Nuprl Lemma : second-derivative-log-contraction
∀a:{a:ℝ| r0 < a} . d((a - e^x/a + e^x)^2)/dx = λx.(((r(-4) * a) * e^x) * (a - e^x)/a + e^x^3) on (-∞, ∞)
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
riiint: (-∞, ∞),
rexp: e^x,
rdiv: (x/y),
rless: x < y,
rnexp: x^k1,
rsub: x - y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
nonzero-on: f[x]≠r0 for x ∈ I,
sq_exists: ∃x:A [B[x]],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
nat_plus: ℕ+,
rless: x < y,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
uiff: uiff(P;Q),
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rgt: x > y,
rfun: I ⟶ℝ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
rneq: x ≠ y,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
req_int_terms: t1 ≡ t2,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
subtract: n - m,
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermMultiply: left "*" right,
rtermVar: rtermVar(var),
rtermConstant: "const",
pi1: fst(t),
true: True,
pi2: snd(t)
Lemmas referenced :
sq_stable__rless,
int-to-real_wf,
i-member_wf,
i-approx_wf,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-less_than,
riiint_wf,
real_wf,
rless_wf,
rleq_wf,
rabs_wf,
radd_wf,
rexp_wf,
nat_plus_wf,
icompact_wf,
trivial-rleq-radd,
rleq_weakening_rless,
rleq_weakening_equal,
rleq_functionality,
req_weakening,
rabs-of-nonneg,
rleq_functionality_wrt_implies,
radd_functionality_wrt_rless1,
rexp-positive,
rsub_wf,
req_functionality,
rsub_functionality,
rexp_functionality,
req_wf,
radd_functionality,
istype-top,
member_riiint_lemma,
subtype_rel_dep_function,
top_wf,
true_wf,
istype-true,
itermSubtract_wf,
itermAdd_wf,
rnexp_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
istype-le,
rmul_wf,
rdiv_wf,
derivative-rdiv,
derivative-sub,
derivative-const,
derivative-rexp,
derivative-add,
rless_functionality_wrt_implies,
rless_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rnexp2,
rmul-is-positive,
derivative_functionality,
itermMultiply_wf,
rdiv_functionality,
req_inversion,
real_term_value_mul_lemma,
derivative-rnexp2,
iproper-riiint,
rmul_functionality,
rnexp_functionality,
rnexp-positive,
rnexp_step,
rless_transitivity1,
rleq_weakening,
assert-rat-term-eq2,
rtermMultiply_wf,
rtermConstant_wf,
rtermDivide_wf,
rtermVar_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
dependent_set_memberFormation_alt,
setElimination,
thin,
rename,
hypothesisEquality,
sqequalHypSubstitution,
hypothesis,
introduction,
extract_by_obid,
dependent_functionElimination,
isectElimination,
natural_numberEquality,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
universeIsType,
dependent_set_memberEquality_alt,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
productIsType,
functionIsType,
inhabitedIsType,
because_Cache,
setIsType,
productElimination,
equalityTransitivity,
equalitySymmetry,
applyEquality,
setEquality,
inlFormation_alt,
inrFormation_alt,
closedConclusion,
minusEquality,
equalityIstype
Latex:
\mforall{}a:\{a:\mBbbR{}| r0 < a\}
d((a - e\^{}x/a + e\^{}x)\^{}2)/dx = \mlambda{}x.(((r(-4) * a) * e\^{}x) * (a - e\^{}x)/a + e\^{}x\^{}3) on (-\minfty{}, \minfty{})
Date html generated:
2019_10_31-AM-06_08_56
Last ObjectModification:
2019_04_03-PM-04_42_12
Theory : reals_2
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