Nuprl Lemma : Q-R-glues-trivial-split
∀[Info:Type]. ∀[P:es:EO+(Info) ─→ E ─→ ℙ].
∀p:∀es:EO+(Info). ∀e:E. Dec(P[es;e])
∀[A,B:Type].
∀Ia:EClass(A). ∀Ib:EClass(B).
(Singlevalued(Ib)
⇒ (∀g:es:EO+(Info) ─→ E(Ib) ─→ E. ∀q:∀es:EO+(Info). ∀e:E. Dec((↑e ∈b Ib) c∧ P[es;g es e]).
((∀es:EO+(Info). ∀e:E. P[es;g es e] supposing ↑e ∈b Ib)
⇒ (∀es:EO+(Info). ∀f:E(Ia) ─→ B.
∀[Q,R:E ─→ E ─→ ℙ]. (g es glues (Ia|p):Q ──f─→ (Ib|q):R ⇒ g es glues (Ia|p):Q ──f─→ Ib:R)))))
Proof
Definitions occuring in Statement :
Q-R-glues: g glues Ia:Qa ──f─→ Ib:Rb,
es-interface-restrict: (I|p),
es-E-interface: E(X),
sv-class: Singlevalued(X),
in-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-E: E,
assert: ↑b,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ─→ B[x],
universe: Type
Lemmas :
Q-R-glues-trivial-restrict,
assert_wf,
in-eclass_wf,
es-interface-subtype_rel2,
es-E_wf,
event-ordering+_subtype,
event-ordering+_wf,
top_wf,
es-interface-restrict_wf,
subtype_rel_dep_function,
es-E-interface_wf,
es-E-interface-restrict,
not_wf,
decidable__assert,
bag_wf,
le_wf,
bag-size_wf,
nat_wf,
nequal_wf,
neg_assert_of_eq_int,
eq_int_wf,
equal-wf-T-base,
bag-size-zero,
decidable__le,
false_wf,
not-le-2,
not-equal-2,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
zero-add,
add_functionality_wrt_le,
add-associates,
le-add-cancel2,
add-swap,
empty-bag_wf
Latex:
\mforall{}[Info:Type]. \mforall{}[P:es:EO+(Info) {}\mrightarrow{} E {}\mrightarrow{} \mBbbP{}].
\mforall{}p:\mforall{}es:EO+(Info). \mforall{}e:E. Dec(P[es;e])
\mforall{}[A,B:Type].
\mforall{}Ia:EClass(A). \mforall{}Ib:EClass(B).
(Singlevalued(Ib)
{}\mRightarrow{} (\mforall{}g:es:EO+(Info) {}\mrightarrow{} E(Ib) {}\mrightarrow{} E. \mforall{}q:\mforall{}es:EO+(Info). \mforall{}e:E. Dec((\muparrow{}e \mmember{}\msubb{} Ib) c\mwedge{} P[es;g es e]).
((\mforall{}es:EO+(Info). \mforall{}e:E. P[es;g es e] supposing \muparrow{}e \mmember{}\msubb{} Ib)
{}\mRightarrow{} (\mforall{}es:EO+(Info). \mforall{}f:E(Ia) {}\mrightarrow{} B.
\mforall{}[Q,R:E {}\mrightarrow{} E {}\mrightarrow{} \mBbbP{}].
(g es glues (Ia|p):Q {}{}f{}\mrightarrow{} (Ib|q):R {}\mRightarrow{} g es glues (Ia|p):Q {}{}f{}\mrightarrow{} Ib:R)))))
Date html generated:
2015_07_21-PM-04_11_26
Last ObjectModification:
2015_01_27-PM-05_42_31
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