Nuprl Lemma : es-interface-extensionality
∀[Info,A:Type]. ∀[X,Y:EClass(A)].
(X = Y ∈ EClass(A)) supposing
((∀es:EO+(Info). ∀e:E. ((↑e ∈b X) ⇒ (↑e ∈b Y) ⇒ (X(e) = Y(e) ∈ A))) and
(∀es:EO+(Info). ∀e:E. (↑e ∈b X ⇐⇒ ↑e ∈b Y)) and
Singlevalued(X) and
Singlevalued(Y))
Proof
Definitions occuring in Statement :
sv-class: Singlevalued(X),
eclass-val: X(e),
in-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-E: E,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Lemmas :
es-E_wf,
event-ordering+_subtype,
all_wf,
event-ordering+_wf,
assert_wf,
in-eclass_wf,
dep-eclass_subtype_rel,
top_wf,
eclass-val_wf,
iff_wf,
sv-class_wf,
eclass_wf,
bag_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
bag-only_wf2,
single-valued-bag-if-le1,
decidable__lt,
false_wf,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
add-commutes,
zero-add,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
bag-size_wf,
nequal-le-implies,
equal-wf-base-T,
true_wf,
nat_wf,
le_wf,
bag-size-one,
bag_only_single_lemma,
and_wf,
single-bag_wf,
decidable__le,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
add-associates,
minus-zero,
le-add-cancel2,
empty-bag_wf,
bag-size-zero
\mforall{}[Info,A:Type]. \mforall{}[X,Y:EClass(A)].
(X = Y) supposing
((\mforall{}es:EO+(Info). \mforall{}e:E. ((\muparrow{}e \mmember{}\msubb{} X) {}\mRightarrow{} (\muparrow{}e \mmember{}\msubb{} Y) {}\mRightarrow{} (X(e) = Y(e)))) and
(\mforall{}es:EO+(Info). \mforall{}e:E. (\muparrow{}e \mmember{}\msubb{} X \mLeftarrow{}{}\mRightarrow{} \muparrow{}e \mmember{}\msubb{} Y)) and
Singlevalued(X) and
Singlevalued(Y))
Date html generated:
2015_07_17-PM-00_46_54
Last ObjectModification:
2015_01_27-PM-11_09_37
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