Nuprl Lemma : arctangent-chain-rule
∀I:Interval. ∀f,f':I ⟶ℝ.
(iproper(I)
⇒ (∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f'[x] = f'[y])))
⇒ d(f[x])/dx = λx.f'[x] on I
⇒ d(arctangent(f[x]))/dx = λx.(f'[x]/r1 + f[x]^2) on I)
Proof
Definitions occuring in Statement :
arctangent: arctangent(x),
derivative: d(f[x])/dx = λz.g[z] on I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
iproper: iproper(I),
interval: Interval,
rdiv: (x/y),
rnexp: x^k1,
req: x = y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
false: False,
rneq: x ≠ y,
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
label: ...$L... t,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
rge: x ≥ y,
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermVar: rtermVar(var),
rtermAdd: left "+" right,
rtermConstant: "const",
pi1: fst(t),
rtermMultiply: left "*" right,
pi2: snd(t),
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x)
Lemmas referenced :
simple-chain-rule,
rnexp2-nonneg,
arctangent_wf,
i-member_wf,
riiint_wf,
rdiv_wf,
int-to-real_wf,
radd_wf,
rnexp_wf,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
rless_wf,
req_functionality,
rdiv_functionality,
req_weakening,
radd_functionality,
rnexp_functionality,
req_wf,
derivative-arctangent,
derivative_wf,
real_wf,
iproper_wf,
rfun_wf,
interval_wf,
trivial-rless-radd,
rless-int,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
rmul_wf,
assert-rat-term-eq2,
rtermMultiply_wf,
rtermDivide_wf,
rtermConstant_wf,
rtermAdd_wf,
rtermVar_wf,
derivative_functionality
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
sqequalRule,
lambdaEquality_alt,
isectElimination,
setElimination,
rename,
setIsType,
universeIsType,
closedConclusion,
natural_numberEquality,
dependent_set_memberEquality_alt,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
isect_memberEquality_alt,
voidElimination,
inrFormation_alt,
productElimination,
applyEquality,
functionIsType,
inhabitedIsType,
independent_pairFormation,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
int_eqEquality
Latex:
\mforall{}I:Interval. \mforall{}f,f':I {}\mrightarrow{}\mBbbR{}.
(iproper(I)
{}\mRightarrow{} (\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x = y) {}\mRightarrow{} (f'[x] = f'[y])))
{}\mRightarrow{} d(f[x])/dx = \mlambda{}x.f'[x] on I
{}\mRightarrow{} d(arctangent(f[x]))/dx = \mlambda{}x.(f'[x]/r1 + f[x]\^{}2) on I)
Date html generated:
2019_10_31-AM-06_05_05
Last ObjectModification:
2019_04_03-AM-00_28_53
Theory : reals_2
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