Nuprl Lemma : fpf-restrict-cap
∀[A:Type]. ∀[P:A ⟶ 𝔹]. ∀[x:A].
∀[f:x:A fp-> Top]. ∀[eq:EqDecider(A)]. ∀[z:Top]. (fpf-restrict(f;P)(x)?z ~ f(x)?z) supposing ↑(P x)
Proof
Definitions occuring in Statement :
fpf-restrict: fpf-restrict(f;P),
fpf-cap: f(x)?z,
fpf: a:A fp-> B[a],
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
fpf-cap: f(x)?z,
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B
Lemmas referenced :
ap_fpf_restrict_lemma,
top_wf,
deq_wf,
fpf_wf,
assert_wf,
bool_wf,
subtype_base_sq,
bool_subtype_base,
iff_imp_equal_bool,
fpf-dom_wf,
fpf-restrict_wf2,
domain_fpf_restrict_lemma,
member-fpf-domain,
and_wf,
l_member_wf,
fpf-domain_wf,
member_filter,
filter_wf5,
subtype_rel_dep_function,
subtype_rel_self,
set_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
because_Cache,
isectElimination,
hypothesisEquality,
lambdaEquality,
applyEquality,
functionEquality,
universeEquality,
isect_memberFormation,
axiomSqEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
cumulativity,
independent_isectElimination,
independent_functionElimination,
productElimination,
independent_pairFormation,
lambdaFormation,
rename,
setElimination,
setEquality,
impliesFunctionality,
addLevel,
lemma_by_obid
Latex:
\mforall{}[A:Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[x:A].
\mforall{}[f:x:A fp-> Top]. \mforall{}[eq:EqDecider(A)]. \mforall{}[z:Top]. (fpf-restrict(f;P)(x)?z \msim{} f(x)?z)
supposing \muparrow{}(P x)
Date html generated:
2019_10_16-AM-11_26_33
Last ObjectModification:
2018_08_29-PM-02_20_40
Theory : finite!partial!functions
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