Nuprl Lemma : member-eclass-simple-comb-2-iff
∀[Info,A,B,C:Type]. ∀[es:EO+(Info)]. ∀[e:E]. ∀[F:bag(A) ─→ bag(B) ─→ bag(C)]. ∀[X:EClass(A)]. ∀[Y:EClass(B)].
(uiff(↑e ∈b F|X, Y|;(↑e ∈b X) ∧ (↑e ∈b Y) ∧ (¬↑bag-null(F {X@e} {Y@e})))) supposing
(single-valued-classrel(es;Y;B) and
single-valued-classrel(es;X;A) and
lifting2-like(A;B;F))
Proof
Definitions occuring in Statement :
lifting2-like: lifting2-like(A;B;f),
simple-comb-2: F|X, Y|,
classfun-res: 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-E: E,
assert: ↑b,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
apply: f a,
function: x:A ─→ B[x],
universe: Type,
bag-null: bag-null(bs),
single-bag: {x},
bag: bag(T)
Lemmas :
assert_witness,
assert_wf,
member-eclass_wf,
simple-comb-2_wf,
not_wf,
bag-null_wf,
single-bag_wf,
classfun-res_wf,
single-valued-classrel_wf,
lifting2-like_wf,
eclass_wf,
es-E_wf,
event-ordering+_subtype,
event-ordering+_wf,
bag_wf,
bnot_wf,
eq_int_wf,
bag-size_wf,
nat_wf,
equal-wf-T-base,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
assert_of_eq_int,
equal_wf,
null-bag-size,
empty-bag-iff-size,
le_wf,
decidable__lt,
false_wf,
not-le-2,
condition-implies-le,
minus-add,
minus-zero,
add-zero,
add-commutes,
zero-add,
add_functionality_wrt_le,
le-add-cancel,
member-eclass-iff-size,
less_than_wf,
bag-member-classfun-res,
bag-member_wf,
neg_assert_of_eq_int
Latex:
\mforall{}[Info,A,B,C:Type]. \mforall{}[es:EO+(Info)]. \mforall{}[e:E]. \mforall{}[F:bag(A) {}\mrightarrow{} bag(B) {}\mrightarrow{} bag(C)]. \mforall{}[X:EClass(A)].
\mforall{}[Y:EClass(B)].
(uiff(\muparrow{}e \mmember{}\msubb{} F|X, Y|;(\muparrow{}e \mmember{}\msubb{} X) \mwedge{} (\muparrow{}e \mmember{}\msubb{} Y) \mwedge{} (\mneg{}\muparrow{}bag-null(F \{X@e\} \{Y@e\})))) supposing
(single-valued-classrel(es;Y;B) and
single-valued-classrel(es;X;A) and
lifting2-like(A;B;F))
Date html generated:
2015_07_23-AM-11_27_27
Last ObjectModification:
2015_01_28-PM-11_17_04
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