Nuprl Lemma : rnexp-convex
∀a,b:ℝ. ((r0 ≤ b) ⇒ (b ≤ a) ⇒ (∀n:ℕ+. (a - b^n ≤ (a^n - b^n))))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rnexp: x^k1,
rsub: x - y,
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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
prop: ℙ,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
cand: A c∧ B,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
rleq: x ≤ y,
rnonneg: rnonneg(x)
Lemmas referenced :
nat_plus_properties,
rleq_wf,
rnexp_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_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_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
rsub_wf,
primrec-wf-nat-plus,
nat_plus_subtype_nat,
nat_plus_wf,
int-to-real_wf,
real_wf,
false_wf,
rleq_weakening_equal,
itermAdd_wf,
int_term_value_add_lemma,
rmul_wf,
rnexp-nonneg,
rleq-implies-rleq,
real_term_polynomial,
itermSubtract_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rleq_functionality,
rpower-one,
rsub_functionality,
req_transitivity,
req_inversion,
rnexp-add,
rmul_functionality,
req_weakening,
rleq_functionality_wrt_implies,
rmul_functionality_wrt_rleq2,
radd_wf,
rminus_wf,
itermMultiply_wf,
itermMinus_wf,
real_term_value_mul_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
radd_functionality,
rminus_functionality,
radd-preserves-rleq,
uiff_transitivity,
rnexp-rleq,
rmul_preserves_rleq2,
less_than'_wf,
radd_functionality_wrt_rleq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
rename,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
dependent_set_memberEquality,
because_Cache,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
applyEquality,
addEquality,
inlFormation,
independent_functionElimination,
productElimination,
productEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberFormation,
independent_pairEquality,
minusEquality,
axiomEquality
Latex:
\mforall{}a,b:\mBbbR{}. ((r0 \mleq{} b) {}\mRightarrow{} (b \mleq{} a) {}\mRightarrow{} (\mforall{}n:\mBbbN{}\msupplus{}. (a - b\^{}n \mleq{} (a\^{}n - b\^{}n))))
Date html generated:
2017_10_03-AM-10_36_21
Last ObjectModification:
2017_07_28-AM-08_13_55
Theory : reals
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