Nuprl Lemma : disjoint-union-classrel
∀[Info,A,B:Type]. ∀[X:EClass(A)]. ∀[Y:EClass(B)]. ∀[es:EO+(Info)]. ∀[e:E].
∀v:A + B. uiff(v ∈ X + Y(e);case v of inl(x) => x ∈ X(e) | inr(x) => x ∈ 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,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
decide: case b of inl(x) => s[x] | inr(y) => t[y],
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
disjoint-union-class: X + Y,
classrel: v ∈ X(e),
simple-comb-2: F|X, Y|,
simple-comb: simple-comb(F;Xs),
select: L[n],
cons: [a / b],
subtract: n - m,
member: t ∈ T,
eclass: EClass(A[eo; e]),
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
prop: ℙ,
sq_stable: SqStable(P),
not: ¬A,
false: False,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
squash: ↓T,
exists: ∃x:A. B[x],
isl: isl(x),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bag-member: x ↓∈ bs,
bfalse: ff,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
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
uiff(v \mmember{} X + Y(e);case v of inl(x) => x \mmember{} X(e) | inr(x) => x \mmember{} Y(e) \mwedge{} (\mforall{}w:A. (\mneg{}w \mmember{} X(e))))
Date html generated:
2016_05_17-AM-09_26_16
Last ObjectModification:
2016_01_17-PM-11_10_49
Theory : classrel!lemmas
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