Nuprl Lemma : Q-R-glues-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(Ia) ∧ Singlevalued(Ib))
⇒ (∀g1:es:EO+(Info) ─→ E(Ib) ─→ E. ∀q:∀es:EO+(Info). ∀e:E. Dec((↑e ∈b Ib) c∧ P[es;g1 es e]). ∀es:EO+(Info).
∀[Q,R:E ─→ E ─→ ℙ].
((∀x,y:E. ((Q x y) ⇒ (P[es;x] ⇐⇒ P[es;y])))
⇒ (∀f:E(Ia) ─→ B. ∀g2:E(Ib) ─→ E.
(g1 es glues (Ia|p):Q ──f─→ (Ib|q):R
⇒ g2 glues (Ia|¬p):Q ──f─→ (Ib|¬q):R
⇒ [λe.P[es;g1 es e]? g1 es : g2] glues Ia:Q ──f─→ Ib:R)))))
Proof
Definitions occuring in Statement :
Q-R-glues: g glues Ia:Qa ──f─→ Ib:Rb,
es-interface-co-restrict: (I|¬p),
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),
conditional: [P? f : g],
es-E: E,
compose: f o g,
assert: ↑b,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
cand: A c∧ B,
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ─→ B[x],
universe: Type
Lemmas :
Q-R-glues_wf,
es-interface-co-restrict_wf,
subtype_rel_dep_function,
es-E-interface_wf,
top_wf,
es-E-interface-co-restrict,
subtype_rel_self,
es-interface-restrict_wf,
es-E_wf,
event-ordering+_subtype,
assert_wf,
in-eclass_wf,
es-E-interface-restrict,
es-interface-subtype_rel2,
all_wf,
iff_wf,
decidable_wf,
sv-class_wf,
eclass_wf,
event-ordering+_wf,
cond-class_wf,
es-E-interface-conditional-subtype_rel,
es-interface-restrict-disjoint,
Q-R-glues-conditional2,
or_wf,
es-is-interface-restrict,
assert_elim,
subtype_base_sq,
bool_wf,
bool_subtype_base,
es-is-interface-co-restrict,
bool-decider_wf,
equal_wf,
es-interface-restrict-conditional,
es-E-interface_functionality,
and_wf,
member_wf,
squash_wf,
true_wf,
equal_functionality_wrt_subtype_rel2,
subtype_rel_weakening,
ext-eq_weakening,
rel_equivalent_wf,
rel_or_wf,
rel-restriction_wf,
es-interface-predicate_wf,
Q-R-glues_functionality,
event-ordering+_cumulative2
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(Ia) \mwedge{} Singlevalued(Ib))
{}\mRightarrow{} (\mforall{}g1: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;g1 es e]). \mforall{}es:EO+(Info).
\mforall{}[Q,R:E {}\mrightarrow{} E {}\mrightarrow{} \mBbbP{}].
((\mforall{}x,y:E. ((Q x y) {}\mRightarrow{} (P[es;x] \mLeftarrow{}{}\mRightarrow{} P[es;y])))
{}\mRightarrow{} (\mforall{}f:E(Ia) {}\mrightarrow{} B. \mforall{}g2:E(Ib) {}\mrightarrow{} E.
(g1 es glues (Ia|p):Q {}{}f{}\mrightarrow{} (Ib|q):R
{}\mRightarrow{} g2 glues (Ia|\mneg{}p):Q {}{}f{}\mrightarrow{} (Ib|\mneg{}q):R
{}\mRightarrow{} [\mlambda{}e.P[es;g1 es e]? g1 es : g2] glues Ia:Q {}{}f{}\mrightarrow{} Ib:R)))))
Date html generated:
2015_07_21-PM-04_11_02
Last ObjectModification:
2015_01_27-PM-06_05_51
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