Nuprl Lemma : mu-dec_wf
∀[A:Type]. ∀[P:A ⟶ ℕ ⟶ ℙ]. ∀[d:a:A ⟶ k:ℕ ⟶ Dec(P[a;k])]. ∀[a:A]. mu-dec(d;a) ∈ ℕ supposing ↓∃k:ℕ. P[a;k]
Proof
Definitions occuring in Statement :
mu-dec: mu-dec(d;a),
nat: ℕ,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
exists: ∃x:A. B[x],
squash: ↓T,
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
squash: ↓T,
member: t ∈ T,
mu-dec: mu-dec(d;a),
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
or: P ∨ Q,
decidable: Dec(P),
true: True,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
isl: isl(x),
false: False,
not: ¬A
Lemmas referenced :
mu_wf,
isl_wf,
not_wf,
nat_wf,
squash_wf,
exists_wf,
decidable_wf,
assert_wf,
subtype_rel_self,
equal_wf
Rules used in proof :
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
imageElimination,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
Error :lambdaEquality_alt,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
hypothesis,
because_Cache,
sqequalRule,
Error :universeIsType,
independent_isectElimination,
Error :functionIsType,
universeEquality,
Error :isect_memberFormation_alt,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :isect_memberEquality_alt,
productElimination,
Error :dependent_pairFormation_alt,
independent_functionElimination,
dependent_functionElimination,
lambdaFormation,
lemma_by_obid,
unionElimination,
natural_numberEquality,
voidElimination,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
instantiate,
Error :equalityIsType1
Latex:
\mforall{}[A:Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbP{}]. \mforall{}[d:a:A {}\mrightarrow{} k:\mBbbN{} {}\mrightarrow{} Dec(P[a;k])]. \mforall{}[a:A].
mu-dec(d;a) \mmember{} \mBbbN{} supposing \mdownarrow{}\mexists{}k:\mBbbN{}. P[a;k]
Date html generated:
2019_06_20-PM-01_17_27
Last ObjectModification:
2018_10_03-PM-10_14_54
Theory : int_2
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