Nuprl Lemma : derivative-cosine
d(cosine(x))/dx = λx.-(sine(x)) on (-∞, ∞)
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
riiint: (-∞, ∞),
cosine: cosine(x),
sine: sine(x),
rminus: -(x)
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
int_seg: {i..j-},
guard: {T},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
nat_plus: ℕ+,
int_nzero: ℤ-o,
so_apply: x[s],
nequal: a ≠ b ∈ T ,
so_apply: x[s1;s2],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
true: True,
int_upper: {i...},
sq_stable: SqStable(P),
squash: ↓T,
rneq: x ≠ y,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
pointwise-req: x[k] = y[k] for k ∈ [n,m],
less_than: a < b,
fact: (n)!,
primrec: primrec(n;b;c),
eq_int: (i =z j),
rless: x < y,
sq_exists: ∃x:{A| B[x]},
real: ℝ,
cand: A c∧ B
Lemmas referenced :
fun-converges-to-cosine,
fun-converges-to-derivative,
riiint_wf,
iproper-riiint,
rsum_wf,
int-rmul_wf,
fastexp_wf,
int_seg_subtype_nat,
false_wf,
int-rdiv_wf,
fact_wf,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
subtype_rel_sets,
less_than_wf,
nequal_wf,
nat_plus_properties,
intformeq_wf,
intformless_wf,
int_formula_prop_eq_lemma,
int_formula_prop_less_lemma,
equal-wf-base,
int_subtype_base,
rnexp_wf,
int_seg_wf,
real_wf,
i-member_wf,
nat_wf,
rminus_wf,
subtract_wf,
subtract-add-cancel,
itermAdd_wf,
int_term_value_add_lemma,
cosine_wf,
sine_wf,
req_functionality,
rminus_functionality,
rsum_functionality2,
int-rmul_functionality,
int-rdiv_functionality,
rnexp_functionality,
nat_plus_wf,
req_weakening,
req_wf,
set_wf,
fun-converges-to-sine,
fun-converges-to-rminus,
decidable__lt,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
int_upper_properties,
sq_stable__icompact,
i-approx_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
int_upper_wf,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
icompact_wf,
subtype_rel_self,
rmul_wf,
rneq-int,
fact-non-zero,
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,
derivative-rsum,
derivative-const-mul,
derivative-rdiv-const,
derivative-rnexp,
mul_nat_plus,
not-equal-2,
minus-zero,
rnexp_zero_lemma,
derivative-const,
derivative_functionality,
rsum_functionality,
req_transitivity,
int-rmul-req,
rmul_functionality,
int-rdiv-req,
rminus-as-rmul,
req_inversion,
rsum_linearity2,
radd_wf,
fact0_redex_lemma,
rsum-split-first,
req-int,
decidable__equal_int,
uiff_transitivity,
rdiv-zero,
rmul-int,
radd_functionality,
rsum-shift,
radd-zero-both,
set_subtype_base,
fact_unroll_1,
mul_preserves_le,
nat_plus_subtype_nat,
rdiv_functionality,
rneq_functionality,
rmul_preserves_rless,
rless_functionality,
rmul-zero-both,
rmul_comm,
sq_stable__less_than,
rmul-rdiv-cancel10,
rmul_assoc,
exp_step,
multiply-is-int-iff,
add-subtract-cancel,
exp_wf2,
exp-fastexp
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
hypothesis,
independent_functionElimination,
sqequalRule,
lambdaEquality,
isectElimination,
natural_numberEquality,
setElimination,
rename,
because_Cache,
minusEquality,
hypothesisEquality,
applyEquality,
addEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
dependent_set_memberEquality,
multiplyEquality,
productElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
setEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
baseClosed,
imageMemberEquality,
imageElimination,
inrFormation,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
baseApply,
closedConclusion
Latex:
d(cosine(x))/dx = \mlambda{}x.-(sine(x)) on (-\minfty{}, \minfty{})
Date html generated:
2017_10_04-PM-10_20_45
Last ObjectModification:
2017_07_28-AM-08_48_07
Theory : reals_2
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