Nuprl Lemma : rcos-reduce-half-pi
∀[n:ℤ]. ∀[x:ℝ].
(rcos(x - n * π/2)
= if (n mod 4 =z 0) then rcos(x)
if (n mod 4 =z 2) then -(rcos(x))
if (n mod 4 =z 1) then rsin(x)
else -(rsin(x))
fi )
Proof
Definitions occuring in Statement :
halfpi: π/2,
rcos: rcos(x),
rsin: rsin(x),
int-rmul: k1 * a,
rsub: x - y,
req: x = y,
rminus: -(x),
real: ℝ,
modulus: a mod n,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
false: False,
int_seg: {i..j-},
and: P ∧ Q,
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
eqmod: a ≡ b mod m,
divides: b | a,
le: A ≤ B,
sq_type: SQType(T),
guard: {T},
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
pi: π,
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2
Lemmas referenced :
mod_bounds,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
mod-eqmod,
modulus_wf_int_mod,
decidable__le,
intformand_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
istype-le,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
int_seg_subtype_special,
int_seg_cases,
eqmod_wf,
req_witness,
rcos_wf,
rsub_wf,
int-rmul_wf,
halfpi_wf,
ifthenelse_wf,
eq_int_wf,
real_wf,
rminus_wf,
rsin_wf,
intformeq_wf,
itermMultiply_wf,
itermMinus_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_minus_lemma,
int_term_value_subtract_lemma,
rmul_wf,
int-to-real_wf,
radd_wf,
itermAdd_wf,
req_weakening,
req_functionality,
rsub_functionality,
int-rmul-req,
radd_functionality,
rmul_functionality,
req_transitivity,
req_inversion,
rmul-int,
squash_wf,
true_wf,
rminus-int,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
pi_wf,
rcos-shift-2n-pi,
rcos_functionality,
int_term_value_add_lemma,
subtract_wf,
rminus-rminus,
rminus_functionality,
radd-int,
rsub-int,
rcos-shift-pi,
rcos-shift-half-pi,
rsin-shift-2n-pi
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality_alt,
natural_numberEquality,
dependent_functionElimination,
hypothesis,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
universeIsType,
because_Cache,
productElimination,
applyEquality,
independent_pairFormation,
imageElimination,
int_eqEquality,
productIsType,
inhabitedIsType,
lambdaFormation_alt,
setElimination,
rename,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
hypothesis_subsumption,
equalityIstype,
isectIsTypeImplies,
multiplyEquality,
minusEquality,
imageMemberEquality,
baseClosed,
addEquality
Latex:
\mforall{}[n:\mBbbZ{}]. \mforall{}[x:\mBbbR{}].
(rcos(x - n * \mpi{}/2)
= if (n mod 4 =\msubz{} 0) then rcos(x)
if (n mod 4 =\msubz{} 2) then -(rcos(x))
if (n mod 4 =\msubz{} 1) then rsin(x)
else -(rsin(x))
fi )
Date html generated:
2019_10_31-AM-06_07_06
Last ObjectModification:
2019_02_03-PM-07_09_41
Theory : reals_2
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