Nuprl Lemma : derivative-rinv-basic
d((r1/x))/dx = λx.(r(-1)/x^2) on (r0, ∞)
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
roiint: (l, ∞),
rdiv: (x/y),
rnexp: x^k1,
int-to-real: r(n),
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
uimplies: b supposing a,
nonzero-on: f[x]≠r0 for x ∈ I,
top: Top,
roiint: (l, ∞),
i-approx: i-approx(I;n),
cand: A c∧ B,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
decidable: Dec(P),
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
nat_plus: ℕ+,
sq_exists: ∃x:A [B[x]],
uiff: uiff(P;Q),
rless: x < y,
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
rge: x ≥ y,
sq_stable: SqStable(P),
squash: ↓T,
nat: ℕ,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermConstant: "const",
rtermVar: rtermVar(var),
pi1: fst(t),
true: True,
rtermMinus: rtermMinus(num),
pi2: snd(t)
Lemmas referenced :
derivative-rinv,
roiint_wf,
int-to-real_wf,
real_wf,
i-member_wf,
req_weakening,
req_wf,
derivative-id,
i-approx_wf,
icompact_wf,
nat_plus_wf,
set_wf,
member_rccint_lemma,
rabs_wf,
all_wf,
radd_wf,
rleq_wf,
int_term_value_mul_lemma,
itermMultiply_wf,
less_than_wf,
rless-int-fractions2,
rless_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_plus_properties,
rless-int,
rdiv_wf,
req-iff-rsub-is-0,
itermAdd_wf,
itermSubtract_wf,
rsub_wf,
rleq-implies-rleq,
rleq_functionality,
rabs-of-nonneg,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma,
rleq_weakening_rless,
trivial-rleq-radd,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rnexp-positive,
member_roiint_lemma,
istype-void,
sq_stable__rless,
decidable__le,
intformle_wf,
istype-int,
int_formula_prop_le_lemma,
istype-le,
rmul_wf,
rnexp_wf,
rless_functionality,
req_inversion,
rnexp2,
rminus_wf,
derivative_functionality,
assert-rat-term-eq2,
rtermDivide_wf,
rtermMinus_wf,
rtermConstant_wf,
rtermVar_wf,
req_functionality,
rdiv_functionality
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
isectElimination,
natural_numberEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
setElimination,
rename,
hypothesisEquality,
setIsType,
universeIsType,
because_Cache,
independent_functionElimination,
lambdaFormation_alt,
independent_isectElimination,
inhabitedIsType,
lambdaEquality,
lambdaFormation,
voidEquality,
voidElimination,
isect_memberEquality,
functionEquality,
productEquality,
multiplyEquality,
dependent_set_memberEquality,
independent_pairFormation,
intEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
unionElimination,
productElimination,
inrFormation,
dependent_set_memberFormation,
isect_memberEquality_alt,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality_alt,
dependent_pairFormation_alt,
closedConclusion,
inrFormation_alt,
minusEquality
Latex:
d((r1/x))/dx = \mlambda{}x.(r(-1)/x\^{}2) on (r0, \minfty{})
Date html generated:
2019_10_30-AM-09_03_30
Last ObjectModification:
2019_04_02-AM-09_46_17
Theory : reals
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