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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