Nuprl Lemma : strong-continuity2-implies-uniform-continuity-nat

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: 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 :  all: x:A. B[x] uall: [x:A]. B[x] member: t ∈ T implies:  Q nat: so_lambda: λ2x.t[x] prop: and: P ∧ Q so_apply: x[s] exists: x:A. B[x] uimplies: supposing a isl: isl(x) sq_exists: x:A [B[x]] so_lambda: λ2y.t[x; y] subtype_rel: A ⊆B so_apply: x[s1;s2] quotient: x,y:A//B[x; y] le: A ≤ B less_than': less_than'(a;b) false: False not: ¬A guard: {T}
Lemmas referenced :  uniform-continuity-from-fan-ext nat_wf istype-nat bool_wf strong-continuity2-no-inner-squash-cantor2 quotient_wf sq_exists_wf int_seg_wf unit_wf2 all_wf exists_wf equal-wf-base-T isect_wf assert_wf istype-assert true_wf union_subtype_base set_subtype_base le_wf istype-int int_subtype_base unit_subtype_base equiv_rel_true btrue_wf bfalse_wf quotient-member-eq subtype_rel_function int_seg_subtype_nat istype-false subtype_rel_self pi1_wf isl_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesis dependent_functionElimination hypothesisEquality independent_functionElimination Error :functionIsType,  Error :universeIsType,  rename pointwiseFunctionalityForEquality closedConclusion functionEquality natural_numberEquality setElimination unionEquality sqequalRule Error :lambdaEquality_alt,  productEquality because_Cache Error :inhabitedIsType,  unionElimination Error :equalityIstype,  equalityTransitivity equalitySymmetry Error :unionIsType,  Error :setIsType,  Error :productIsType,  applyEquality independent_isectElimination intEquality Error :inlEquality_alt,  sqequalBase Error :isectIsType,  isectEquality pertypeElimination promote_hyp productElimination independent_pairFormation Error :dependent_pairEquality_alt,  Error :dependent_set_memberEquality_alt

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_49
Last ObjectModification: 2019_02_06-PM-05_39_34

Theory : continuity


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