### Nuprl Lemma : weak-continuity-implies-strong-cantor-unique

`∀F:(ℕ ⟶ 𝔹) ⟶ ℕ`
`  ∃M:n:ℕ ⟶ (ℕn ⟶ 𝔹) ⟶ (ℕ?)`
`   ∀f:ℕ ⟶ 𝔹. ∃n:ℕ. (((M n f) = (inl (F f)) ∈ (ℕ?)) ∧ (∀m:ℕ. ((↑isl(M m f)) `` (m = n ∈ ℕ))))`

Proof

Definitions occuring in Statement :  int_seg: `{i..j-}` nat: `ℕ` assert: `↑b` isl: `isl(x)` bool: `𝔹` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` 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 :  all: `∀x:A. B[x]` member: `t ∈ T` exists: `∃x:A. B[x]` uall: `∀[x:A]. B[x]` nat: `ℕ` and: `P ∧ Q` prop: `ℙ` so_lambda: `λ2x.t[x]` 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` implies: `P `` Q` decidable: `Dec(P)` or: `P ∨ Q` cand: `A c∧ B` squash: `↓T` true: `True` guard: `{T}` uiff: `uiff(P;Q)` top: `Top` assert: `↑b` ifthenelse: `if b then t else f fi ` bfalse: `ff` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` isl: `isl(x)` sq_type: `SQType(T)` btrue: `tt` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  weak-continuity-implies-strong-cantor nat_wf bool_wf strong-continuity-test_wf int_seg_wf all_wf exists_wf equal_wf unit_wf2 subtype_rel_dep_function int_seg_subtype_nat false_wf subtype_rel_self assert_wf isl_wf decidable__assert strong-continuity-test-prop1 decidable__lt assert_functionality_wrt_uiff squash_wf true_wf isr-not-isl subtype_rel_union top_wf decidable__equal_int nat_properties 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 intformless_wf int_formula_prop_less_lemma not-isl-assert-isr strong-continuity-test-prop2 and_wf btrue_wf subtype_base_sq bool_subtype_base iff_weakening_equal strong-continuity-test-prop3
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality productElimination functionEquality hypothesis dependent_pairFormation lambdaEquality isectElimination functionExtensionality applyEquality natural_numberEquality setElimination rename because_Cache sqequalRule productEquality unionEquality independent_isectElimination independent_pairFormation inlEquality equalityTransitivity equalitySymmetry unionElimination independent_functionElimination imageElimination cumulativity universeEquality imageMemberEquality baseClosed isect_memberEquality voidElimination voidEquality int_eqEquality intEquality computeAll dependent_set_memberEquality applyLambdaEquality instantiate

Latex:
\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  \mBbbN{}
\mexists{}M:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  (\mBbbN{}?)
\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}.  \mexists{}n:\mBbbN{}.  (((M  n  f)  =  (inl  (F  f)))  \mwedge{}  (\mforall{}m:\mBbbN{}.  ((\muparrow{}isl(M  m  f))  {}\mRightarrow{}  (m  =  n))))

Date html generated: 2017_04_17-AM-10_00_15
Last ObjectModification: 2017_02_27-PM-05_53_10

Theory : continuity

Home Index