Nuprl Lemma : strong-continuity2-implies-uniform-continuity-nat
∀F:(ℕ ⟶ 𝔹) ⟶ ℕ. ⇃(∃n:ℕ. ∀f,g:ℕ ⟶ 𝔹. ((f = g ∈ (ℕn ⟶ 𝔹)) ⇒ ((F f) = (F g) ∈ ℕ)))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
true: True,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
nat: ℕ,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
exists: ∃x:A. B[x],
uimplies: b supposing a,
isl: isl(x),
sq_exists: ∃x:A [B[x]],
so_lambda: λ2x y.t[x; y],
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
quotient: x,y:A//B[x; y],
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
guard: {T}
Lemmas referenced :
uniform-continuity-from-fan-ext,
nat_wf,
istype-nat,
bool_wf,
strong-continuity2-no-inner-squash-cantor2,
quotient_wf,
sq_exists_wf,
int_seg_wf,
unit_wf2,
all_wf,
exists_wf,
equal-wf-base-T,
isect_wf,
assert_wf,
istype-assert,
true_wf,
union_subtype_base,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
unit_subtype_base,
equiv_rel_true,
btrue_wf,
bfalse_wf,
quotient-member-eq,
subtype_rel_function,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
pi1_wf,
isl_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
Error :functionIsType,
Error :universeIsType,
rename,
pointwiseFunctionalityForEquality,
closedConclusion,
functionEquality,
natural_numberEquality,
setElimination,
unionEquality,
sqequalRule,
Error :lambdaEquality_alt,
productEquality,
because_Cache,
Error :inhabitedIsType,
unionElimination,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
Error :unionIsType,
Error :setIsType,
Error :productIsType,
applyEquality,
independent_isectElimination,
intEquality,
Error :inlEquality_alt,
sqequalBase,
Error :isectIsType,
isectEquality,
pertypeElimination,
promote_hyp,
productElimination,
independent_pairFormation,
Error :dependent_pairEquality_alt,
Error :dependent_set_memberEquality_alt
Latex:
\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbN{}. \00D9(\mexists{}n:\mBbbN{}. \mforall{}f,g:\mBbbN{} {}\mrightarrow{} \mBbbB{}. ((f = g) {}\mRightarrow{} ((F f) = (F g))))
Date html generated:
2019_06_20-PM-02_52_49
Last ObjectModification:
2019_02_06-PM-05_39_34
Theory : continuity
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