Nuprl Lemma : decidable-cantor-to-int-ext
∀[R:ℤ ⟶ ℤ ⟶ ℙ]. ((∀x,y:ℤ. Dec(R[x;y])) ⇒ (∀F:(ℕ ⟶ 𝔹) ⟶ ℤ. Dec(∃f,g:ℕ ⟶ 𝔹. R[F f;F g])))
Proof
Definitions occuring in Statement :
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],
int: ℤ
Definitions unfolded in proof :
member: t ∈ T,
bfalse: ff,
it: ⋅,
btrue: tt,
subtract: n - m,
ifthenelse: if b then t else f fi ,
let: let,
lt_int: i <z j,
so_lambda: λ2x.t[x],
spreadn: spread3,
decidable-cantor-to-int,
cantor-to-int-uniform-continuity,
decidable-finite-cantor-to-int,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
uimplies: b supposing a,
so_apply: x[s]
Lemmas referenced :
decidable-cantor-to-int,
lifting-strict-spread,
istype-void,
strict4-spread,
lifting-strict-callbyvalue,
lifting-strict-less,
strict4-decide,
cantor-to-int-uniform-continuity,
decidable-finite-cantor-to-int
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination
Latex:
\mforall{}[R:\mBbbZ{} {}\mrightarrow{} \mBbbZ{} {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x,y:\mBbbZ{}. Dec(R[x;y])) {}\mRightarrow{} (\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbZ{}. Dec(\mexists{}f,g:\mBbbN{} {}\mrightarrow{} \mBbbB{}. R[F f;F g])))
Date html generated:
2019_10_15-AM-10_26_38
Last ObjectModification:
2019_08_05-PM-02_14_27
Theory : continuity
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