Nuprl Lemma : rsin-pi-over-4
rsin((π/r(4))) = (r1/rsqrt(r(2)))
Proof
Definitions occuring in Statement :
pi: π,
rsin: rsin(x),
rsqrt: rsqrt(x),
rdiv: (x/y),
req: x = y,
int-to-real: r(n),
natural_number: $n
Definitions unfolded in proof :
prop: ℙ,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
sq_type: SQType(T),
false: False,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
decidable: Dec(P),
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
pi: π,
le: A ≤ B,
nat: ℕ,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
rnexp: x^k1,
subtract: n - m,
fact: (n)!,
cbv_list_accum: cbv_list_accum(x,a.f[x; a];y;L),
reg-seq-list-add: reg-seq-list-add(L),
length: ||as||,
radd-list: radd-list(L),
it: ⋅,
nil: [],
efficient-exp-ext,
fastexp: i^n,
cons: [a / b],
from-upto: [n, m),
list_ind: list_ind,
map: map(f;as),
evalall: evalall(t),
callbyvalueall: callbyvalueall,
expfact: expfact(n;x;p;b),
rsum: Σ{x[k] | n≤k≤m},
int-rdiv: (a)/k1,
sine-exists-ext,
pi1: fst(t),
sine: sine(x),
reg-seq-mul: reg-seq-mul(x;y),
reg-seq-inv: reg-seq-inv(x),
primrec: primrec(n;b;c),
fastpi: fastpi(n),
bfalse: ff,
bnot: ¬bb,
le_int: i ≤z j,
cubic_converge: cubic_converge(b;m),
halfpi: π/2,
canonical-bound: canonical-bound(r),
imax: imax(a;b),
eq_int: (i =z j),
btrue: tt,
absval: |i|,
lt_int: i <z j,
ifthenelse: if b then t else f fi ,
mu-ge: mu-ge(f;n),
rinv: rinv(x),
int-rmul: k1 * a,
rmul: a * b,
accelerate: accelerate(k;f),
approx-arg: approx-arg(f;B;x),
rsin: rsin(x),
int-to-real: r(n),
sq_exists: ∃x:A [B[x]],
rless: x < y
Lemmas referenced :
rcos-radd,
rless_wf,
rless-int,
int-to-real_wf,
pi_wf,
rdiv_wf,
int-rmul-req,
req_weakening,
rmul_functionality,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
int-rinv-cancel,
req_transitivity,
req_functionality,
rmul_comm,
nequal_wf,
true_wf,
equal-wf-base,
int_formula_prop_wf,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
intformeq_wf,
intformnot_wf,
full-omega-unsat,
decidable__equal_int,
int_subtype_base,
subtype_base_sq,
req-iff-rsub-is-0,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
itermSubtract_wf,
rinv_wf2,
int-rmul_wf,
rmul_wf,
rmul_preserves_req,
halfpi_wf,
radd_wf,
rcos_functionality,
rnexp2,
req_inversion,
rsub_functionality,
rcos-halfpi,
rsin-rcos-pythag,
le_wf,
false_wf,
rnexp_wf,
rsin_wf,
rsub_wf,
rcos_wf,
rinv-mul-as-rdiv,
rsin_functionality,
rnexp_functionality,
radd-zero,
radd-preserves-req,
radd_functionality,
rleq_wf,
less_than_wf,
rleq-int-fractions2,
rsqrt-unique,
rleq_weakening_rless,
equal_wf,
real_wf,
rmul-rinv,
req-implies-req,
req_wf,
rleq-int,
rsqrt_wf,
rsqrt-positive,
rdiv_functionality,
rsqrt-rdiv,
rsqrt1,
efficient-exp-ext,
sine-exists-ext
Rules used in proof :
baseClosed,
hypothesisEquality,
imageMemberEquality,
independent_pairFormation,
independent_functionElimination,
productElimination,
because_Cache,
dependent_functionElimination,
inrFormation,
sqequalRule,
independent_isectElimination,
natural_numberEquality,
hypothesis,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
int_eqEquality,
lambdaFormation,
addLevel,
dependent_set_memberEquality,
equalitySymmetry,
equalityTransitivity,
voidEquality,
voidElimination,
isect_memberEquality,
lambdaEquality,
dependent_pairFormation,
approximateComputation,
unionElimination,
intEquality,
cumulativity,
instantiate,
multiplyEquality,
applyEquality,
addEquality,
productEquality,
setEquality,
rename,
setElimination
Latex:
rsin((\mpi{}/r(4))) = (r1/rsqrt(r(2)))
Date html generated:
2018_05_22-PM-03_00_39
Last ObjectModification:
2018_05_18-PM-04_43_27
Theory : reals_2
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