Nuprl Lemma : fpf-split
∀[A:Type]
∀eq:EqDecider(A)
∀[B:A ⟶ Type]
∀f:a:A fp-> B[a]
∀[P:A ⟶ ℙ]
((∀a:A. Dec(P[a]))
⇒ (∃fp,fnp:a:A fp-> B[a]
((f ⊆ fp ⊕ fnp ∧ fp ⊕ fnp ⊆ f)
∧ ((∀a:A. P[a] supposing ↑a ∈ dom(fp)) ∧ (∀a:A. ¬P[a] supposing ↑a ∈ dom(fnp)))
∧ fpf-domain(fp) ⊆ fpf-domain(f)
∧ fpf-domain(fnp) ⊆ fpf-domain(f))))
Proof
Definitions occuring in Statement :
fpf-join: f ⊕ g,
fpf-sub: f ⊆ g,
fpf-domain: fpf-domain(f),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
sublist: L1 ⊆ L2,
deq: EqDecider(T),
assert: ↑b,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
fpf: a:A fp-> B[a],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
top: Top,
not: ¬A,
false: False,
guard: {T},
fpf-join: f ⊕ g,
fpf-sub: f ⊆ g,
pi1: fst(t),
fpf-cap: f(x)?z,
fpf-dom: x ∈ dom(f),
rev_implies: P ⇐ Q,
or: P ∨ Q,
decidable: Dec(P),
fpf-domain: fpf-domain(f)
Lemmas referenced :
all_wf,
decidable_wf,
fpf_wf,
deq_wf,
l_member_wf,
filter_wf5,
dcdr-to-bool_wf,
subtype_rel_dep_function,
subtype_rel_sets,
member_filter_2,
subtype_rel_self,
set_wf,
bnot_wf,
assert_witness,
fpf-dom_wf,
top_wf,
assert_wf,
fpf-sub_wf,
fpf-join_wf,
isect_wf,
subtype-fpf2,
not_wf,
sublist_wf,
fpf-domain_wf,
exists_wf,
fpf_ap_pair_lemma,
assert-deq-member,
append_wf,
deq-member_wf,
trivial-ifthenelse,
trivial-equal,
member_append,
member_filter,
or_wf,
dcdr-to-bool-equivalence,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
filter_is_sublist
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
functionEquality,
cumulativity,
universeEquality,
dependent_pairEquality,
setEquality,
setElimination,
functionExtensionality,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
isect_memberEquality,
voidElimination,
voidEquality,
instantiate,
productEquality,
independent_pairEquality,
axiomEquality,
addLevel,
orFunctionality,
impliesFunctionality,
andLevelFunctionality,
impliesLevelFunctionality,
unionElimination,
inlFormation,
inrFormation,
dependent_set_memberEquality,
promote_hyp
Latex:
\mforall{}[A:Type]
\mforall{}eq:EqDecider(A)
\mforall{}[B:A {}\mrightarrow{} Type]
\mforall{}f:a:A fp-> B[a]
\mforall{}[P:A {}\mrightarrow{} \mBbbP{}]
((\mforall{}a:A. Dec(P[a]))
{}\mRightarrow{} (\mexists{}fp,fnp:a:A fp-> B[a]
((f \msubseteq{} fp \moplus{} fnp \mwedge{} fp \moplus{} fnp \msubseteq{} f)
\mwedge{} ((\mforall{}a:A. P[a] supposing \muparrow{}a \mmember{} dom(fp)) \mwedge{} (\mforall{}a:A. \mneg{}P[a] supposing \muparrow{}a \mmember{} dom(fnp)))
\mwedge{} fpf-domain(fp) \msubseteq{} fpf-domain(f)
\mwedge{} fpf-domain(fnp) \msubseteq{} fpf-domain(f))))
Date html generated:
2018_05_21-PM-09_24_58
Last ObjectModification:
2018_05_19-PM-04_38_11
Theory : finite!partial!functions
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