Nuprl Lemma : simple-finite-cantor-decider_wf
∀[T:Type]. ∀[R:T ⟶ ℙ]. ∀[dcdr:∀x:T. Dec(R[x])]. ∀[n:ℕ]. ∀[F:(ℕn ⟶ 𝔹) ⟶ T].
(FiniteCantorDecide(dcdr;n;F) ∈ Dec(∃f:ℕn ⟶ 𝔹. R[F f]))
Proof
Definitions occuring in Statement :
simple-finite-cantor-decider: FiniteCantorDecide(dcdr;n;F),
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
prop: ℙ,
simple-decidable-finite-cantor-ext,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x]
Lemmas referenced :
int_seg_wf,
bool_wf,
nat_wf,
all_wf,
decidable_wf,
simple-decidable-finite-cantor-ext,
uall_wf,
exists_wf,
isect_wf,
equal_wf,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
extract_by_obid,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
isect_memberEquality,
because_Cache,
lambdaEquality,
applyEquality,
cumulativity,
universeEquality,
instantiate,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination,
isectEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} \mBbbP{}]. \mforall{}[dcdr:\mforall{}x:T. Dec(R[x])]. \mforall{}[n:\mBbbN{}]. \mforall{}[F:(\mBbbN{}n {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} T].
(FiniteCantorDecide(dcdr;n;F) \mmember{} Dec(\mexists{}f:\mBbbN{}n {}\mrightarrow{} \mBbbB{}. R[F f]))
Date html generated:
2019_06_20-PM-02_49_56
Last ObjectModification:
2018_09_26-AM-09_54_17
Theory : continuity
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