### Nuprl Lemma : not-decidable-zero-sequence

`¬(∀s:ℕ ⟶ ℕ. ((s = (λx.0) ∈ (ℕ ⟶ ℕ)) ∨ (¬(s = (λx.0) ∈ (ℕ ⟶ ℕ)))))`

Proof

Definitions occuring in Statement :  nat: `ℕ` all: `∀x:A. B[x]` not: `¬A` or: `P ∨ Q` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  not: `¬A` implies: `P `` Q` all: `∀x:A. B[x]` member: `t ∈ T` or: `P ∨ Q` uall: `∀[x:A]. B[x]` prop: `ℙ` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` half-squash-stable: `half-squash-stable(P)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` exists: `∃x:A. B[x]` subtype_rel: `A ⊆r B` guard: `{T}` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` uimplies: `b supposing a` less_than: `a < b` top: `Top` true: `True` squash: `↓T` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` int_seg: `{i..j-}` ge: `i ≥ j ` lelt: `i ≤ j < k` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)`
Lemmas referenced :  nat_wf not_wf equal-wf-T-base strong-continuity2-implies-weak istype-false le_wf sq_stable-implies-half-squash-stable false_wf sq_stable_from_decidable decidable__false implies-quotient-true exists_wf all_wf equal-wf-base-T equal-wf-base int_seg_wf int_subtype_base lt_int_wf eqtt_to_assert assert_of_lt_int istype-top istype-void eqff_to_assert bool_cases_sqequal subtype_base_sq bool_wf bool_subtype_base assert-bnot iff_weakening_uiff assert_wf less_than_wf int_seg_properties nat_properties decidable__equal_int full-omega-unsat intformnot_wf intformeq_wf itermConstant_wf istype-int int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_constant_lemma int_formula_prop_wf set_subtype_base lelt_wf intformand_wf intformless_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma subtype_rel_function int_seg_subtype_nat subtype_rel_self less_than_anti-reflexive
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  rename sqequalRule Error :functionIsType,  Error :universeIsType,  cut introduction extract_by_obid hypothesis Error :inhabitedIsType,  hypothesisEquality Error :unionIsType,  Error :equalityIsType3,  thin baseClosed sqequalHypSubstitution isectElimination functionEquality dependent_functionElimination Error :lambdaEquality_alt,  because_Cache unionElimination Error :equalityIsType1,  equalityTransitivity equalitySymmetry independent_functionElimination Error :dependent_set_memberEquality_alt,  natural_numberEquality independent_pairFormation productElimination Error :productIsType,  Error :equalityIsType2,  setElimination Error :equalityIsType4,  applyEquality functionExtensionality voidElimination equalityElimination independent_isectElimination lessCases Error :isect_memberFormation_alt,  axiomSqEquality Error :isect_memberEquality_alt,  imageMemberEquality imageElimination Error :dependent_pairFormation_alt,  baseApply closedConclusion promote_hyp instantiate cumulativity Error :functionExtensionality_alt,  approximateComputation intEquality int_eqEquality applyLambdaEquality

Latex:
\mneg{}(\mforall{}s:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  ((s  =  (\mlambda{}x.0))  \mvee{}  (\mneg{}(s  =  (\mlambda{}x.0)))))

Date html generated: 2019_06_20-PM-02_56_57
Last ObjectModification: 2018_10_05-PM-08_21_08

Theory : continuity

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