### Nuprl Lemma : weak-continuity-equipollent

`∀T:Type. (T ~ ℕ `` (∀F:(ℕ ⟶ T) ⟶ ℕ. ∀f:ℕ ⟶ T.  ⇃(∃n:ℕ. ∀g:ℕ ⟶ T. ((f = g ∈ (ℕn ⟶ T)) `` ((F f) = (F g) ∈ ℕ)))))`

Proof

Definitions occuring in Statement :  equipollent: `A ~ B` quotient: `x,y:A//B[x; y]` int_seg: `{i..j-}` nat: `ℕ` 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` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` equipollent: `A ~ B` exists: `∃x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` and: `P ∧ Q` so_lambda: `λ2x.t[x]` prop: `ℙ` nat: `ℕ` subtype_rel: `A ⊆r B` so_apply: `x[s]` uimplies: `b supposing a` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` guard: `{T}` compose: `f o g` int_seg: `{i..j-}` ge: `i ≥ j ` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  biject-inverse nat_wf strong-continuity2-implies-weak compose_wf equipollent_wf exists_wf all_wf equal_wf int_seg_wf subtype_rel_dep_function int_seg_subtype_nat false_wf subtype_rel_self implies-quotient-true int_seg_properties nat_properties decidable__le le_wf full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformnot_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_not_lemma int_formula_prop_wf and_wf squash_wf true_wf iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation sqequalHypSubstitution productElimination thin rename cut introduction extract_by_obid isectElimination hypothesisEquality hypothesis independent_functionElimination dependent_functionElimination lambdaEquality applyEquality functionExtensionality functionEquality cumulativity sqequalRule universeEquality because_Cache natural_numberEquality setElimination independent_isectElimination independent_pairFormation dependent_pairFormation dependent_set_memberEquality unionElimination equalityTransitivity equalitySymmetry applyLambdaEquality approximateComputation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality hyp_replacement imageElimination imageMemberEquality baseClosed

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

Date html generated: 2017_09_29-PM-06_05_56
Last ObjectModification: 2017_07_05-PM-05_53_35

Theory : continuity

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