Nuprl Lemma : rec-comb-es-sv
∀[Info,B:Type]. ∀[n:ℕ]. ∀[A:ℕn ─→ Type]. ∀[Xs:k:ℕn ─→ EClass(A k)]. ∀[F:Id ─→ (k:ℕn ─→ bag(A k)) ─→ bag(B) ─→ bag(B)].
∀[init:Id ─→ bag(B)]. ∀[es:EO+(Info)].
(es-sv-class(es;rec-comb(Xs;F;init))) supposing
((∀bs:k:ℕn ─→ bag(A k). ∀l:Id. ∀b:bag(B). ((∀k:ℕn. (#(bs k) ≤ 1)) ⇒ (#(b) ≤ 1) ⇒ (#(F l bs b) ≤ 1))) and
(∀k:ℕn. es-sv-class(es;Xs k)) and
(∀l:Id. (#(init l) ≤ 1)))
Proof
Definitions occuring in Statement :
rec-comb: rec-comb(X;f;init),
es-sv-class: es-sv-class(es;X),
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
Id: Id,
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ─→ B[x],
natural_number: $n,
universe: Type,
bag-size: #(bs),
bag: bag(T)
Lemmas :
es-causl-swellfnd,
event-ordering+_subtype,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
bag-size_wf,
rec-comb_wf2,
nat_wf,
es-E_wf,
le_wf,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
decidable__lt,
es-causl_wf,
eclass_wf,
event-ordering+_wf,
es-loc_wf,
primed-class-opt_wf,
primed-class-opt-es-sv0,
es-locl_wf,
zero-le-nat,
add-mul-special,
zero-mul,
all_wf,
int_seg_wf,
bag_wf,
Id_wf,
es-sv-class_wf
Latex:
\mforall{}[Info,B:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[A:\mBbbN{}n {}\mrightarrow{} Type]. \mforall{}[Xs:k:\mBbbN{}n {}\mrightarrow{} EClass(A k)]. \mforall{}[F:Id
{}\mrightarrow{} (k:\mBbbN{}n {}\mrightarrow{} bag(A k))
{}\mrightarrow{} bag(B)
{}\mrightarrow{} bag(B)].
\mforall{}[init:Id {}\mrightarrow{} bag(B)]. \mforall{}[es:EO+(Info)].
(es-sv-class(es;rec-comb(Xs;F;init))) supposing
((\mforall{}bs:k:\mBbbN{}n {}\mrightarrow{} bag(A k). \mforall{}l:Id. \mforall{}b:bag(B).
((\mforall{}k:\mBbbN{}n. (\#(bs k) \mleq{} 1)) {}\mRightarrow{} (\#(b) \mleq{} 1) {}\mRightarrow{} (\#(F l bs b) \mleq{} 1))) and
(\mforall{}k:\mBbbN{}n. es-sv-class(es;Xs k)) and
(\mforall{}l:Id. (\#(init l) \mleq{} 1)))
Date html generated:
2015_07_22-PM-00_15_27
Last ObjectModification:
2015_01_28-AM-10_45_46
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