Nuprl Lemma : mu-wf2
∀[P:ℕ ⟶ ℙ]. ∀[d:∀n:ℕ. Dec(P[n])]. mu(d) ∈ ℕ supposing ∃n:ℕ. P[n]
Proof
Definitions occuring in Statement :
mu: mu(f),
nat: ℕ,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
top: Top,
mu: mu(f),
exists: ∃x:A. B[x],
nat: ℕ,
int_upper: {i...},
isl: isl(x),
bfalse: ff,
true: True,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
guard: {T}
Lemmas referenced :
mu-ge_wf2,
subtype_rel_dep_function,
nat_wf,
decidable_wf,
int_upper_wf,
top_wf,
upper_subtype_nat,
istype-void,
subtype_rel_union,
not_wf,
subtype_rel_self,
assert_wf,
btrue_wf,
bfalse_wf,
equal_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
natural_numberEquality,
Error :isect_memberFormation_alt,
hypothesis,
hypothesisEquality,
applyEquality,
sqequalRule,
Error :lambdaEquality_alt,
Error :universeIsType,
because_Cache,
unionEquality,
independent_isectElimination,
independent_pairFormation,
Error :lambdaFormation_alt,
Error :isect_memberEquality_alt,
voidElimination,
Error :unionIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :productIsType,
Error :functionIsType,
universeEquality,
productElimination,
Error :dependent_pairFormation_alt,
functionExtensionality,
Error :inhabitedIsType,
unionElimination,
Error :equalityIsType1,
dependent_functionElimination,
independent_functionElimination,
lambdaFormation,
lemma_by_obid
Latex:
\mforall{}[P:\mBbbN{} {}\mrightarrow{} \mBbbP{}]. \mforall{}[d:\mforall{}n:\mBbbN{}. Dec(P[n])]. mu(d) \mmember{} \mBbbN{} supposing \mexists{}n:\mBbbN{}. P[n]
Date html generated:
2019_06_20-PM-01_17_09
Last ObjectModification:
2018_10_06-AM-11_21_47
Theory : int_2
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