Nuprl Lemma : fpf-join-list-dom
∀[A:Type]. ∀eq:EqDecider(A). ∀[B:A ⟶ Type]. ∀L:a:A fp-> B[a] List. ∀x:A. (↑x ∈ dom(⊕(L)) ⇐⇒ (∃f∈L. ↑x ∈ dom(f)))
Proof
Definitions occuring in Statement :
fpf-join-list: ⊕(L),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
l_exists: (∃x∈L. P[x]),
list: T List,
deq: EqDecider(T),
assert: ↑b,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
prop: ℙ,
implies: P ⇒ Q,
fpf-join-list: ⊕(L),
fpf-empty: ⊗,
fpf-dom: x ∈ dom(f),
pi1: fst(t),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
and: P ∧ Q,
false: False,
rev_implies: P ⇐ Q,
or: P ∨ Q,
guard: {T}
Lemmas referenced :
list_induction,
fpf_wf,
all_wf,
iff_wf,
assert_wf,
fpf-dom_wf,
fpf-join-list_wf,
top_wf,
subtype_rel_list,
subtype-fpf2,
l_exists_wf,
l_member_wf,
list_wf,
deq_wf,
reduce_nil_lemma,
deq_member_nil_lemma,
false_wf,
l_exists_nil,
l_exists_wf_nil,
l_exists_cons,
cons_wf,
or_wf,
reduce_cons_lemma,
fpf-join-dom,
fpf-join_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
cumulativity,
because_Cache,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
setElimination,
rename,
setEquality,
independent_functionElimination,
dependent_functionElimination,
functionEquality,
universeEquality,
introduction,
independent_pairFormation,
productElimination,
independent_pairEquality,
addLevel,
allFunctionality,
impliesFunctionality,
unionElimination,
inlFormation,
inrFormation
Latex:
\mforall{}[A:Type]
\mforall{}eq:EqDecider(A)
\mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}L:a:A fp-> B[a] List. \mforall{}x:A. (\muparrow{}x \mmember{} dom(\moplus{}(L)) \mLeftarrow{}{}\mRightarrow{} (\mexists{}f\mmember{}L. \muparrow{}x \mmember{} dom(f)))
Date html generated:
2018_05_21-PM-09_22_40
Last ObjectModification:
2018_02_09-AM-10_18_51
Theory : finite!partial!functions
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