Nuprl Lemma : derivative-log-contraction
∀a:{a:ℝ| r0 < a} . d(log-contraction(a;x))/dx = λx.(a - e^x/a + e^x)^2 on (-∞, ∞)
Proof
Definitions occuring in Statement :
log-contraction: log-contraction(a;x),
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,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
nonzero-on: f[x]≠r0 for x ∈ I,
sq_exists: ∃x:{A| B[x]},
and: P ∧ Q,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
prop: ℙ,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
uimplies: b supposing a,
rev_implies: P ⇐ Q,
rge: x ≥ y,
rgt: x > y,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_uimplies: rev_uimplies(P;Q),
r-ap: f(x),
rfun-eq: rfun-eq(I;f;g),
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
rneq: x ≠ y,
rfun: I ⟶ℝ,
log-contraction: log-contraction(a;x),
rsub: x - y,
rdiv: (x/y),
true: True
Lemmas referenced :
real_wf,
sq_stable__rless,
int-to-real_wf,
i-member_wf,
i-approx_wf,
less_than_wf,
riiint_wf,
rless_wf,
all_wf,
rleq_wf,
rabs_wf,
radd_wf,
rexp_wf,
set_wf,
nat_plus_wf,
icompact_wf,
rleq_weakening_rless,
rleq_weakening_equal,
rless_functionality_wrt_implies,
radd_functionality_wrt_rless1,
rexp-positive,
rless_functionality,
req_weakening,
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,
rleq_functionality,
rabs-of-nonneg,
rleq_functionality_wrt_implies,
rmul-is-positive,
derivative_functionality,
derivative-rexp,
derivative-const,
derivative-sub,
derivative-rdiv,
derivative-const-mul,
derivative-id,
derivative-add,
le_wf,
false_wf,
rnexp_wf,
true_wf,
top_wf,
subtype_rel_dep_function,
member_riiint_lemma,
radd_functionality,
req_wf,
rexp_functionality,
rsub_functionality,
req_functionality,
rsub_wf,
rdiv_wf,
rmul_wf,
rnexp2,
rneq_functionality,
rmul_comm,
rdiv_functionality,
rmul_functionality,
equal_wf,
radd-int,
rmul-identity1,
rmul-distrib2,
rmul-rdiv-cancel2,
rminus-as-rmul,
rmul-int,
radd-zero-both,
radd-ac,
radd_comm,
radd-assoc,
rminus-zero,
rminus-radd,
rmul-one-both,
rmul-assoc,
req_inversion,
rmul-zero-both,
rmul_over_rminus,
rminus_functionality,
rmul-distrib,
req_transitivity,
uiff_transitivity,
rminus_wf,
rmul_preserves_req,
rmul_assoc,
log-contraction_wf,
rinv_wf2,
rinv-of-rmul,
itermMultiply_wf,
real_term_value_mul_lemma,
rmul-rinv3,
rmul-rinv,
squash_wf,
iff_weakening_equal
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
extract_by_obid,
hypothesis,
dependent_set_memberFormation,
setElimination,
thin,
rename,
hypothesisEquality,
sqequalHypSubstitution,
dependent_functionElimination,
isectElimination,
natural_numberEquality,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
dependent_set_memberEquality,
because_Cache,
productEquality,
lambdaEquality,
functionEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
computeAll,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
applyEquality,
inlFormation,
inrFormation,
setEquality,
addEquality,
multiplyEquality,
minusEquality,
universeEquality
Latex:
\mforall{}a:\{a:\mBbbR{}| r0 < a\} . d(log-contraction(a;x))/dx = \mlambda{}x.(a - e\^{}x/a + e\^{}x)\^{}2 on (-\minfty{}, \minfty{})
Date html generated:
2017_10_04-PM-10_27_27
Last ObjectModification:
2017_07_28-AM-08_50_12
Theory : reals_2
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