### Nuprl Lemma : strong-continuity2-implies-uniform-continuity-ext

F:(ℕ ⟶ 𝔹) ⟶ 𝔹. ⇃(∃n:ℕ. ∀f,g:ℕ ⟶ 𝔹.  ((f g ∈ (ℕn ⟶ 𝔹))  g))

Proof

Definitions occuring in Statement :  quotient: x,y:A//B[x; y] int_seg: {i..j-} nat: bool: 𝔹 all: x:A. B[x] exists: x:A. B[x] implies:  Q true: True apply: a function: x:A ⟶ B[x] natural_number: \$n equal: t ∈ T
Definitions unfolded in proof :  member: t ∈ T ifthenelse: if then else fi  compose: g pi1: fst(t) strong-continuity-test: strong-continuity-test(M;n;f;b) isl: isl(x) lt_int: i <j let: let strong-continuity2-implies-uniform-continuity uniform-continuity-from-fan-ext implies-quotient-true2 trivial-quotient-true strong-continuity2-no-inner-squash-cantor4 implies-quotient-true strong-continuity2-half-squash-surject-biject retraction-nat-nsub surject-nat-bool biject-bool-nsub2 strong-continuity2_biject_retract-ext bool_cases_sqequal any: any x decidable__int_equal strong-continuity2_functionality_surject strong-continuity2-half-squash strong-continuity2-iff-3 strong-continuity3_functionality_surject basic-implies-strong-continuity2-ext strong-continuity2-implies-3 surject-inverse 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: λ2y.t[x; y] top: Top so_apply: x[s1;s2] uimplies: supposing a so_lambda: λ2x.t[x] so_apply: x[s] all: x:A. B[x] implies:  Q has-value: (a)↓ strict4: strict4(F) and: P ∧ Q prop: or: P ∨ Q squash: T btrue: tt bfalse: ff
Lemmas referenced :  strong-continuity2-implies-uniform-continuity lifting-strict-decide istype-void strict4-spread strict4-decide has-value_wf_base is-exception_wf lifting-strict-callbyvalue lifting-strict-int_eq lifting-strict-isint value-type-has-value int-value-type istype-base lifting-strict-less uniform-continuity-from-fan-ext implies-quotient-true2 trivial-quotient-true strong-continuity2-no-inner-squash-cantor4 implies-quotient-true strong-continuity2-half-squash-surject-biject retraction-nat-nsub surject-nat-bool biject-bool-nsub2 strong-continuity2_biject_retract-ext bool_cases_sqequal decidable__int_equal strong-continuity2_functionality_surject strong-continuity2-half-squash strong-continuity2-iff-3 strong-continuity3_functionality_surject basic-implies-strong-continuity2-ext strong-continuity2-implies-3 surject-inverse
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 independent_pairFormation callbyvalueIntEq productElimination intEquality Error :universeIsType,  int_eqExceptionCases Error :inrFormation_alt,  imageMemberEquality imageElimination Error :inlFormation_alt,  because_Cache

Latex:
\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  \mBbbB{}.  \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_46
Last ObjectModification: 2019_03_12-PM-04_21_49

Theory : continuity

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