Nuprl Lemma : strong-continuity2-implies-uniform-continuity2-int
∀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 :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uniform-continuity-pi: ucA(T;F;n),
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
nat: ℕ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cand: A c∧ B,
quotient: x,y:A//B[x; y],
squash: ↓T,
sq_type: SQType(T),
guard: {T},
uniform-continuity-pi-pi: ucpB(T;F;n)
Lemmas referenced :
istype-nat,
bool_wf,
istype-int,
strong-continuity2-implies-uniform-continuity-int-ext,
uniform-continuity-pi-pi-prop2,
decidable__equal_int,
int_seg_wf,
subtype_rel_function,
nat_wf,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
int_subtype_base,
true_wf,
quotient_wf,
exists_wf,
uniform-continuity-pi-pi_wf,
equiv_rel_true,
quotient-member-eq,
member_wf,
squash_wf,
istype-universe,
prop-truncation-implies,
uniform-continuity-pi-pi-prop,
subtype_base_sq,
set_subtype_base,
le_wf,
equal_wf,
equal-wf-base,
uniform-continuity-pi_wf,
le_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
Error :functionIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
Error :universeIsType,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
intEquality,
because_Cache,
independent_functionElimination,
Error :inhabitedIsType,
sqequalRule,
productElimination,
independent_pairFormation,
rename,
Error :productIsType,
Error :equalityIstype,
isectElimination,
natural_numberEquality,
setElimination,
applyEquality,
independent_isectElimination,
sqequalBase,
equalitySymmetry,
promote_hyp,
Error :lambdaEquality_alt,
closedConclusion,
productEquality,
pointwiseFunctionality,
pertypeElimination,
imageElimination,
equalityTransitivity,
instantiate,
universeEquality,
imageMemberEquality,
baseClosed,
cumulativity,
Error :dependent_pairEquality_alt,
independent_pairEquality,
Error :functionExtensionality_alt,
functionExtensionality,
functionEquality,
equalityElimination
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_28
Last ObjectModification:
2018_11_28-AM-09_01_29
Theory : continuity
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