Nuprl Lemma : weak-continuity-implies-strong1

`(∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ⇃(∃M:(ℕ ⟶ ℕ) ⟶ ℕ. ∀f,g:ℕ ⟶ ℕ.  ((f = g ∈ (ℕM f ⟶ ℕ)) `` ((F f) = (F g) ∈ ℕ))))`
` (∀F:(ℕ ⟶ ℕ) ⟶ ℕ`
`      ⇃(∃M:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ (ℕ?)`
`         ∀f:ℕ ⟶ ℕ`
`           ((∃n:ℕ. ((M n f) = (inl (F f)) ∈ (ℕ?))) ∧ (∀n:ℕ. (M n f) = (inl (F f)) ∈ (ℕ?) supposing ↑isl(M n f)))))`

Proof

Definitions occuring in Statement :  quotient: `x,y:A//B[x; y]` int_seg: `{i..j-}` nat: `ℕ` assert: `↑b` isl: `isl(x)` uimplies: `b supposing a` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` true: `True` unit: `Unit` apply: `f a` function: `x:A ⟶ B[x]` inl: `inl x` union: `left + right` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  implies: `P `` Q` all: `∀x:A. B[x]` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` nat: `ℕ` so_apply: `x[s]` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` exists: `∃x:A. B[x]` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` exposed-it: `exposed-it` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` bfalse: `ff` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` ge: `i ≥ j ` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` ext2Baire: `ext2Baire(n;f;d)` int_seg: `{i..j-}` lelt: `i ≤ j < k` less_than: `a < b` squash: `↓T` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` isl: `isl(x)` true: `True` cand: `A c∧ B` quotient: `x,y:A//B[x; y]`
Lemmas referenced :  nat_wf all_wf quotient_wf exists_wf equal_wf int_seg_wf subtype_rel_dep_function int_seg_subtype_nat false_wf subtype_rel_self true_wf equiv_rel_true implies-prop-truncation unit_wf2 isect_wf assert_wf isl_wf le_int_wf ext2Baire_wf le_wf bool_wf eqtt_to_assert assert_of_le_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot imax_wf imax_nat nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_formula_prop_wf lt_int_wf assert_of_lt_int less_than_wf int_seg_properties decidable__lt intformless_wf int_formula_prop_less_lemma imax_strict_ub set_subtype_base int_subtype_base imax_ub decidable__equal_int prop-truncation-quot quotient-member-eq equal-wf-base member_wf squash_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation rename cut hypothesis sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality functionEquality introduction extract_by_obid isectElimination sqequalRule lambdaEquality because_Cache natural_numberEquality applyEquality functionExtensionality setElimination independent_isectElimination independent_pairFormation productElimination comment unionEquality productEquality inlEquality independent_functionElimination dependent_pairFormation dependent_set_memberEquality unionElimination equalityElimination equalityTransitivity equalitySymmetry promote_hyp instantiate cumulativity voidElimination inrEquality axiomEquality applyLambdaEquality int_eqEquality intEquality isect_memberEquality voidEquality computeAll imageElimination inlFormation inrFormation isect_memberFormation pointwiseFunctionality pertypeElimination imageMemberEquality baseClosed

Latex:
(\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbN{}.  \00D9(\mexists{}M:(\mBbbN{}  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbN{}.  \mforall{}f,g:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.    ((f  =  g)  {}\mRightarrow{}  ((F  f)  =  (F  g)))))
{}\mRightarrow{}  (\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbN{}
\00D9(\mexists{}M:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  (\mBbbN{}?)
\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}
((\mexists{}n:\mBbbN{}.  ((M  n  f)  =  (inl  (F  f))))  \mwedge{}  (\mforall{}n:\mBbbN{}.  (M  n  f)  =  (inl  (F  f))  supposing  \muparrow{}isl(M  n  f)))))

Date html generated: 2017_04_17-AM-09_59_59
Last ObjectModification: 2017_02_27-PM-05_55_15

Theory : continuity

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