### 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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