Nuprl Lemma : mconverges-to_wf

[X:Type]. ∀[d:X ⟶ X ⟶ ℝ]. ∀[x:ℕ ⟶ X]. ∀[y:X].  (lim n→∞.x[n] y ∈ ℙ)


Proof




Definitions occuring in Statement :  mconverges-to: lim n→∞.x[n] y real: nat: uall: [x:A]. B[x] prop: so_apply: x[s] member: t ∈ T function: x:A ⟶ B[x] universe: Type
Definitions unfolded in proof :  mconverges-to: lim n→∞.x[n] y mdist: mdist(d;x;y) uall: [x:A]. B[x] member: t ∈ T so_lambda: λ2x.t[x] implies:  Q prop: nat: so_apply: x[s] nat_plus: + uimplies: supposing a rneq: x ≠ y guard: {T} or: P ∨ Q all: x:A. B[x] iff: ⇐⇒ Q and: P ∧ Q rev_implies:  Q ge: i ≥  decidable: Dec(P) not: ¬A satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] false: False top: Top
Lemmas referenced :  all_wf nat_plus_wf sq_exists_wf nat_wf le_wf rleq_wf rdiv_wf int-to-real_wf rless-int nat_properties nat_plus_properties decidable__lt full-omega-unsat intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf rless_wf istype-nat real_wf istype-universe
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation_alt introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesis lambdaEquality_alt because_Cache functionEquality setElimination rename applyEquality hypothesisEquality closedConclusion natural_numberEquality independent_isectElimination inrFormation_alt dependent_functionElimination productElimination independent_functionElimination unionElimination approximateComputation dependent_pairFormation_alt int_eqEquality isect_memberEquality_alt voidElimination independent_pairFormation universeIsType inhabitedIsType axiomEquality equalityTransitivity equalitySymmetry isectIsTypeImplies functionIsType instantiate universeEquality

Latex:
\mforall{}[X:Type].  \mforall{}[d:X  {}\mrightarrow{}  X  {}\mrightarrow{}  \mBbbR{}].  \mforall{}[x:\mBbbN{}  {}\mrightarrow{}  X].  \mforall{}[y:X].    (lim  n\mrightarrow{}\minfty{}.x[n]  =  y  \mmember{}  \mBbbP{})



Date html generated: 2019_10_30-AM-06_38_06
Last ObjectModification: 2019_10_02-AM-10_51_07

Theory : reals


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