Nuprl Lemma : can-apply-mu'
∀[A:Type]. ∀P:A ⟶ ℕ ⟶ 𝔹. ∀d:∀x:A. Dec(∃n:ℕ. (↑(P x n))). ∀x:A. (↑can-apply(mu'(P);x) ⇐⇒ ∃n:ℕ. (↑(P x n)))
Proof
Definitions occuring in Statement :
mu': mu'(P),
can-apply: can-apply(f;x),
nat: ℕ,
assert: ↑b,
bool: 𝔹,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
mu': mu'(P),
can-apply: can-apply(f;x),
p-mu-decider,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
pi1: fst(t),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
p-mu: p-mu(P;x),
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
bfalse: ff,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
not: ¬A
Lemmas referenced :
all_wf,
decidable_wf,
exists_wf,
nat_wf,
assert_wf,
bool_wf,
p-mu-decider,
top_wf,
p-mu_wf,
true_wf,
false_wf,
not_over_exists,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalRule,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
lambdaEquality,
hypothesis,
applyEquality,
functionEquality,
universeEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
isectEquality,
cumulativity,
unionEquality,
functionExtensionality,
because_Cache,
productElimination,
unionElimination,
independent_pairFormation,
dependent_pairFormation,
natural_numberEquality,
voidElimination,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}[A:Type]
\mforall{}P:A {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbB{}. \mforall{}d:\mforall{}x:A. Dec(\mexists{}n:\mBbbN{}. (\muparrow{}(P x n))). \mforall{}x:A. (\muparrow{}can-apply(mu'(P);x) \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}. (\muparrow{}(P x n)))
Date html generated:
2018_05_21-PM-06_29_49
Last ObjectModification:
2018_05_19-PM-04_40_29
Theory : general
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