Nuprl Lemma : fpf-join-dom
∀[A:Type]. ∀[B:A ⟶ Type].
∀eq:EqDecider(A). ∀f,g:a:A fp-> B[a]. ∀x:A. (↑x ∈ dom(f ⊕ g) ⇐⇒ (↑x ∈ dom(f)) ∨ (↑x ∈ dom(g)))
Proof
Definitions occuring in Statement :
fpf-join: f ⊕ g,
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
assert: ↑b,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
fpf-dom: x ∈ dom(f),
fpf-join: f ⊕ g,
fpf: a:A fp-> B[a],
pi1: fst(t),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
bfalse: ff,
or: P ∨ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
false: False,
not: ¬A,
cand: A c∧ B,
true: True
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type].
\mforall{}eq:EqDecider(A). \mforall{}f,g:a:A fp-> B[a]. \mforall{}x:A. (\muparrow{}x \mmember{} dom(f \moplus{} g) \mLeftarrow{}{}\mRightarrow{} (\muparrow{}x \mmember{} dom(f)) \mvee{} (\muparrow{}x \mmember{} dom(g)))
Date html generated:
2016_05_16-AM-11_10_30
Last ObjectModification:
2015_12_29-AM-09_19_22
Theory : event-ordering
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