Nuprl Lemma : rcos-shift-2n-pi
∀x:ℝ. ∀M:ℤ. (rcos(x + (r(M) * 2 * π)) = rcos(x))
Proof
Definitions occuring in Statement :
pi: π,
rcos: rcos(x),
int-rmul: k1 * a,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
true: True,
squash: ↓T
Lemmas referenced :
real_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
req_witness,
rcos_wf,
radd_wf,
rmul_wf,
int-to-real_wf,
int-rmul_wf,
pi_wf,
subtract-1-ge-0,
istype-nat,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
req_weakening,
req_functionality,
rcos_functionality,
radd_functionality,
rmul_functionality,
int-rmul-req,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
rcos-shift-2pi,
subtract_wf,
req_inversion,
rsub_wf,
rsub-int,
rsub_functionality,
decidable__le,
istype-le,
intformnot_wf,
itermMinus_wf,
int_formula_prop_not_lemma,
int_term_value_minus_lemma,
rminus_wf,
req_transitivity,
squash_wf,
true_wf,
rminus-int,
real_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
universeIsType,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
functionIsTypeImplies,
inhabitedIsType,
because_Cache,
productElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
unionElimination,
dependent_set_memberEquality_alt,
minusEquality,
applyEquality,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}x:\mBbbR{}. \mforall{}M:\mBbbZ{}. (rcos(x + (r(M) * 2 * \mpi{})) = rcos(x))
Date html generated:
2019_10_31-AM-06_06_28
Last ObjectModification:
2019_02_03-PM-05_41_08
Theory : reals_2
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