Nuprl Lemma : loop-class-memory-classrel
∀[Info,B:Type]. ∀[X:EClass(B ─→ B)]. ∀[init:Id ─→ bag(B)]. ∀[es:EO+(Info)]. ∀[e:E]. ∀[v:B].
uiff(v ∈ loop-class-memory(X;init)(e);↓if first(e)
then v ↓∈ init loc(e)
else ∃b:B
(b ∈ loop-class-memory(X;init)(pred(e))
∧ if pred(e) ∈b X
then ∃f:B ─→ B. (f ∈ X(pred(e)) ∧ (v = (f b) ∈ B))
else v = b ∈ B
fi )
fi )
Proof
Definitions occuring in Statement :
loop-class-memory: loop-class-memory(X;init),
classrel: v ∈ X(e),
member-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
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 ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
and: P ∧ Q,
apply: f a,
function: x:A ─→ B[x],
universe: Type,
equal: s = t ∈ T,
bag-member: x ↓∈ bs,
bag: bag(T)
Lemmas :
es-first_wf2,
bool_wf,
eqtt_to_assert,
primed-class-opt-classrel,
es-locl-first,
assert_elim,
btrue_neq_bfalse,
classrel_wf,
es-locl_wf,
exists_wf,
es-p-local-pred_wf,
squash_wf,
uiff_transitivity,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
eqff_to_assert,
assert_of_bnot,
member-eclass_wf,
es-pred_wf,
eclass3-classrel,
loop-class-memory-exists,
es-loc-pred,
less_than_wf,
bag-size_wf,
nat_wf,
loop-class-memory_wf,
event-ordering+_subtype,
bag-member_wf,
es-loc_wf,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
es-E_wf,
event-ordering+_wf,
Id_wf,
bag_wf,
eclass_wf,
es-pred_property,
es-pred-locl,
es-causl_weakening,
assert-member-eclass,
eclass3_wf,
true_wf,
iff_weakening_equal,
bag-member-size,
es-pred-loc-base,
es-locl_transitivity2,
es-le_weakening_eq,
es-locl_irreflexivity,
es-causl_transitivity2,
es-causle_weakening,
es-causl_irreflexivity,
all_wf,
es-le_weakening,
and_wf
Latex:
\mforall{}[Info,B:Type]. \mforall{}[X:EClass(B {}\mrightarrow{} B)]. \mforall{}[init:Id {}\mrightarrow{} bag(B)]. \mforall{}[es:EO+(Info)]. \mforall{}[e:E]. \mforall{}[v:B].
uiff(v \mmember{} loop-class-memory(X;init)(e);\mdownarrow{}if first(e)
then v \mdownarrow{}\mmember{} init loc(e)
else \mexists{}b:B
(b \mmember{} loop-class-memory(X;init)(pred(e))
\mwedge{} if pred(e) \mmember{}\msubb{} X
then \mexists{}f:B {}\mrightarrow{} B. (f \mmember{} X(pred(e)) \mwedge{} (v = (f b)))
else v = b
fi )
fi )
Date html generated:
2015_07_21-PM-02_36_00
Last ObjectModification:
2015_02_04-PM-06_22_13
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