Nuprl Lemma : rcos-is-1-iff
∀x:ℝ. (rcos(x) = r1 ⇐⇒ ∃n:ℤ. (x = 2 * n * π))
Proof
Definitions occuring in Statement :
pi: π,
rcos: rcos(x),
int-rmul: k1 * a,
req: x = y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
multiply: n * m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
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,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
true: True,
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
guard: {T},
rge: x ≥ y,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
less_than': less_than'(a;b),
int-to-real: r(n),
int-rmul: k1 * a,
pi: π,
halfpi: π/2,
divide: n ÷ m,
cubic_converge: cubic_converge(b;m),
ifthenelse: if b then t else f fi ,
le_int: i ≤z j,
bnot: ¬bb,
lt_int: i <z j,
bfalse: ff,
btrue: tt,
fastpi: fastpi(n),
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
subtype_rel: A ⊆r B,
real: ℝ,
rneq: x ≠ y,
rdiv: (x/y),
so_lambda: λ2x.t[x],
so_apply: x[s],
rat_term_to_real: rat_term_to_real(f;t),
rtermMultiply: left "*" right,
rat_term_ind: rat_term_ind,
rtermVar: rtermVar(var),
rtermDivide: num "/" denom,
pi1: fst(t),
pi2: snd(t)
Lemmas referenced :
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,
le_witness_for_triv,
subtract-1-ge-0,
istype-nat,
req_wf,
rcos_wf,
int-to-real_wf,
real_wf,
rcos-1-implies-at-least-2pi,
rless_wf,
rmul_wf,
int-rmul_wf,
pi_wf,
itermSubtract_wf,
itermMultiply_wf,
rleq_wf,
radd_wf,
itermAdd_wf,
rless_functionality,
req_weakening,
iff_weakening_uiff,
rleq_functionality,
req_transitivity,
rmul_functionality,
req_inversion,
radd-int,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma,
rcos-shift-2n-pi,
rsub_wf,
rcos_functionality,
rminus_wf,
itermMinus_wf,
req_functionality,
squash_wf,
true_wf,
rminus-int,
real_term_value_minus_lemma,
rless-implies-rless,
radd-preserves-rleq,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rabs_wf,
rcos-is-1,
rcos-rabs,
decidable__lt,
intformnot_wf,
int_formula_prop_not_lemma,
rleq_weakening_rless,
rmul_preserves_rless,
rdiv_wf,
rmul_preserves_rleq,
rinv_wf2,
rless_transitivity1,
rleq_weakening,
rabs-of-nonneg,
rabs-rdiv,
rdiv_functionality,
rmul-rinv3,
r-archimedean-rabs-ext,
subtract_wf,
primrec-wf2,
rleq_antisymmetry,
zero-rleq-rabs,
rabs-is-zero,
rless-cases,
rless-int,
int_term_value_subtract_lemma,
nat_plus_properties,
decidable__le,
istype-le,
subtract-add-cancel,
rabs-of-nonpos,
not-rless,
rless_transitivity2,
rmul_preserves_req,
rminus-as-rmul,
rless_irreflexivity,
assert-rat-term-eq2,
rtermVar_wf,
rtermMultiply_wf,
rtermDivide_wf,
int-rmul-req,
rmul-int
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
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,
universeIsType,
productElimination,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
dependent_set_memberEquality_alt,
because_Cache,
addEquality,
minusEquality,
applyEquality,
imageElimination,
imageMemberEquality,
baseClosed,
productIsType,
multiplyEquality,
dependent_set_memberFormation_alt,
unionElimination,
closedConclusion,
inrFormation_alt,
functionIsType,
equalityIstype,
setIsType,
instantiate,
functionEquality,
productEquality,
intEquality,
inlFormation_alt
Latex:
\mforall{}x:\mBbbR{}. (rcos(x) = r1 \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbZ{}. (x = 2 * n * \mpi{}))
Date html generated:
2019_10_31-AM-06_06_39
Last ObjectModification:
2019_05_17-PM-03_51_07
Theory : reals_2
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