Nuprl Lemma : rcos-reduce2
∀[x:ℝ]. (rcos(x) = (rcos((x)/2)^2 - rsin((x)/2)^2))
Proof
Definitions occuring in Statement :
rcos: rcos(x),
rsin: rsin(x),
rnexp: x^k1,
int-rdiv: (a)/k1,
rsub: x - y,
req: x = y,
real: ℝ,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
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,
prop: ℙ,
false: False,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rneq: x ≠ y,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Lemmas referenced :
req_witness,
rcos_wf,
rsub_wf,
rnexp_wf,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
int-rdiv_wf,
subtype_base_sq,
int_subtype_base,
nequal_wf,
rsin_wf,
real_wf,
rmul_wf,
req_weakening,
req_wf,
radd_wf,
rcos-radd,
req_functionality,
rsub_functionality,
rnexp2,
iff_weakening_uiff,
req_inversion,
rcos_functionality,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
rmul_preserves_req,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
rinv_wf2,
itermAdd_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
radd_functionality,
int-rdiv-req,
req_transitivity,
int-rinv-cancel,
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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality_alt,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
universeIsType,
lambdaFormation_alt,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
baseClosed,
sqequalBase,
because_Cache,
inhabitedIsType,
productElimination,
closedConclusion,
inrFormation_alt,
independent_pairFormation,
imageMemberEquality,
int_eqEquality
Latex:
\mforall{}[x:\mBbbR{}]. (rcos(x) = (rcos((x)/2)\^{}2 - rsin((x)/2)\^{}2))
Date html generated:
2019_10_30-AM-11_41_58
Last ObjectModification:
2019_02_02-PM-01_53_45
Theory : reals_2
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