Nuprl Lemma : fpf-restrict-dom
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[f:x:A fp-> B[x]]. ∀[P:A ⟶ 𝔹]. ∀[x:A].
uiff(↑x ∈ dom(fpf-restrict(f;P));{(↑x ∈ dom(f)) ∧ (↑(P x))})
Proof
Definitions occuring in Statement :
fpf-restrict: fpf-restrict(f;P),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
guard: {T},
so_apply: x[s],
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
iff: P ⇐⇒ Q,
uimplies: b supposing a,
implies: P ⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
rev_implies: P ⇐ Q,
top: Top
Lemmas referenced :
assert_witness,
fpf-dom_wf,
assert_wf,
fpf-restrict_wf2,
top_wf,
bool_wf,
fpf_wf,
deq_wf,
domain_fpf_restrict_lemma,
and_wf,
l_member_wf,
fpf-domain_wf,
subtype-fpf2,
member_filter,
filter_wf5,
subtype_rel_dep_function,
subtype_rel_self,
set_wf,
iff_wf,
member-fpf-domain
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
independent_pairFormation,
productElimination,
independent_functionElimination,
sqequalRule,
independent_pairEquality,
lemma_by_obid,
isectElimination,
applyEquality,
because_Cache,
lambdaEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality,
lambdaFormation,
voidElimination,
voidEquality,
independent_isectElimination,
addLevel,
impliesFunctionality,
setEquality,
setElimination,
rename,
andLevelFunctionality
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f:x:A fp-> B[x]]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[x:A].
uiff(\muparrow{}x \mmember{} dom(fpf-restrict(f;P));\{(\muparrow{}x \mmember{} dom(f)) \mwedge{} (\muparrow{}(P x))\})
Date html generated:
2018_05_21-PM-09_31_20
Last ObjectModification:
2018_02_09-AM-10_25_45
Theory : finite!partial!functions
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