Nuprl Lemma : rnexp-nonneg
∀x:ℝ. ((r0 ≤ x) ⇒ (∀n:ℕ. (r0 ≤ x^n)))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rnexp: x^k1,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
rev_uimplies: rev_uimplies(P;Q),
itermConstant: "const",
req_int_terms: t1 ≡ t2
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
less_than'_wf,
rsub_wf,
rnexp_wf,
nat_plus_properties,
int-to-real_wf,
nat_plus_wf,
le_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
rleq_wf,
real_wf,
false_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
rmul_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
rleq-int,
rleq_functionality,
req_weakening,
rnexp-req,
rmul_preserves_rleq2,
rleq-implies-rleq,
real_term_polynomial,
itermMultiply_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
req_transitivity,
rmul_functionality,
rmul-identity1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
productElimination,
independent_pairEquality,
applyEquality,
because_Cache,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
unionElimination,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
isect_memberFormation
Latex:
\mforall{}x:\mBbbR{}. ((r0 \mleq{} x) {}\mRightarrow{} (\mforall{}n:\mBbbN{}. (r0 \mleq{} x\^{}n)))
Date html generated:
2017_10_03-AM-08_32_45
Last ObjectModification:
2017_07_28-AM-07_27_54
Theory : reals
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