Nuprl Lemma : decidable-finite-cantor-to-int
∀[R:ℤ ⟶ ℤ ⟶ ℙ]. ((∀x,y:ℤ. Dec(R[x;y])) ⇒ (∀n:ℕ. ∀F:(ℕn ⟶ 𝔹) ⟶ ℤ. Dec(∃f,g:ℕn ⟶ 𝔹. R[F f;F g])))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
nat: ℕ,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
finite-cantor-decider_wf,
int_seg_wf,
bool_wf,
nat_wf,
all_wf,
decidable_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
rename,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
functionEquality,
natural_numberEquality,
setElimination,
cumulativity,
universeEquality
Latex:
\mforall{}[R:\mBbbZ{} {}\mrightarrow{} \mBbbZ{} {}\mrightarrow{} \mBbbP{}]
((\mforall{}x,y:\mBbbZ{}. Dec(R[x;y])) {}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}F:(\mBbbN{}n {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbZ{}. Dec(\mexists{}f,g:\mBbbN{}n {}\mrightarrow{} \mBbbB{}. R[F f;F g])))
Date html generated:
2019_06_20-PM-02_49_53
Last ObjectModification:
2018_09_26-AM-09_54_21
Theory : continuity
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