Nuprl Lemma : rpower-greater-one
∀x,q:ℝ. ((r0 < x) ⇒ ((r1 + x) < q) ⇒ (∀n:ℕ. (r1 + (r(n) * x)) < q^n supposing 1 < n))
Proof
Definitions occuring in Statement :
rless: x < y,
rnexp: x^k1,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
and: P ∧ Q,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
not: ¬A,
uiff: uiff(P;Q),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
so_lambda: λ2x.t[x],
nat: ℕ,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
subtype_rel: A ⊆r B,
real: ℝ,
sq_stable: SqStable(P),
nat_plus: ℕ+,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
so_apply: x[s],
cand: A c∧ B,
rge: x ≥ y,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
member-less_than,
less_than_wf,
rleq-int,
false_wf,
radd-preserves-rleq,
int-to-real_wf,
rleq_functionality,
radd_wf,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rless_transitivity1,
rleq_weakening_rless,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
isect_wf,
subtract_wf,
rless_wf,
rmul_wf,
rnexp_wf,
sq_stable__less_than,
nat_plus_properties,
real_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
set_wf,
primrec-wf2,
nat_wf,
rmul_functionality_wrt_rleq2,
rleq_wf,
rleq_transitivity,
rleq_weakening_equal,
rless_functionality_wrt_implies,
rpower-two,
req_weakening,
rless_functionality,
rmul-one-both,
radd-preserves-rless,
rminus-as-rmul,
rmul-distrib,
rminus-radd,
rminus_wf,
rmul-int,
rmul-distrib2,
rmul-identity1,
req_inversion,
radd-assoc,
radd-int,
rmul_functionality,
rmul-zero-both,
radd-ac,
req_transitivity,
radd-zero-both,
radd_functionality,
radd_assoc,
rmul_preserves_rless,
decidable__lt,
intformeq_wf,
int_formula_prop_eq_lemma,
rmul_functionality_wrt_rless2,
rnexp-nonneg,
radd_comm,
rless_transitivity2,
rless-int,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
rnexp-req,
rless-implies-rless,
itermMultiply_wf,
real_term_value_mul_lemma,
rsub_wf,
req_functionality,
rsub-int,
itermMinus_wf,
real_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
isect_memberFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
sqequalRule,
imageElimination,
productElimination,
hypothesis,
voidElimination,
independent_isectElimination,
rename,
natural_numberEquality,
setElimination,
hypothesisEquality,
dependent_functionElimination,
because_Cache,
independent_functionElimination,
independent_pairFormation,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
unionElimination,
instantiate,
cumulativity,
dependent_set_memberEquality,
addEquality,
applyEquality,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
productEquality,
inlFormation,
equalitySymmetry,
equalityTransitivity,
multiplyEquality,
levelHypothesis,
minusEquality,
addLevel,
promote_hyp,
equalityElimination
Latex:
\mforall{}x,q:\mBbbR{}. ((r0 < x) {}\mRightarrow{} ((r1 + x) < q) {}\mRightarrow{} (\mforall{}n:\mBbbN{}. (r1 + (r(n) * x)) < q\^{}n supposing 1 < n))
Date html generated:
2017_10_03-AM-08_34_00
Last ObjectModification:
2017_07_28-AM-07_28_33
Theory : reals
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