Nuprl Lemma : loop-class-memory-fun-eq
∀[Info,B:Type]. ∀[X:EClass(B ─→ B)]. ∀[init:Id ─→ bag(B)]. ∀[es:EO+(Info)]. ∀[e:E].
(loop-class-memory(X;init)(e)
= if first(e) then sv-bag-only(init loc(e))
if pred(e) ∈b X then X(pred(e)) loop-class-memory(X;init)(pred(e))
else loop-class-memory(X;init)(pred(e))
fi
∈ B) supposing
(single-valued-classrel(es;X;B ─→ B) and
(∀l:Id. single-valued-bag(init l;B)) and
(∀l:Id. (1 ≤ #(init l))))
Proof
Definitions occuring in Statement :
loop-class-memory: loop-class-memory(X;init),
classfun: X(e),
single-valued-classrel: single-valued-classrel(es;X;T),
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 ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
apply: f a,
function: x:A ─→ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T,
sv-bag-only: sv-bag-only(b),
single-valued-bag: single-valued-bag(b;T),
bag-size: #(bs),
bag: bag(T)
Lemmas :
es-local-pred_wf,
lt_int_wf,
bag-size_wf,
eclass3_wf,
loop-class-memory_wf,
eclass_wf,
es-E_wf,
event-ordering+_subtype,
event-ordering+_wf,
nat_wf,
or_wf,
sq_exists_wf,
es-locl_wf,
assert_wf,
all_wf,
not_wf,
es-locl-first,
assert_elim,
btrue_neq_bfalse,
es-loc_wf,
sv-bag-only_wf,
decidable__lt,
false_wf,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
equal_wf,
assert_of_lt_int,
member-eclass-iff-size,
es-pred_property,
es-pred-locl,
es-causl_weakening,
es-pred_wf,
es-loc-pred,
less_than_wf,
member-eclass_wf,
loop-class-memory-member,
loop-class-memory-single-val,
classfun-res_wf,
assert_functionality_wrt_uiff,
squash_wf,
true_wf,
member-eclass-eclass3,
classfun-res-eclass3,
iff_weakening_equal,
single-valued-classrel_wf,
es-locl-trichotomy,
bag_wf,
es-locl_transitivity2,
es-le_weakening_eq,
es-locl_irreflexivity,
es-causl_transitivity2,
es-causle_weakening,
es-causl_irreflexivity,
es-le_weakening,
and_wf,
eclass3-single-val,
Id_wf,
es-pred-loc-base,
single-valued-bag_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].
(loop-class-memory(X;init)(e)
= if first(e) then sv-bag-only(init loc(e))
if pred(e) \mmember{}\msubb{} X then X(pred(e)) loop-class-memory(X;init)(pred(e))
else loop-class-memory(X;init)(pred(e))
fi ) supposing
(single-valued-classrel(es;X;B {}\mrightarrow{} B) and
(\mforall{}l:Id. single-valued-bag(init l;B)) and
(\mforall{}l:Id. (1 \mleq{} \#(init l))))
Date html generated:
2015_07_23-AM-11_29_54
Last ObjectModification:
2015_02_04-PM-04_46_21
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