Nuprl Lemma : reduce-halfpi_wf
∀[x:ℝ]. (reduce-halfpi(x) ∈ {n:ℤ| |x - r(n) * π/2| ≤ r(2)} )
Proof
Definitions occuring in Statement :
reduce-halfpi: reduce-halfpi(x),
halfpi: π/2,
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
reduce-halfpi: reduce-halfpi(x),
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
rless: x < y,
sq_exists: ∃x:A [B[x]],
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,
false: False,
real: ℝ,
int-to-real: r(n),
int-rmul: k1 * a,
MachinPi4: MachinPi4(),
rsub: x - y,
radd: a + b,
accelerate: accelerate(k;f),
reg-seq-list-add: reg-seq-list-add(L),
cbv_list_accum: cbv_list_accum(x,a.f[x; a];y;L),
cons: [a / b],
atan: atan(a;x),
atan_approx: atan_approx(a;x;M),
atan-log: atan-log(a;M),
gen_log_aux: gen_log_aux(p;c;x;i;n;M),
ifthenelse: if b then t else f fi ,
le_int: i ≤z j,
bnot: ¬bb,
lt_int: i <z j,
exp: i^n,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
subtract: n - m,
btrue: tt,
bfalse: ff,
atan-approx: atan-approx(k;x;N),
poly-approx: poly-approx(a;x;k;N),
rmul: a * b,
int-rdiv: (a)/k1,
imax: imax(a;b),
absval: |i|,
reg-seq-mul: reg-seq-mul(x;y),
poly-approx-aux: poly-approx-aux(a;x;xM;M;n;k),
eq_int: (i =z j),
rminus: -(x),
nil: [],
it: ⋅,
guard: {T},
rdiv: (x/y),
rinv: rinv(x),
mu-ge: mu-ge(f;n),
reg-seq-inv: reg-seq-inv(x),
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
reduce-real_wf,
istype-less_than,
real_wf,
int-rmul_wf,
MachinPi4_wf,
rless_wf,
int-to-real_wf,
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,
rleq_wf,
rabs_wf,
rsub_wf,
rmul_wf,
halfpi_wf,
radd_wf,
rdiv_wf,
rless-int,
rless_transitivity2,
rleq_weakening_rless,
rleq_functionality,
rabs_functionality,
rsub_functionality,
req_weakening,
rmul_functionality,
req_inversion,
2-MachinPi4
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality_alt,
closedConclusion,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
hypothesis,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
dependent_set_memberFormation_alt,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
isect_memberEquality_alt,
voidElimination,
addEquality,
applyLambdaEquality,
imageElimination,
because_Cache,
inrFormation_alt,
productElimination
Latex:
\mforall{}[x:\mBbbR{}]. (reduce-halfpi(x) \mmember{} \{n:\mBbbZ{}| |x - r(n) * \mpi{}/2| \mleq{} r(2)\} )
Date html generated:
2019_10_31-AM-06_06_59
Last ObjectModification:
2019_02_03-PM-04_44_49
Theory : reals_2
Home
Index