Nuprl Lemma : rexp-of-nonneg-stronger
∀x:ℝ. ((r0 ≤ x) ⇒ ((r1 + x) ≤ e^x))
Proof
Definitions occuring in Statement :
rexp: e^x,
rleq: x ≤ y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
series-sum: Σn.x[n] = a,
converges-to: lim n→∞.x[n] = y,
sq_exists: ∃x:{A| B[x]},
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uimplies: b supposing a,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
so_apply: x[s],
nat_plus: ℕ+,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
rev_uimplies: rev_uimplies(P;Q),
eq_int: (i =z j),
nequal: a ≠ b ∈ T ,
int_upper: {i...},
ml-term-to-poly: ml-term-to-poly(t),
nil: [],
has-value: (a)↓,
req_int_terms: t1 ≡ t2,
absval: |i|,
rdiv: (x/y),
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
lelt: i ≤ j < k,
int_nzero: ℤ-o,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
less_than: a < b,
subtract: n - m,
primrec: primrec(n;b;c),
fact: (n)!
Lemmas referenced :
rexp-is-limit,
rleq_wf,
int-to-real_wf,
real_wf,
false_wf,
le_wf,
nat_wf,
all_wf,
rabs_wf,
rsub_wf,
rsum_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,
int_seg_wf,
radd_wf,
rdiv_wf,
rless-int,
nat_properties,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
nat_plus_wf,
radd_functionality,
rsum-split-first,
req_functionality,
req_weakening,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
ifthenelse_wf,
int_upper_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
int_upper_properties,
rsum_functionality2,
rsum-zero,
radd-zero,
itermSubtract_wf,
itermAdd_wf,
rleq_functionality,
rabs_functionality,
rsub_functionality,
req_transitivity,
real_polynomial_null,
evalall-sqequal,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rabs-int,
minus-zero,
rmul_preserves_rleq,
rmul_wf,
itermMultiply_wf,
rinv_wf2,
rleq-int,
real_term_value_mul_lemma,
rmul-rinv,
int-rdiv-req,
rsum_functionality_wrt_rleq,
rexp_wf,
rnexp_wf,
int_subtype_base,
equal-wf-base,
int_seg_properties,
nequal_wf,
less_than_wf,
subtype_rel_sets,
int_seg_subtype_nat,
fact_wf,
int-rdiv_wf,
rleq-limit,
nequal-le-implies,
fact0_redex_lemma,
rnexp_zero_lemma,
rmul-ident-div,
rmul-int,
uiff_transitivity,
rleq_weakening_equal,
rnexp1,
rdiv_functionality,
rmul-one-both,
rmul-rdiv-cancel2,
set_subtype_base,
fact-non-zero,
rneq-int,
zero-mul,
rnexp-nonneg
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
natural_numberEquality,
hypothesis,
dependent_set_memberFormation,
dependent_set_memberEquality,
sqequalRule,
independent_pairFormation,
setElimination,
rename,
lambdaEquality,
functionEquality,
because_Cache,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
addEquality,
inrFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
minusEquality,
baseClosed,
sqleReflexivity,
mlComputation,
applyLambdaEquality,
setEquality,
applyEquality,
multiplyEquality,
imageMemberEquality
Latex:
\mforall{}x:\mBbbR{}. ((r0 \mleq{} x) {}\mRightarrow{} ((r1 + x) \mleq{} e\^{}x))
Date html generated:
2017_10_03-AM-09_30_26
Last ObjectModification:
2017_07_28-AM-07_49_12
Theory : reals
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