Nuprl Lemma : increasing-sequence-prop1
∀a:ℕ ⟶ ℕ. (increasing-sequence(a) ⇒ (∀n,m:ℕ. ((n ≤ m) ⇒ ((a n) ≤ (a m)))))
Proof
Definitions occuring in Statement :
increasing-sequence: increasing-sequence(a),
nat: ℕ,
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
increasing-sequence: increasing-sequence(a),
subtype_rel: A ⊆r B,
le: A ≤ B,
guard: {T},
sq_type: SQType(T),
prop: ℙ,
and: P ∧ Q,
top: Top,
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
uall: ∀[x:A]. B[x],
nat: ℕ,
member: t ∈ T,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
and_wf,
less_than'_wf,
less_than_wf,
ge_wf,
int_formula_prop_less_lemma,
intformless_wf,
increasing-sequence_wf,
nat_wf,
int_subtype_base,
subtype_base_sq,
equal_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
itermAdd_wf,
intformeq_wf,
decidable__equal_int,
le_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
subtract_wf
Rules used in proof :
applyLambdaEquality,
hyp_replacement,
axiomEquality,
independent_pairEquality,
intWeakElimination,
functionEquality,
applyEquality,
functionExtensionality,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
cumulativity,
instantiate,
promote_hyp,
productElimination,
addEquality,
computeAll,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
independent_isectElimination,
unionElimination,
natural_numberEquality,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
because_Cache,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
dependent_set_memberEquality,
dependent_pairFormation,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a:\mBbbN{} {}\mrightarrow{} \mBbbN{}. (increasing-sequence(a) {}\mRightarrow{} (\mforall{}n,m:\mBbbN{}. ((n \mleq{} m) {}\mRightarrow{} ((a n) \mleq{} (a m)))))
Date html generated:
2017_04_20-AM-07_36_37
Last ObjectModification:
2017_04_18-AM-05_58_51
Theory : continuity
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