### Nuprl Lemma : strong-continuity2-implies-uniform-continuity2-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 :  int_seg: `{i..j-}` nat: `ℕ` bool: `𝔹` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` subtype_rel: `A ⊆r B` nat: `ℕ` implies: `P `` Q` uniform-continuity-pi: `ucA(T;F;n)` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` prop: `ℙ` so_apply: `x[s]` uimplies: `b supposing a` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` exists: `∃x:A. B[x]` true: `True` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` cand: `A c∧ B` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` guard: `{T}` quotient: `x,y:A//B[x; y]` squash: `↓T` sq_type: `SQType(T)` uniform-continuity-pi-pi: `ucpB(T;F;n)`
Lemmas referenced :  nat_wf bool_wf strong-continuity2-implies-uniform-continuity-int uniform-continuity-pi-pi-prop2 decidable__equal_nat exists_wf all_wf equal_wf int_seg_wf subtype_rel_dep_function int_seg_subtype_nat false_wf subtype_rel_self true_wf quotient_wf uniform-continuity-pi-pi_wf equiv_rel_true quotient-member-eq equal_subtype nat_properties decidable__equal_int satisfiable-full-omega-tt intformand_wf intformnot_wf intformeq_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_formula_prop_wf decidable__le intformle_wf itermConstant_wf int_formula_prop_le_lemma int_term_value_constant_lemma le_wf equal-wf-base member_wf squash_wf prop-truncation-implies uniform-continuity-pi-pi-prop subtype_base_sq set_subtype_base int_subtype_base uniform-continuity-pi_wf less_than'_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation functionEquality cut introduction extract_by_obid hypothesis sqequalHypSubstitution dependent_functionElimination thin functionExtensionality applyEquality hypothesisEquality lambdaEquality setElimination rename sqequalRule independent_functionElimination because_Cache productElimination independent_pairFormation isectElimination natural_numberEquality independent_isectElimination intEquality promote_hyp equalityTransitivity equalitySymmetry dependent_pairEquality unionElimination dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality computeAll dependent_set_memberEquality applyLambdaEquality pointwiseFunctionality pertypeElimination productEquality imageElimination imageMemberEquality baseClosed instantiate cumulativity independent_pairEquality equalityElimination axiomEquality

Latex:
\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  \mBbbN{}.  \mexists{}n:\mBbbN{}.  \mforall{}f,g:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}.    ((f  =  g)  {}\mRightarrow{}  ((F  f)  =  (F  g)))

Date html generated: 2017_04_17-AM-09_59_33
Last ObjectModification: 2017_02_27-PM-05_52_48

Theory : continuity

Home Index