### Nuprl Lemma : Rice-theorem-for-Type_3

`∀F:Type ⟶ 𝔹`
`  ((∀X,Y:Type.  (X ~ Y `` F X = F Y)) `` weak-continuity(𝔹;𝔹) `` ((∀X:Type. (↑(F X))) ∨ (∀X:Type. (¬↑(F X)))))`

Proof

Definitions occuring in Statement :  weak-continuity: `weak-continuity(T;V)` equipollent: `A ~ B` assert: `↑b` bool: `𝔹` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` or: `P ∨ Q` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` implies: `P `` Q` prop: `ℙ` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` or: `P ∨ Q` false: `False` weak-continuity: `weak-continuity(T;V)` decidable: `Dec(P)` squash: `↓T` exists: `∃x:A. B[x]` isl: `isl(x)` iff: `P `⇐⇒` Q` and: `P ∧ Q` true: `True` subtype_rel: `A ⊆r B` uimplies: `b supposing a` guard: `{T}` rev_implies: `P `` Q` not: `¬A` nat: `ℕ` le: `A ≤ B` less_than': `less_than'(a;b)` int_seg: `{i..j-}` ge: `i ≥ j ` lelt: `i ≤ j < k` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` assert: `↑b` ifthenelse: `if b then t else f fi ` bfalse: `ff` uiff: `uiff(P;Q)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` sq_type: `SQType(T)`
Lemmas referenced :  Rice-theorem-for-Type_2 weak-continuity_wf bool_wf all_wf equipollent_wf equal_wf nat_wf decidable_wf equal-wf-T-base isl_wf not_wf squash_wf true_wf bfalse_wf iff_weakening_equal equal-wf-base btrue_neq_bfalse int_seg_wf int_seg_subtype_nat false_wf iff_wf lt_int_wf iff_imp_equal_bool int_seg_properties nat_properties satisfiable-full-omega-tt intformand_wf intformless_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_formula_prop_wf less_than_wf assert_of_lt_int assert_wf decidable__le intformnot_wf intformle_wf itermConstant_wf itermAdd_wf int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_add_lemma le_wf eqff_to_assert bnot_wf iff_transitivity iff_weakening_uiff assert_of_bnot subtype_base_sq bool_subtype_base assert_functionality_wrt_uiff or_wf btrue_wf
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation hypothesis sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality independent_functionElimination isectElimination instantiate universeEquality sqequalRule lambdaEquality cumulativity functionEquality applyEquality functionExtensionality unionElimination voidElimination because_Cache rename baseClosed equalityTransitivity equalitySymmetry imageElimination productElimination dependent_pairFormation independent_pairFormation natural_numberEquality imageMemberEquality independent_isectElimination setElimination int_eqEquality intEquality isect_memberEquality voidEquality computeAll addLevel impliesFunctionality dependent_set_memberEquality addEquality equalityElimination inlFormation orFunctionality allFunctionality allLevelFunctionality impliesLevelFunctionality inrFormation

Latex:
\mforall{}F:Type  {}\mrightarrow{}  \mBbbB{}
((\mforall{}X,Y:Type.    (X  \msim{}  Y  {}\mRightarrow{}  F  X  =  F  Y))
{}\mRightarrow{}  weak-continuity(\mBbbB{};\mBbbB{})
{}\mRightarrow{}  ((\mforall{}X:Type.  (\muparrow{}(F  X)))  \mvee{}  (\mforall{}X:Type.  (\mneg{}\muparrow{}(F  X)))))

Date html generated: 2017_10_01-AM-08_29_47
Last ObjectModification: 2017_07_26-PM-04_24_09

Theory : basic

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