Nuprl Lemma : radd_rcos-deriv-seq
finite-deriv-seq((-∞, ∞);3;i,x.if (i =z 0) then radd_rcos(x)
if (i =z 1) then r1 - rsin(x)
if (i =z 2) then -(rcos(x))
else rsin(x)
fi )
Proof
Definitions occuring in Statement :
radd_rcos: radd_rcos(x),
rcos: rcos(x),
rsin: rsin(x),
finite-deriv-seq: finite-deriv-seq(I;k;i,x.F[i; x]),
riiint: (-∞, ∞),
rsub: x - y,
rminus: -(x),
int-to-real: r(n),
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
natural_number: $n
Definitions unfolded in proof :
finite-deriv-seq: finite-deriv-seq(I;k;i,x.F[i; x]),
all: ∀x:A. B[x],
member: t ∈ T,
int_seg: {i..j-},
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
rsub: x - y,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
derivative-radd_rcos,
int_seg_subtype,
false_wf,
int_seg_cases,
derivative-minus-minus,
riiint_wf,
rcos_wf,
real_wf,
i-member_wf,
rsin_wf,
deriviative-rcos,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
int-to-real_wf,
rsub_wf,
rminus_wf,
req_weakening,
set_wf,
radd_wf,
derivative-sub,
derivative-const,
deriviative-rsin,
derivative_functionality,
req_functionality,
radd-zero-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
hypothesis_subsumption,
addEquality,
independent_pairFormation,
lambdaEquality,
setEquality,
productElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll
Latex:
finite-deriv-seq((-\minfty{}, \minfty{});3;i,x.if (i =\msubz{} 0) then radd\_rcos(x)
if (i =\msubz{} 1) then r1 - rsin(x)
if (i =\msubz{} 2) then -(rcos(x))
else rsin(x)
fi )
Date html generated:
2016_10_26-PM-00_17_10
Last ObjectModification:
2016_09_12-PM-05_41_22
Theory : reals_2
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