### Nuprl Lemma : monotone-upper-bound-function

`∀f:ℕ ⟶ ℤ. ∃g:ℕ ⟶ ℤ. ((∀i,j:ℕ.  ((i ≤ j) `` ((g i) ≤ (g j)))) ∧ (∀n:ℕ. ((f n) ≤ (g n))))`

Proof

Definitions occuring in Statement :  nat: `ℕ` le: `A ≤ B` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` int: `ℤ`
Definitions unfolded in proof :  all: `∀x:A. B[x]` exists: `∃x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` nat: `ℕ` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` prop: `ℙ` top: `Top` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` cand: `A c∧ B` l_subset: `l_subset(T;as;bs)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  equal_wf member_upto member_map l_exists_iff imax-list-ub upto_iseg int_seg_subtype subtype_rel_list iseg-map l_member_wf iseg_member imax-list-subset all_wf and_wf int_formula_prop_less_lemma intformless_wf decidable__lt le_wf int_formula_prop_wf int_term_value_var_lemma int_term_value_add_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermAdd_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le nat_properties length_upto map-length upto_wf subtype_rel_self false_wf int_seg_subtype_nat nat_wf subtype_rel_dep_function int_seg_wf map_wf imax-list_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation dependent_pairFormation lambdaEquality cut lemma_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality addEquality setElimination rename hypothesisEquality hypothesis intEquality applyEquality sqequalRule independent_isectElimination independent_pairFormation because_Cache isect_memberEquality voidElimination voidEquality dependent_set_memberEquality dependent_functionElimination unionElimination int_eqEquality computeAll functionEquality introduction independent_functionElimination productElimination setEquality productEquality

Latex:
\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbZ{}.  \mexists{}g:\mBbbN{}  {}\mrightarrow{}  \mBbbZ{}.  ((\mforall{}i,j:\mBbbN{}.    ((i  \mleq{}  j)  {}\mRightarrow{}  ((g  i)  \mleq{}  (g  j))))  \mwedge{}  (\mforall{}n:\mBbbN{}.  ((f  n)  \mleq{}  (g  n))))

Date html generated: 2016_05_14-PM-03_19_04
Last ObjectModification: 2016_01_15-AM-07_17_10

Theory : list_1

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