Nuprl Lemma : rnexp-rless
∀x,y:ℝ. ((r0 ≤ x) ⇒ (x < y) ⇒ (∀n:ℕ+. (x^n < y^n)))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rless: x < y,
rnexp: x^k1,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
req_int_terms: t1 ≡ t2,
rge: x ≥ y,
cand: A c∧ B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtract: n - m,
eq_int: (i =z j),
true: True,
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
bfalse: ff,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
less_than': less_than'(a;b),
le: A ≤ B,
so_apply: x[s],
so_lambda: λ2x.t[x],
and: P ∧ Q,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
squash: ↓T,
sq_stable: SqStable(P),
real: ℝ,
sq_exists: ∃x:A [B[x]],
rless: x < y,
nat: ℕ,
nat_plus: ℕ+,
prop: ℙ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
rless_transitivity2,
rnexp-positive,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req-iff-rsub-is-0,
itermMultiply_wf,
itermSubtract_wf,
rsub_wf,
rless-implies-rless,
rmul_preserves_rless,
rleq_weakening_equal,
rmul_functionality_wrt_rleq2,
rless_functionality_wrt_implies,
rnexp-nonneg,
rleq_weakening_rless,
rnexp_unroll,
rless_functionality,
add-subtract-cancel,
subtract_wf,
int_term_value_add_lemma,
itermAdd_wf,
int_subtype_base,
int_formula_prop_eq_lemma,
intformeq_wf,
rmul_wf,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
false_wf,
primrec-wf-nat-plus,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
sq_stable__less_than,
nat_plus_properties,
real_wf,
int-to-real_wf,
rleq_wf,
nat_plus_wf,
rless_wf,
nat_plus_subtype_nat,
rnexp_wf,
rlessw_wf
Rules used in proof :
productEquality,
inlFormation,
cumulativity,
instantiate,
promote_hyp,
equalitySymmetry,
equalityTransitivity,
productElimination,
equalityElimination,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
approximateComputation,
independent_isectElimination,
unionElimination,
imageElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
addEquality,
setElimination,
rename,
natural_numberEquality,
because_Cache,
dependent_set_memberEquality,
sqequalRule,
hypothesis,
applyEquality,
hypothesisEquality,
isectElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}x,y:\mBbbR{}. ((r0 \mleq{} x) {}\mRightarrow{} (x < y) {}\mRightarrow{} (\mforall{}n:\mBbbN{}\msupplus{}. (x\^{}n < y\^{}n)))
Date html generated:
2018_05_22-PM-01_33_01
Last ObjectModification:
2018_05_21-AM-00_08_06
Theory : reals
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