Nuprl Lemma : rcos-pi-over-4
rcos((π/r(4))) = (r1/rsqrt(r(2)))
Proof
Definitions occuring in Statement :
pi: π,
rcos: rcos(x),
rsqrt: rsqrt(x),
rdiv: (x/y),
req: x = y,
int-to-real: r(n),
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
int-to-real: r(n),
rsqrt: rsqrt(x),
rroot: rroot(i;x),
ifthenelse: if b then t else f fi ,
isEven: isEven(n),
eq_int: (i =z j),
modulus: a mod n,
remainder: n rem m,
btrue: tt,
rroot-abs: rroot-abs(i;x),
fastexp: i^n,
efficient-exp-ext,
genrec: genrec,
subtract: n - m,
rabs: |x|,
absval: |i|,
iroot: iroot(n;x),
integer-nth-root-ext,
exp: i^n,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
genrec-ap: genrec-ap,
divide: n ÷ m,
true: True,
and: P ∧ Q,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
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,
subtype_rel: A ⊆r B,
real: ℝ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
rneq: x ≠ y,
guard: {T},
nat: ℕ,
uiff: uiff(P;Q),
rcos: rcos(x),
approx-arg: approx-arg(f;B;x),
accelerate: accelerate(k;f),
rdiv: (x/y),
rmul: a * b,
pi: π,
int-rmul: k1 * a,
rinv: rinv(x),
mu-ge: mu-ge(f;n),
lt_int: i <z j,
imax: imax(a;b),
halfpi: π/2,
cubic_converge: cubic_converge(b;m),
le_int: i ≤z j,
bnot: ¬bb,
bfalse: ff,
fastpi: fastpi(n),
reg-seq-inv: reg-seq-inv(x),
reg-seq-mul: reg-seq-mul(x;y),
cosine: cosine(x),
pi1: fst(t),
cosine-exists-ext,
int-rdiv: (a)/k1,
canonical-bound: canonical-bound(r),
rsum: Σ{x[k] | n≤k≤m},
expfact: expfact(n;x;p;b),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
map: map(f;as),
list_ind: list_ind,
from-upto: [n, m),
cons: [a / b],
nil: [],
it: ⋅,
radd-list: radd-list(L),
length: ||as||,
reg-seq-list-add: reg-seq-list-add(L),
cbv_list_accum: cbv_list_accum(x,a.f[x; a];y;L),
fact: (n)!,
rnexp: x^k1,
canon-bnd: canon-bnd(x),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermConstant: "const",
rtermSubtract: left "-" right,
pi2: snd(t)
Lemmas referenced :
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,
int-to-real_wf,
rsqrt_wf,
rleq-int,
istype-false,
rleq_wf,
rsin-rcos-pythag,
rdiv_wf,
pi_wf,
rless-int,
rless_wf,
radd_wf,
rnexp_wf,
nat_plus_properties,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
istype-le,
rsin_wf,
rcos_wf,
req_functionality,
radd_functionality,
req_weakening,
rnexp_functionality,
rsin-pi-over-4,
rsqrt-unique,
rleq-int-fractions2,
rleq_weakening_rless,
rmul_wf,
req_inversion,
rnexp2,
rsqrt-rnexp-2,
rsqrt-positive-iff,
rdiv_functionality,
rnexp-one,
rneq_functionality,
rnexp-rdiv,
req-implies-req,
rsub_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
assert-rat-term-eq2,
rtermSubtract_wf,
rtermConstant_wf,
rtermDivide_wf,
rsqrt1,
false_wf,
less_than_wf,
rsqrt-rdiv,
efficient-exp-ext,
integer-nth-root-ext,
cosine-exists-ext
Rules used in proof :
cut,
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
independent_pairFormation,
natural_numberEquality,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
sqequalHypSubstitution,
hypothesis,
dependent_set_memberEquality_alt,
extract_by_obid,
dependent_functionElimination,
unionElimination,
isectElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
universeIsType,
addEquality,
applyEquality,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
productElimination,
lambdaFormation_alt,
because_Cache,
closedConclusion,
inrFormation_alt,
int_eqEquality,
multiplyEquality,
lambdaFormation,
dependent_set_memberEquality,
inrFormation
Latex:
rcos((\mpi{}/r(4))) = (r1/rsqrt(r(2)))
Date html generated:
2019_10_30-AM-11_44_11
Last ObjectModification:
2019_04_03-AM-00_21_28
Theory : reals_2
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