Nuprl Lemma : fpf-union-join_wf
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[f,g:a:A fp-> B[a] List]. ∀[R:⋂a:A. ((B[a] List) ⟶ B[a] ⟶ 𝔹)].
(fpf-union-join(eq;R;f;g) ∈ a:A fp-> B[a] List)
Proof
Definitions occuring in Statement :
fpf-union-join: fpf-union-join(eq;R;f;g),
fpf: a:A fp-> B[a],
list: T List,
deq: EqDecider(T),
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fpf-union-join: fpf-union-join(eq;R;f;g),
fpf: a:A fp-> B[a],
pi1: fst(t),
all: ∀x:A. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
top: Top
Lemmas referenced :
append_wf,
filter_wf5,
l_member_wf,
bnot_wf,
fpf-dom_wf,
subtype-fpf2,
list_wf,
top_wf,
fpf-union_wf,
bool_wf,
fpf_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_pairEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
productElimination,
sqequalRule,
because_Cache,
lambdaEquality,
lambdaFormation,
hypothesis,
setElimination,
rename,
applyEquality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
setEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
axiomEquality,
isectEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f,g:a:A fp-> B[a] List]. \mforall{}[R:\mcap{}a:A
((B[a] List)
{}\mrightarrow{} B[a]
{}\mrightarrow{} \mBbbB{})].
(fpf-union-join(eq;R;f;g) \mmember{} a:A fp-> B[a] List)
Date html generated:
2018_05_21-PM-09_23_19
Last ObjectModification:
2018_02_09-AM-10_19_15
Theory : finite!partial!functions
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