seq:⊦ allReadsOK(h1 + (W(t, l, v), false) ' + h2) → allReadsOK(h1 + (W(t, l, v), true) ' + persistDependentReads(l, h2)), subst:[h2, l, v, t, h1] → [gs.h.drop(firstVolWrite(l, gs.h) + 1), l, v, t0, gs.h.take(firstVolWrite(l, gs.h))], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: v, gs.h.drop(firstVolWrite(l, gs.h) + 1), l, gs.h.take(firstVolWrite(l, gs.h)), t0

Proof step 12 from proof step-APersist (previous step) (back to tree)

⊦ allReadsOK(h1 + (W(t, l, v), false) ' + h2) → allReadsOK(h1 + (W(t, l, v), true) ' + persistDependentReads(l, h2))

A spec-lemma

Proof step 13 from proof step-APersist (previous step) (back to tree)

gs.h = gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1), gs.pm = ∅ ⊕ ((gs.h.take(firstVolWrite(l, gs.h)) | true) + (gs.h.drop(firstVolWrite(l, gs.h) + 1) | true)), gs.vm = ∅ ⊕ gs.h, PLIBLInv(gs, lsf(t), pcf(t)), allReadsOK(gs.h), allReadsOK(gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1)), allWritesOK(gs.h) ⊦ true ∧ allWritesOK(gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), true) ' + persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1))) ∧ gs.pm(l := v) = ∅ ⊕ ((gs.h.take(firstVolWrite(l, gs.h)) | true) + (W(t0, l, v), false) ' + (gs.h.drop(firstVolWrite(l, gs.h) + 1) | true))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 14 from proof step-APersist (previous step) (back to tree)

gs.h = gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1), gs.pm = ∅ ⊕ ((gs.h.take(firstVolWrite(l, gs.h)) | true) + (gs.h.drop(firstVolWrite(l, gs.h) + 1) | true)), gs.vm = ∅ ⊕ gs.h, PLIBLInv(gs, lsf(t), pcf(t)), allReadsOK(gs.h), allWritesOK(gs.h) ⊦ allWritesOK(gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), true) ' + persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1))) ∧ gs.pm(l := v) = ∅ ⊕ ((gs.h.take(firstVolWrite(l, gs.h)) | true) + (W(t0, l, v), false) ' + (gs.h.drop(firstVolWrite(l, gs.h) + 1) | true))

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: PLIB-preds, name:allwritesOK-persist-one-write, seq:⊦ allWritesOK(h1 + (W(t, l, v), false) ' + h2) → (allWritesOK(h1 + (W(t, l, v), true) ' + h2) ↔ (h1 |W | l) | false = []), subst:[h2, l, v, t, h1] → [persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1)), l, v, t0, gs.h.take(firstVolWrite(l, gs.h))], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: v, persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1)), l, gs.h.take(firstVolWrite(l, gs.h)), t0

Proof step 15 from proof step-APersist (previous step) (back to tree)

⊦ allWritesOK(h1 + (W(t, l, v), false) ' + h2) → (allWritesOK(h1 + (W(t, l, v), true) ' + h2) ↔ (h1 |W | l) | false = [])

A spec-lemma

Proof step 16 from proof step-APersist (previous step) (back to tree)

gs.h = gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1), gs.pm = ∅ ⊕ ((gs.h.take(firstVolWrite(l, gs.h)) | true) + (gs.h.drop(firstVolWrite(l, gs.h) + 1) | true)), gs.vm = ∅ ⊕ gs.h, PLIBLInv(gs, lsf(t), pcf(t)), allReadsOK(gs.h), allWritesOK(gs.h), ¬ allWritesOK(gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1))) ⊦ allWritesOK(gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), true) ' + persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1))) ∧ gs.pm(l := v) = ∅ ⊕ ((gs.h.take(firstVolWrite(l, gs.h)) | true) + (W(t0, l, v), false) ' + (gs.h.drop(firstVolWrite(l, gs.h) + 1) | true))

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: PLIB-preds, name:allwritesOK-persist-dependent-reads, seq: ⊦ allWritesOK(h1 + h2) → allWritesOK(h1 + persistDependentReads(l, h2)), subst:[h1, l, h2] → [gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ', l, gs.h.drop(firstVolWrite(l, gs.h) + 1)], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ', l, gs.h.drop(firstVolWrite(l, gs.h) + 1)

Proof step 17 from proof step-APersist (previous step) (back to tree)

⊦ allWritesOK(h1 + h2) → allWritesOK(h1 + persistDependentReads(l, h2))

A spec-lemma

Proof step 18 from proof step-APersist (previous step) (back to tree)

gs.h = gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1), gs.pm = ∅ ⊕ ((gs.h.take(firstVolWrite(l, gs.h)) | true) + (gs.h.drop(firstVolWrite(l, gs.h) + 1) | true)), gs.vm = ∅ ⊕ gs.h, PLIBLInv(gs, lsf(t), pcf(t)), allReadsOK(gs.h), allWritesOK(gs.h), allWritesOK(gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1))) ⊦ ((gs.h.take(firstVolWrite(l, gs.h)))|W | l) | false = [] ∧ gs.pm(l := v) = ∅ ⊕ ((gs.h.take(firstVolWrite(l, gs.h)) | true) + (W(t0, l, v), false) ' + (gs.h.drop(firstVolWrite(l, gs.h) + 1) | true))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ((h.take(firstVolWrite(l, h)))|W | l) | false = []; (s)

Proof step 19 from proof step-APersist (previous step) (back to tree)

gs.h = gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1), gs.pm = ∅ ⊕ ((gs.h.take(firstVolWrite(l, gs.h)) | true) + (gs.h.drop(firstVolWrite(l, gs.h) + 1) | true)), gs.vm = ∅ ⊕ gs.h, gs.pm(l := v) ≠ ∅ ⊕ ((gs.h.take(firstVolWrite(l, gs.h)) | true) + (W(t0, l, v), false) ' + (gs.h.drop(firstVolWrite(l, gs.h) + 1) | true)), PLIBLInv(gs, lsf(t), pcf(t)), allReadsOK(gs.h), allWritesOK(gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1))), allWritesOK(gs.h) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 2, flags: discard

Additional information

Proof step 20 from proof step-APersist (previous step) (back to tree)

gs.h = gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1), gs.vm = ∅ ⊕ gs.h, (∅ ⊕ ((gs.h.take(firstVolWrite(l, gs.h)) | true) + (gs.h.drop(firstVolWrite(l, gs.h) + 1) | true)))(l := v) ≠ ∅ ⊕ ((gs.h.take(firstVolWrite(l, gs.h)) | true) + (W(t0, l, v), false) ' + (gs.h.drop(firstVolWrite(l, gs.h) + 1) | true)), PLIBLInv(gs, lsf(t), pcf(t)), allReadsOK(gs.h), allWritesOK(gs.h), allWritesOK(gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1))) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: PLIB-preds, name:lastwrite-to-update, seq:⊦ (h2 |W | l) | true = [] → mem ⊕ (h1 + (W(t, l, v), b) ' + (h2 | true)) = (mem ⊕ (h1 + (h2 | true)))(l := v), subst:[b, h2, l, mem, v, t, h1] → [false, gs.h.drop(firstVolWrite(l, gs.h) + 1), l, ∅, v, t0, gs.h.take(firstVolWrite(l, gs.h)) | true], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: v, gs.h.drop(firstVolWrite(l, gs.h) + 1), false, ∅, l, gs.h.take(firstVolWrite(l, gs.h)) | true, t0

Proof step 21 from proof step-APersist (previous step) (back to tree)

⊦ (h2 |W | l) | true = [] → mem ⊕ (h1 + (W(t, l, v), b) ' + (h2 | true)) = (mem ⊕ (h1 + (h2 | true)))(l := v)

A spec-lemma

Proof step 22 from proof step-APersist (previous step) (back to tree)

gs.h = gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + gs.h.drop(firstVolWrite(l, gs.h) + 1), gs.vm = ∅ ⊕ gs.h, (∅ ⊕ ((gs.h.take(firstVolWrite(l, gs.h)) | true) + (gs.h.drop(firstVolWrite(l, gs.h) + 1) | true)))(l := v) ≠ ∅ ⊕ ((gs.h.take(firstVolWrite(l, gs.h)) | true) + (W(t0, l, v), false) ' + (gs.h.drop(firstVolWrite(l, gs.h) + 1) | true)), ((gs.h.drop(firstVolWrite(l, gs.h) + 1))|W | l) | true ≠ [], PLIBLInv(gs, lsf(t), pcf(t)), allReadsOK(gs.h), allWritesOK(gs.h.take(firstVolWrite(l, gs.h)) + (W(t0, l, v), false) ' + persistDependentReads(l, gs.h.drop(firstVolWrite(l, gs.h) + 1))), allWritesOK(gs.h) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: PLIB-preds, name:allwritesOK-one-unpersisted-write-no-persisted-after, seq: ⊦ allWritesOK(h1 + (W(t, l, v), false) ' + h2) → (h2 |W | l) | true = [], subst:[h2, l, v, t, h1] → [gs.h.drop(firstVolWrite(l, gs.h) + 1), l, v, t0, gs.h.take(firstVolWrite(l, gs.h))]

Additional information

Used terms: v, gs.h.drop(firstVolWrite(l, gs.h) + 1), l, gs.h.take(firstVolWrite(l, gs.h)), t0

Proof step 23 from proof step-APersist (previous step) (back to tree)

⊦ allWritesOK(h1 + (W(t, l, v), false) ' + h2) → (h2 |W | l) | true = []

A spec-lemma

Proof Tree for step-APersist (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