Nuprl Lemma : strong-continuity2-implies-uniform-continuity-nat-ext
∀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 :
member: t ∈ T,
compose: f o g,
ifthenelse: if b then t else f fi ,
pi1: fst(t),
strong-continuity-test: strong-continuity-test(M;n;f;b),
isl: isl(x),
is_int: is_int(x),
mu: mu(f),
mu-ge: mu-ge(f;n),
subtract: n - m,
let: let,
strong-continuity2-implies-uniform-continuity-nat,
uniform-continuity-from-fan-ext,
strong-continuity2-no-inner-squash-cantor2,
strong-continuity2-half-squash-surject-biject,
surject-nat-bool,
trivial-biject-exists,
implies-quotient-true2,
trivial-quotient-true,
strong-continuity2_biject_retract-ext,
id-biject,
strong-continuity2_functionality_surject,
bool_cases_sqequal,
any: any x,
strong-continuity2-half-squash,
implies-quotient-true,
strong-continuity2-iff-3,
strong-continuity3_functionality_surject,
basic-implies-strong-continuity2-ext,
strong-continuity2-implies-3,
surject-inverse,
decidable__assert,
strong-continuity-test-prop1,
strong-continuity-test-prop2,
not-isl-assert-isr,
bool_cases,
eqtt_to_assert,
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],
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
btrue: tt,
bfalse: ff
Lemmas referenced :
strong-continuity2-implies-uniform-continuity-nat,
lifting-strict-decide,
istype-void,
strict4-spread,
strict4-decide,
has-value_wf_base,
is-exception_wf,
lifting-strict-callbyvalue,
lifting-strict-isint,
uniform-continuity-from-fan-ext,
strong-continuity2-no-inner-squash-cantor2,
strong-continuity2-half-squash-surject-biject,
surject-nat-bool,
trivial-biject-exists,
implies-quotient-true2,
trivial-quotient-true,
strong-continuity2_biject_retract-ext,
id-biject,
strong-continuity2_functionality_surject,
bool_cases_sqequal,
strong-continuity2-half-squash,
implies-quotient-true,
strong-continuity2-iff-3,
strong-continuity3_functionality_surject,
basic-implies-strong-continuity2-ext,
strong-continuity2-implies-3,
surject-inverse,
decidable__assert,
strong-continuity-test-prop1,
strong-continuity-test-prop2,
not-isl-assert-isr,
bool_cases,
eqtt_to_assert
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,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
sqequalSqle,
divergentSqle,
callbyvalueDecide,
hypothesisEquality,
unionElimination,
sqleReflexivity,
Error :equalityIstype,
dependent_functionElimination,
independent_functionElimination,
decideExceptionCases,
axiomSqleEquality,
exceptionSqequal,
baseApply,
closedConclusion
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_52
Last ObjectModification:
2019_03_12-PM-09_29_30
Theory : continuity
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