Nuprl Lemma : not-LPO

`¬(∀f:ℕ ⟶ ℕ. ((∀n:ℕ. ((f n) = 0 ∈ ℤ)) ∨ (¬(∀n:ℕ. ((f n) = 0 ∈ ℤ)))))`

Proof

Definitions occuring in Statement :  nat: `ℕ` all: `∀x:A. B[x]` not: `¬A` or: `P ∨ Q` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  not: `¬A` implies: `P `` Q` all: `∀x:A. B[x]` member: `t ∈ T` or: `P ∨ Q` exists: `∃x:A. B[x]` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` prop: `ℙ` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` squash: `↓T` true: `True` subtype_rel: `A ⊆r B` uimplies: `b supposing a` guard: `{T}` rev_implies: `P `` Q` less_than: `a < b` so_lambda: `λ2x.t[x]` so_apply: `x[s]` pi1: `fst(t)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` int_seg: `{i..j-}` ge: `i ≥ j ` lelt: `i ≤ j < k` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top`
Lemmas referenced :  false_wf le_wf equal_wf squash_wf true_wf iff_weakening_equal nat_wf less_than_wf all_wf equal-wf-T-base iff_wf or_wf not_wf exists_wf weak-continuity-nat-nat squash-from-quotient equal-wf-base-T subtype_rel_dep_function int_seg_wf int_seg_subtype_nat subtype_rel_self lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot int_seg_properties nat_properties decidable__equal_int full-omega-unsat intformnot_wf intformeq_wf itermConstant_wf int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_constant_lemma int_formula_prop_wf intformand_wf intformless_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_subtype_base decidable__lt bool_cases iff_transitivity iff_weakening_uiff assert_of_bnot
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut hypothesis sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality unionElimination dependent_pairFormation dependent_set_memberEquality natural_numberEquality sqequalRule independent_pairFormation introduction extract_by_obid isectElimination applyEquality lambdaEquality imageElimination equalityTransitivity equalitySymmetry universeEquality intEquality imageMemberEquality baseClosed independent_isectElimination productElimination independent_functionElimination because_Cache functionExtensionality setElimination rename voidElimination functionEquality promote_hyp equalityElimination instantiate cumulativity approximateComputation isect_memberEquality voidEquality int_eqEquality applyLambdaEquality impliesFunctionality

Latex:
\mneg{}(\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  ((\mforall{}n:\mBbbN{}.  ((f  n)  =  0))  \mvee{}  (\mneg{}(\mforall{}n:\mBbbN{}.  ((f  n)  =  0)))))

Date html generated: 2018_05_21-PM-01_17_55
Last ObjectModification: 2017_10_14-PM-09_14_13

Theory : continuity

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