Nuprl Lemma : ucont_wf
∀[T:Type]. ∀[F:(ℕ ⟶ 𝔹) ⟶ T]. ∀[M:⇃(∃M:n:ℕ ⟶ (ℕn ⟶ 𝔹) ⟶ (T?) [(∀f:ℕ ⟶ 𝔹
((∃n:ℕ. ((M n f) = (inl (F f)) ∈ (T?)))
∧ (∀n:ℕ
(M n f) = (inl (F f)) ∈ (T?)
supposing ↑isl(M n f))))])].
(ucont(F;M) ∈ ⇃(∃n:ℕ. ∀f,g:ℕ ⟶ 𝔹. ((f = g ∈ (ℕn ⟶ 𝔹)) ⇒ ((F f) = (F g) ∈ T))))
Proof
Definitions occuring in Statement :
ucont: ucont(F;M),
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
isl: isl(x),
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
true: True,
unit: Unit,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
inl: inl x,
union: left + right,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ucont: ucont(F;M),
uniform-continuity-from-fan-ext,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
nat: ℕ,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
exists: ∃x:A. B[x],
isl: isl(x),
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
uniform-continuity-from-fan-ext,
bool_wf,
quotient_wf,
sq_exists_wf,
nat_wf,
int_seg_wf,
unit_wf2,
equal_wf,
assert_wf,
btrue_wf,
bfalse_wf,
istype-nat,
true_wf,
istype-assert,
equiv_rel_true,
isl_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
Error :lambdaEquality_alt,
isectElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
Error :isectIsType,
Error :inhabitedIsType,
Error :functionIsType,
because_Cache,
Error :universeIsType,
sqequalHypSubstitution,
functionEquality,
natural_numberEquality,
setElimination,
rename,
unionEquality,
productEquality,
Error :inlEquality_alt,
isectEquality,
Error :lambdaFormation_alt,
unionElimination,
Error :equalityIstype,
dependent_functionElimination,
independent_functionElimination,
Error :unionIsType,
Error :setIsType,
Error :productIsType,
independent_isectElimination,
axiomEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} T]. \mforall{}[M:\00D9(\mexists{}M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} (T?) [(\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}
((\mexists{}n:\mBbbN{}
((M n f) = (inl (F f))))
\mwedge{} (\mforall{}n:\mBbbN{}
(M n f) = (inl (F f))
supposing \muparrow{}isl(M n
f))))])].
(ucont(F;M) \mmember{} \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_39
Last ObjectModification:
2019_01_26-PM-06_13_12
Theory : continuity
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