Nuprl Lemma : cosine-rminus
∀x:ℝ. (cosine(-(x)) = cosine(x))
Proof
Definitions occuring in Statement :
cosine: cosine(x),
req: x = y,
rminus: -(x),
real: ℝ,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
nequal: a ≠ b ∈ T ,
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
squash: ↓T
Lemmas referenced :
cosine-is-limit,
rminus_wf,
real_wf,
int-rmul_wf,
fastexp_wf,
int-rdiv_wf,
fact_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
subtype_rel_sets,
less_than_wf,
nequal_wf,
nat_plus_properties,
intformeq_wf,
intformless_wf,
int_formula_prop_eq_lemma,
int_formula_prop_less_lemma,
equal-wf-base,
int_subtype_base,
rnexp_wf,
nat_wf,
isOdd_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
cosine_wf,
nat_plus_wf,
series-sum_functionality,
int-rmul_functionality,
int-rdiv_functionality,
rnexp-rminus,
req_weakening,
bfalse_wf,
odd-iff-not-even,
assert-isEven,
equal-wf-T-base,
series-sum_wf,
squash_wf,
true_wf,
int_nzero_wf,
ifthenelse_wf,
iff_weakening_equal,
series-sum-unique,
req_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
hypothesis,
lambdaEquality,
minusEquality,
natural_numberEquality,
dependent_set_memberEquality,
multiplyEquality,
setElimination,
rename,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
applyEquality,
because_Cache,
setEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
baseClosed,
independent_functionElimination,
equalityElimination,
productElimination,
promote_hyp,
instantiate,
cumulativity,
baseApply,
closedConclusion,
imageElimination,
functionEquality,
universeEquality,
imageMemberEquality
Latex:
\mforall{}x:\mBbbR{}. (cosine(-(x)) = cosine(x))
Date html generated:
2017_10_03-AM-09_29_24
Last ObjectModification:
2017_07_28-AM-07_48_18
Theory : reals
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