Nuprl Lemma : cantor-to-int-uniform-continuity
∀F:(ℕ ⟶ 𝔹) ⟶ ℤ. ∃n:ℕ. ∀f,g:ℕ ⟶ 𝔹. ((f = g ∈ (ℕn ⟶ 𝔹)) ⇒ ((F f) = (F g) ∈ ℤ))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
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,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
subtract: n - m,
ifthenelse: if b then t else f fi ,
ext2Cantor: ext2Cantor(n;f;d),
lt_int: i <z j,
it: ⋅,
bfalse: ff,
btrue: tt,
strong-continuity2-implies-uniform-continuity2-int,
uniform-continuity-pi-pi-prop2,
decidable__equal_int,
prop-truncation-implies,
strong-continuity2-implies-uniform-continuity-int-ext,
uniform-continuity-pi-dec,
decidable__int_equal,
sq_stable__all,
sq_stable__le,
any: any x,
decidable_functionality,
uniform-continuity-pi2-dec-ext,
iff_weakening_equal,
iff_preserves_decidability,
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_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
strong-continuity2-implies-uniform-continuity2-int,
lifting-strict-spread,
istype-void,
strict4-spread,
lifting-strict-callbyvalue,
lifting-strict-decide,
strict4-decide,
lifting-strict-int_eq,
lifting-strict-less,
uniform-continuity-pi-pi-prop2,
decidable__equal_int,
prop-truncation-implies,
strong-continuity2-implies-uniform-continuity-int-ext,
uniform-continuity-pi-dec,
decidable__int_equal,
sq_stable__all,
sq_stable__le,
decidable_functionality,
uniform-continuity-pi2-dec-ext,
iff_weakening_equal,
iff_preserves_decidability
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
Error :isect_memberEquality_alt,
voidElimination,
independent_isectElimination
Latex:
\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbZ{}. \mexists{}n:\mBbbN{}. \mforall{}f,g:\mBbbN{} {}\mrightarrow{} \mBbbB{}. ((f = g) {}\mRightarrow{} ((F f) = (F g)))
Date html generated:
2019_06_20-PM-02_53_31
Last ObjectModification:
2019_03_12-PM-05_57_02
Theory : continuity
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