Nuprl Lemma : p-mu-decider
∀[A:Type]. ∀P:A ⟶ ℕ ⟶ 𝔹. ((∀x:A. Dec(∃n:ℕ. (↑(P x n)))) ⇒ (∀x:A. ∃y:ℕ + Top. p-mu(P x;y)))
Proof
Definitions occuring in Statement :
p-mu: p-mu(P;x),
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],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
p-mu-exists,
all_wf,
decidable_wf,
exists_wf,
nat_wf,
assert_wf,
bool_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
applyEquality,
hypothesisEquality,
independent_functionElimination,
hypothesis,
isectElimination,
sqequalRule,
lambdaEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}P:A {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbB{}. ((\mforall{}x:A. Dec(\mexists{}n:\mBbbN{}. (\muparrow{}(P x n)))) {}\mRightarrow{} (\mforall{}x:A. \mexists{}y:\mBbbN{} + Top. p-mu(P x;y)))
Date html generated:
2016_05_15-PM-03_32_51
Last ObjectModification:
2015_12_27-PM-01_12_00
Theory : general
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