Nuprl Lemma : State-comb-fun-eq
∀[Info,B,A:Type]. ∀[f:A ─→ B ─→ B]. ∀[init:Id ─→ bag(B)]. ∀[X:EClass(A)]. ∀[es:EO+(Info)]. ∀[e:E].
(State-comb(init;f;X)(e)
= if e ∈b X then if first(e) then f X(e) sv-bag-only(init loc(e)) else f X(e) State-comb(init;f;X)(pred(e)) fi
if first(e) then sv-bag-only(init loc(e))
else State-comb(init;f;X)(pred(e))
fi
∈ B) supposing
(single-valued-classrel(es;X;A) and
(∀l:Id. single-valued-bag(init l;B)) and
(∀l:Id. (1 ≤ #(init l))))
Proof
Definitions occuring in Statement :
State-comb: State-comb(init;f;X),
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 :
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
single-valued-classrel_wf,
all_wf,
Id_wf,
single-valued-bag_wf,
le_wf,
bag-size_wf,
nat_wf,
es-E_wf,
event-ordering+_subtype,
event-ordering+_wf,
eclass_wf,
bag_wf,
bag-null_wf,
assert-bag-null,
equal-wf-T-base,
assert_wf,
bnot_wf,
eq_int_wf,
bag_size_empty_lemma,
es-local-pred_wf,
lt_int_wf,
or_wf,
sq_exists_wf,
es-locl_wf,
es-locl-first,
assert_elim,
btrue_neq_bfalse,
not_wf,
rec-comb_wf,
false_wf,
int_seg_wf,
select_wf,
cons_wf,
nil_wf,
sq_stable__le,
length_wf,
length_nil,
non_neg_length,
length_wf_nil,
length_cons,
length_wf_nat,
lelt_wf,
lifting-2_wf,
es-loc_wf,
sv-bag-only-combine,
bag-combine_wf,
single-bag_wf,
single-valued-classrel-implies-bag,
member-eclass-iff-size,
single-valued-bag-single,
single-valued-bag-combine,
sv-bag-only_wf,
decidable__lt,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
bag-member-sv-bag-only,
bag_size_single_lemma,
bag-combine-size-bound2,
sv-bag-only-single,
es-first_wf2,
bool_cases,
assert_of_bnot,
State-comb_wf,
es-pred_wf,
less_than_wf,
assert_of_lt_int,
iff_transitivity,
iff_weakening_uiff,
State-comb-single-val,
iff_weakening_equal,
State-comb-exists,
squash_wf,
true_wf,
reduce_hd_cons_lemma,
State-comb-exists-iff,
not-gt-2,
bag-member-size,
member-eclass-iff-non-empty,
State-comb-functional,
btrue_wf,
and_wf,
isl_wf,
bfalse_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].
(State-comb(init;f;X)(e)
= if e \mmember{}\msubb{} X
then if first(e)
then f X(e) sv-bag-only(init loc(e))
else f X(e) State-comb(init;f;X)(pred(e))
fi
if first(e) then sv-bag-only(init loc(e))
else State-comb(init;f;X)(pred(e))
fi ) supposing
(single-valued-classrel(es;X;A) and
(\mforall{}l:Id. single-valued-bag(init l;B)) and
(\mforall{}l:Id. (1 \mleq{} \#(init l))))
Date html generated:
2015_07_22-PM-00_22_00
Last ObjectModification:
2015_02_04-PM-04_44_00
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