Nuprl Lemma : general-cantor-to-int-bounded
∀B:ℕ ⟶ ℕ+. ∀F:(n:ℕ ⟶ ℕB[n]) ⟶ ℤ. ∃bnd:ℕ. ∀f:n:ℕ ⟶ ℕB[n]. (|F f| ≤ bnd)
Proof
Definitions occuring in Statement :
absval: |i|,
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
true: True,
prop: ℙ,
squash: ↓T,
surject: Surj(A;B;f),
nat_plus: ℕ+,
nat: ℕ,
compose: f o g,
subtype_rel: A ⊆r B,
so_apply: x[s],
uall: ∀[x:A]. B[x],
and: P ∧ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
iff_weakening_equal,
subtype_rel_self,
true_wf,
squash_wf,
le_wf,
istype-int,
nat_plus_wf,
istype-nat,
absval_wf,
istype-le,
int_seg_wf,
bool_wf,
nat_wf,
compose_wf,
cantor-to-int-bounded,
cantor-to-general-cantor
Rules used in proof :
independent_functionElimination,
independent_isectElimination,
universeEquality,
instantiate,
baseClosed,
imageMemberEquality,
inhabitedIsType,
equalitySymmetry,
equalityTransitivity,
imageElimination,
universeIsType,
rename,
setElimination,
lambdaEquality_alt,
functionIsType,
dependent_pairFormation_alt,
intEquality,
sqequalRule,
because_Cache,
applyEquality,
natural_numberEquality,
hypothesis,
functionEquality,
isectElimination,
productElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}B:\mBbbN{} {}\mrightarrow{} \mBbbN{}\msupplus{}. \mforall{}F:(n:\mBbbN{} {}\mrightarrow{} \mBbbN{}B[n]) {}\mrightarrow{} \mBbbZ{}. \mexists{}bnd:\mBbbN{}. \mforall{}f:n:\mBbbN{} {}\mrightarrow{} \mBbbN{}B[n]. (|F f| \mleq{} bnd)
Date html generated:
2019_10_15-AM-10_26_35
Last ObjectModification:
2019_10_03-PM-06_59_43
Theory : continuity
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