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
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
cand: A c∧ B,
uimplies: b supposing a,
top: Top,
es-E-interface: E(X),
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False,
guard: {T},
in-eclass: e ∈b X,
sv-class: Singlevalued(X),
eclass: EClass(A[eo; e]),
nat: ℕ,
uiff: uiff(P;Q),
and: P ∧ Q,
nequal: a ≠ b ∈ T ,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
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:
2016_05_17-AM-07_53_59
Last ObjectModification:
2016_01_17-PM-02_46_44
Theory : event-ordering
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