### Nuprl Lemma : Rice-theorem-for-Type_2

`∀F:Type ⟶ 𝔹. ((∀X,Y:Type.  (X ~ Y `` F X = F Y)) `` (∀X,Y:Type.  (F X = F Y ∨ (∀f:ℕ ⟶ 𝔹. Dec(∀n:ℕ. f n = ff)))))`

Proof

Definitions occuring in Statement :  equipollent: `A ~ B` nat: `ℕ` bfalse: `ff` bool: `𝔹` decidable: `Dec(P)` all: `∀x:A. B[x]` 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` or: `P ∨ Q` prop: `ℙ` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` guard: `{T}` exists: `∃x:A. B[x]` and: `P ∧ Q` nat-inf: `ℕ∞` nat: `ℕ` ge: `i ≥ j ` decidable: `Dec(P)` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` int_seg: `{i..j-}` lelt: `i ≤ j < k` squash: `↓T` nat2inf: `n∞` less_than: `a < b` sq_type: `SQType(T)` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` true: `True` bool: `𝔹` unit: `Unit` it: `⋅` bfalse: `ff` bnot: `¬bb` nat-inf-infinity: `∞`
Lemmas referenced :  Rice-theorem-for-Type_1 all_wf nat_wf bool_wf decidable_wf equal-wf-T-base exists_wf nat-inf_wf not_wf assert_wf nat2inf_wf nat-inf-infinity_wf equal_wf equipollent_wf bnot_wf b-exists_wf nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermAdd_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_add_lemma int_term_value_var_lemma int_formula_prop_wf le_wf int_seg_subtype_nat false_wf int_seg_wf assert_of_bnot assert-b-exists decidable__lt intformless_wf int_formula_prop_less_lemma lelt_wf int_seg_properties decidable__equal_int subtract_wf int_seg_subtype itermSubtract_wf intformeq_wf int_term_value_subtract_lemma int_formula_prop_eq_lemma set_wf less_than_wf primrec-wf2 decidable__exists_int_seg decidable__equal_bool btrue_wf subtract-add-cancel iff_imp_equal_bool lt_int_wf assert_of_lt_int iff_wf subtype_base_sq bool_subtype_base true_wf eqtt_to_assert eqff_to_assert bool_cases_sqequal assert-bnot assert_functionality_wrt_uiff bfalse_wf assert_elim btrue_neq_bfalse
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation hypothesis sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality independent_functionElimination unionElimination inlFormation isectElimination functionEquality sqequalRule lambdaEquality applyEquality functionExtensionality baseClosed inrFormation productElimination productEquality universeEquality cumulativity instantiate dependent_set_memberEquality addEquality setElimination rename because_Cache natural_numberEquality independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll promote_hyp addLevel impliesFunctionality equalityTransitivity equalitySymmetry applyLambdaEquality levelHypothesis hypothesis_subsumption imageMemberEquality imageElimination impliesLevelFunctionality equalityElimination hyp_replacement

Latex:
\mforall{}F:Type  {}\mrightarrow{}  \mBbbB{}
((\mforall{}X,Y:Type.    (X  \msim{}  Y  {}\mRightarrow{}  F  X  =  F  Y))
{}\mRightarrow{}  (\mforall{}X,Y:Type.    (F  X  =  F  Y  \mvee{}  (\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}.  Dec(\mforall{}n:\mBbbN{}.  f  n  =  ff)))))

Date html generated: 2017_10_01-AM-08_29_44
Last ObjectModification: 2017_07_26-PM-04_24_07

Theory : basic

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