Nuprl Lemma : State-loc-comb-invariant-sv2
∀[Info,A,S:Type].
∀es:EO+(Info). ∀P:E ─→ S ─→ ℙ. ∀init:Id ─→ bag(S). ∀f:Id ─→ A ─→ S ─→ S. ∀X:EClass(A). ∀e:E. ∀v:S.
(single-valued-bag(init loc(e);S)
⇒ single-valued-classrel(es;X;A)
⇒ (∀s:S. ∀e':E.
(e' ≤loc e
⇒ if first(e') then s ↓∈ init loc(e') else s ∈ State-loc-comb(init;f;X)(pred(e')) ∧ P[pred(e');s] fi
⇒ if e' ∈b X then ∀a:A. (a ∈ X(e') ⇒ P[e';f loc(e') a s]) else P[e';s] fi ))
⇒ v ∈ State-loc-comb(init;f;X)(e)
⇒ P[e;v])
Proof
Definitions occuring in Statement :
State-loc-comb: State-loc-comb(init;f;X),
single-valued-classrel: single-valued-classrel(es;X;T),
classrel: v ∈ X(e),
member-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-le: e ≤loc e' ,
es-first: first(e),
es-pred: pred(e),
es-loc: loc(e),
es-E: E,
Id: Id,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ─→ B[x],
universe: Type,
single-valued-bag: single-valued-bag(b;T),
bag-member: x ↓∈ bs,
bag: bag(T)
Lemmas :
member-eclass_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bool_cases,
es-first_wf2,
event-ordering+_subtype,
assert_of_bnot,
State-loc-comb-classrel2,
es-pred_wf,
es-le-loc,
es-loc-pred,
and_wf,
Id_wf,
iterated-classrel_wf,
bag-member_wf,
es-loc_wf,
sq_stable__single-valued-iterated-classrel
Latex:
\mforall{}[Info,A,S:Type].
\mforall{}es:EO+(Info). \mforall{}P:E {}\mrightarrow{} S {}\mrightarrow{} \mBbbP{}. \mforall{}init:Id {}\mrightarrow{} bag(S). \mforall{}f:Id {}\mrightarrow{} A {}\mrightarrow{} S {}\mrightarrow{} S. \mforall{}X:EClass(A). \mforall{}e:E. \mforall{}v:S.
(single-valued-bag(init loc(e);S)
{}\mRightarrow{} single-valued-classrel(es;X;A)
{}\mRightarrow{} (\mforall{}s:S. \mforall{}e':E.
(e' \mleq{}loc e
{}\mRightarrow{} if first(e')
then s \mdownarrow{}\mmember{} init loc(e')
else s \mmember{} State-loc-comb(init;f;X)(pred(e')) \mwedge{} P[pred(e');s]
fi
{}\mRightarrow{} if e' \mmember{}\msubb{} X then \mforall{}a:A. (a \mmember{} X(e') {}\mRightarrow{} P[e';f loc(e') a s]) else P[e';s] fi ))
{}\mRightarrow{} v \mmember{} State-loc-comb(init;f;X)(e)
{}\mRightarrow{} P[e;v])
Date html generated:
2015_07_22-PM-00_24_38
Last ObjectModification:
2015_01_28-AM-10_10_37
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