### Nuprl Lemma : unary-strong-almost-full-has-strict-inc

`∀[A:ℕ ⟶ ℙ]. ((∀s:StrictInc. ∃n:ℕ. A[s n]) `` (∃s:StrictInc. ∀n:ℕ. A[s n]))`

Proof

Definitions occuring in Statement :  strict-inc: `StrictInc` nat: `ℕ` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` apply: `f a` function: `x:A ⟶ B[x]`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` all: `∀x:A. B[x]` member: `t ∈ T` exists: `∃x:A. B[x]` so_apply: `x[s]` strict-inc: `StrictInc` subtype_rel: `A ⊆r B` prop: `ℙ` nat: `ℕ` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` top: `Top` and: `P ∧ Q` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` less_than: `a < b` squash: `↓T` cand: `A c∧ B` pi1: `fst(t)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` subtract: `n - m` true: `True` so_lambda: `λ2x.t[x]`
Lemmas referenced :  strict-inc_wf istype-nat subtype_rel_self nat_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermAdd_wf itermVar_wf istype-int int_formula_prop_and_lemma istype-void 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 istype-le int_seg_properties decidable__lt intformless_wf int_formula_prop_less_lemma int_seg_wf istype-less_than nat_wf implies-strict-inc primrec_wf primrec-unroll lt_int_wf eqtt_to_assert assert_of_lt_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_wf bool_subtype_base assert-bnot iff_weakening_uiff assert_wf less_than_wf add-associates add-swap add-commutes zero-add add-subtract-cancel squash_wf true_wf istype-universe primrec0_lemma subtract_wf itermSubtract_wf int_term_value_subtract_lemma primrec-wf2
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  Error :lambdaFormation_alt,  cut sqequalRule Error :functionIsType,  Error :universeIsType,  introduction extract_by_obid hypothesis Error :productIsType,  applyEquality hypothesisEquality sqequalHypSubstitution setElimination thin rename instantiate isectElimination universeEquality dependent_functionElimination Error :dependent_set_memberEquality_alt,  Error :lambdaEquality_alt,  addEquality natural_numberEquality unionElimination independent_isectElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  int_eqEquality Error :isect_memberEquality_alt,  voidElimination independent_pairFormation Error :inhabitedIsType,  productElimination imageElimination because_Cache promote_hyp functionExtensionality Error :equalityIstype,  equalityTransitivity equalitySymmetry equalityElimination cumulativity hyp_replacement imageMemberEquality baseClosed applyLambdaEquality Error :setIsType

Latex:
\mforall{}[A:\mBbbN{}  {}\mrightarrow{}  \mBbbP{}].  ((\mforall{}s:StrictInc.  \mexists{}n:\mBbbN{}.  A[s  n])  {}\mRightarrow{}  (\mexists{}s:StrictInc.  \mforall{}n:\mBbbN{}.  A[s  n]))

Date html generated: 2019_06_20-PM-02_57_25
Last ObjectModification: 2019_02_06-PM-03_51_07

Theory : continuity

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