Nuprl Lemma : mu'_wf
∀[A:Type]. ∀[P:A ⟶ ℕ ⟶ 𝔹]. ∀[d:∀x:A. Dec(∃n:ℕ. (↑(P x n)))]. (mu'(P) ∈ A ⟶ (ℕ + Top))
Proof
Definitions occuring in Statement :
mu': mu'(P),
nat: ℕ,
assert: ↑b,
bool: 𝔹,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
top: Top,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
mu': mu'(P),
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
p-mu-decider,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
top: Top
Lemmas referenced :
all_wf,
decidable_wf,
exists_wf,
nat_wf,
assert_wf,
bool_wf,
p-mu-decider,
top_wf,
p-mu_wf,
pi1_wf_top,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
isect_memberEquality,
because_Cache,
functionEquality,
universeEquality,
instantiate,
isectEquality,
cumulativity,
unionEquality,
functionExtensionality,
lambdaFormation,
productElimination,
independent_pairEquality,
voidElimination,
voidEquality,
dependent_functionElimination,
independent_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)))]. (mu'(P) \mmember{} A {}\mrightarrow{} (\mBbbN{} + Top))
Date html generated:
2018_05_21-PM-06_29_44
Last ObjectModification:
2018_05_19-PM-04_40_21
Theory : general
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