### Nuprl Lemma : weak-continuity-nat-int-bool

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

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` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` uall: `∀[x:A]. B[x]` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` nat: `ℕ` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` prop: `ℙ` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` so_apply: `x[s]` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top`
Lemmas referenced :  weak-continuity-nat-int nat_wf bool_wf eqtt_to_assert false_wf le_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot exists_wf all_wf int_seg_wf subtype_rel_dep_function int_seg_subtype_nat subtype_rel_self implies-quotient-true btrue_wf equal-wf-base nat_properties full-omega-unsat intformeq_wf itermConstant_wf int_formula_prop_eq_lemma int_term_value_constant_lemma int_formula_prop_wf bfalse_wf
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation hypothesis sqequalHypSubstitution dependent_functionElimination thin lambdaEquality applyEquality functionExtensionality hypothesisEquality functionEquality intEquality because_Cache unionElimination equalityElimination sqequalRule isectElimination productElimination independent_isectElimination dependent_set_memberEquality natural_numberEquality independent_pairFormation equalityTransitivity equalitySymmetry dependent_pairFormation promote_hyp instantiate cumulativity independent_functionElimination voidElimination setElimination rename baseClosed applyLambdaEquality approximateComputation isect_memberEquality voidEquality

Latex:
\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  \mBbbZ{})  {}\mrightarrow{}  \mBbbB{}.  \mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbZ{}.    \00D9(\mexists{}n:\mBbbN{}.  \mforall{}g:\mBbbN{}  {}\mrightarrow{}  \mBbbZ{}.  ((f  =  g)  {}\mRightarrow{}  F  f  =  F  g))

Date html generated: 2017_09_29-PM-06_06_35
Last ObjectModification: 2017_07_05-PM-06_21_55

Theory : continuity

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