Nuprl Lemma : glues-iff
∀[Info:Type]
∀es:EO+(Info)
∀[A,B:Type].
∀Ia:EClass(A). ∀Ib:EClass(B). ∀f:E(Ia) ⟶ B. ∀g:E(Ib) ⟶ E(Ia).
(g glues Ia ──f⟶ Ib
⇐⇒ {Bij(E(Ib);E(Ia);g)
∧ (∀e:E(Ib). g e c≤ e)
∧ (∀e,e':E(Ib). (g e ≤loc g e' ⇒ e ≤loc e' ))
∧ (∀e:E(Ib). ((f (g e)) = Ib(e) ∈ B))})
Proof
Definitions occuring in Statement :
glues: g glues Ia ──f⟶ Ib,
es-E-interface: E(X),
eclass-val: X(e),
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-causle: e c≤ e',
es-le: e ≤loc e' ,
biject: Bij(A;B;f),
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
true: True,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
sq_type: SQType(T),
guard: {T},
rev_implies: P ⇐ Q,
es-E-interface: E(X),
so_apply: x[s],
so_lambda: λ2x.t[x],
top: Top,
uimplies: b supposing a,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
subtype_rel: A ⊆r B,
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
cand: A c∧ B,
glues: g glues Ia ──f⟶ Ib,
Q-R-glues: g glues Ia:Qa ──f⟶ Ib:Rb,
weak-antecedent-surjection: Q ←←= f== P,
weak-antecedent-function: Q ←==f== P,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
exists: ∃x:A. B[x],
surject: Surj(A;B;f),
es-interface-predicate: {I},
Q-R-pre-preserving: f is Q-R-pre-preserving on P
Latex:
\mforall{}[Info:Type]
\mforall{}es:EO+(Info)
\mforall{}[A,B:Type].
\mforall{}Ia:EClass(A). \mforall{}Ib:EClass(B). \mforall{}f:E(Ia) {}\mrightarrow{} B. \mforall{}g:E(Ib) {}\mrightarrow{} E(Ia).
(g glues Ia {}{}f{}\mrightarrow{} Ib
\mLeftarrow{}{}\mRightarrow{} \{Bij(E(Ib);E(Ia);g)
\mwedge{} (\mforall{}e:E(Ib). g e c\mleq{} e)
\mwedge{} (\mforall{}e,e':E(Ib). (g e \mleq{}loc g e' {}\mRightarrow{} e \mleq{}loc e' ))
\mwedge{} (\mforall{}e:E(Ib). ((f (g e)) = Ib(e)))\})
Date html generated:
2016_05_17-AM-07_59_10
Last ObjectModification:
2015_12_28-PM-11_24_55
Theory : event-ordering
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