Nuprl Lemma : disjoint-union-classrel-ite-weak
∀[Info,A,B:Type]. ∀[X:EClass(A)]. ∀[Y:EClass(B)]. ∀[es:EO+(Info)]. ∀[e:E]. ∀[v:A + B].
(v ∈ X + Y(e) ⇐⇒ ↓((↑isl(v)) ∧ outl(v) ∈ X(e)) ∨ ((¬↑isl(v)) ∧ outr(v) ∈ Y(e) ∧ (∀w:A. (¬w ∈ X(e)))))
Proof
Definitions occuring in Statement :
disjoint-union-class: X + Y,
classrel: v ∈ X(e),
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-E: E,
outr: outr(x),
outl: outl(x),
assert: ↑b,
isl: isl(x),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
squash: ↓T,
or: P ∨ Q,
and: P ∧ Q,
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
rev_implies: P ⇐ Q,
outl: outl(x),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
false: False,
outr: outr(x),
not: ¬A,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
all: ∀x:A. B[x],
or: P ∨ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
classrel: v ∈ X(e),
bag-member: x ↓∈ bs
Latex:
\mforall{}[Info,A,B:Type]. \mforall{}[X:EClass(A)]. \mforall{}[Y:EClass(B)]. \mforall{}[es:EO+(Info)]. \mforall{}[e:E]. \mforall{}[v:A + B].
(v \mmember{} X + Y(e)
\mLeftarrow{}{}\mRightarrow{} \mdownarrow{}((\muparrow{}isl(v)) \mwedge{} outl(v) \mmember{} X(e)) \mvee{} ((\mneg{}\muparrow{}isl(v)) \mwedge{} outr(v) \mmember{} Y(e) \mwedge{} (\mforall{}w:A. (\mneg{}w \mmember{} X(e)))))
Date html generated:
2016_05_17-AM-09_26_28
Last ObjectModification:
2016_01_17-PM-11_10_12
Theory : classrel!lemmas
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