Nuprl Lemma : uniform-continuity-from-fan-ext
∀[T:Type]
∀F:(ℕ ⟶ 𝔹) ⟶ T
(⇃(∃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))))])
⇒ ⇃(∃n:ℕ. ∀f,g:ℕ ⟶ 𝔹. ((f = g ∈ (ℕn ⟶ 𝔹)) ⇒ ((F f) = (F g) ∈ T))))
Proof
Definitions occuring in Statement :
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,
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 :
member: t ∈ T,
isl: isl(x),
btrue: tt,
it: ⋅,
bfalse: ff,
subtract: n - m,
ifthenelse: if b then t else f fi ,
ucont: ucont(F;M),
uniform-continuity-from-fan,
implies-quotient-true2,
trivial-quotient-true,
fan_theorem-ext,
decidable__assert,
implies-quotient-true,
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 :
uniform-continuity-from-fan,
lifting-strict-callbyvalue,
istype-void,
strict4-spread,
lifting-strict-decide,
strict4-decide,
implies-quotient-true2,
trivial-quotient-true,
fan_theorem-ext,
decidable__assert,
implies-quotient-true
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{}[T:Type]
\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} T
(\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))))])
{}\mRightarrow{} \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_36
Last ObjectModification:
2019_03_12-PM-06_00_00
Theory : continuity
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