Nuprl Lemma : eclass-cond-classrel
∀[Info,B:Type]. ∀[X:EClass(B ⟶ B)]. ∀[Y:EClass(B)]. ∀[es:EO+(Info)]. ∀[e:E]. ∀[v:B].
uiff(v ∈ eclass-cond(X;Y)(
e);↓if e ∈b X then ∃f:B ⟶ B. ∃b:B. (f ∈ X(e) ∧ b ∈ Y(e) ∧ (v = (f b) ∈ B)) else v ∈ Y(e) fi )
Proof
Definitions occuring in Statement :
eclass-cond: eclass-cond(X;Y),
classrel: v ∈ X(e),
member-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-E: E,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
classrel: v ∈ X(e),
bag-member: x ↓∈ bs,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
so_lambda: λ2x.t[x],
so_apply: x[s],
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
eclass-cond: eclass-cond(X;Y),
class-ap: X(e),
not: ¬A,
sq_stable: SqStable(P)
Latex:
\mforall{}[Info,B:Type]. \mforall{}[X:EClass(B {}\mrightarrow{} B)]. \mforall{}[Y:EClass(B)]. \mforall{}[es:EO+(Info)]. \mforall{}[e:E]. \mforall{}[v:B].
uiff(v \mmember{} eclass-cond(X;Y)(e);\mdownarrow{}if e \mmember{}\msubb{} X
then \mexists{}f:B {}\mrightarrow{} B. \mexists{}b:B. (f \mmember{} X(e) \mwedge{} b \mmember{} Y(e) \mwedge{} (v = (f b)))
else v \mmember{} Y(e)
fi )
Date html generated:
2016_05_16-PM-02_14_41
Last ObjectModification:
2016_01_17-PM-07_39_35
Theory : event-ordering
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