Nuprl Lemma : State-comb-classrel-mem2
∀[Info,B,A:Type]. ∀[f:A ─→ B ─→ B]. ∀[init:Id ─→ bag(B)].
∀X:EClass(A). ∀es:EO+(Info). ∀e:E.
∀[v:B]
(v ∈ State-comb(init;f;X)(e)
⇐⇒ if e ∈b X
then ↓∃w:B. ∃a:A. (w ∈ Memory-class(f;init;X)(e) ∧ (v = (f a w) ∈ B) ∧ a ∈ X(e))
else v ∈ Memory-class(f;init;X)(e)
fi )
Proof
Definitions occuring in Statement :
State-comb: State-comb(init;f;X),
Memory-class: Memory-class(f;init;X),
classrel: v ∈ X(e),
member-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-E: E,
Id: Id,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
squash: ↓T,
and: P ∧ Q,
apply: f a,
function: x:A ─→ B[x],
universe: Type,
equal: s = t ∈ T,
bag: bag(T)
Lemmas :
classrel_wf,
State-comb_wf,
member-eclass_wf,
bool_wf,
eqtt_to_assert,
squash_wf,
exists_wf,
Memory-class_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
es-E_wf,
event-ordering+_subtype,
event-ordering+_wf,
eclass_wf,
Id_wf,
bag_wf,
State-comb-classrel,
es-first_wf2,
bool_cases,
assert_of_bnot,
sq_stable__squash,
uiff_transitivity,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
sq_stable__classrel,
assert-member-eclass,
Memory-classrel,
iterated_classrel_wf,
es-pred_wf,
and_wf,
bag-member_wf,
es-loc_wf,
all_wf,
or_wf
Latex:
\mforall{}[Info,B,A:Type]. \mforall{}[f:A {}\mrightarrow{} B {}\mrightarrow{} B]. \mforall{}[init:Id {}\mrightarrow{} bag(B)].
\mforall{}X:EClass(A). \mforall{}es:EO+(Info). \mforall{}e:E.
\mforall{}[v:B]
(v \mmember{} State-comb(init;f;X)(e)
\mLeftarrow{}{}\mRightarrow{} if e \mmember{}\msubb{} X
then \mdownarrow{}\mexists{}w:B. \mexists{}a:A. (w \mmember{} Memory-class(f;init;X)(e) \mwedge{} (v = (f a w)) \mwedge{} a \mmember{} X(e))
else v \mmember{} Memory-class(f;init;X)(e)
fi )
Date html generated:
2015_07_22-PM-00_21_24
Last ObjectModification:
2015_01_28-AM-10_14_47
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