Nuprl Lemma : cosine0
cosine(r0) = r1
Proof
Definitions occuring in Statement :
cosine: cosine(x),
req: x = y,
int-to-real: r(n),
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
le: A ≤ B,
less_than': less_than'(a;b),
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
rat_term_to_real: rat_term_to_real(f;t),
rtermMultiply: left "*" right,
rat_term_ind: rat_term_ind,
rtermConstant: "const",
rtermDivide: num "/" denom,
pi1: fst(t),
pi2: snd(t),
rev_uimplies: rev_uimplies(P;Q),
nat_plus: ℕ+
Lemmas referenced :
cosine-is-limit,
int-to-real_wf,
series-sum-constant,
ifthenelse_wf,
eq_int_wf,
real_wf,
istype-nat,
int-rmul_wf,
fastexp_wf,
int-rdiv_wf,
fact_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-le,
nat_plus_inc_int_nzero,
rnexp_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
upper_subtype_nat,
istype-false,
nequal-le-implies,
zero-add,
req_weakening,
series-sum_functionality,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
rnexp_zero_lemma,
fact0_redex_lemma,
exp-fastexp,
exp0_lemma,
nequal_wf,
rmul_wf,
rdiv_wf,
rless-int,
rless_wf,
assert-rat-term-eq2,
rtermConstant_wf,
rtermMultiply_wf,
rtermDivide_wf,
req_functionality,
req_transitivity,
int-rmul-req,
rmul_functionality,
int-rdiv-req,
req-int,
fact-non-zero,
rneq-int,
req_wf,
nat_plus_wf,
le-add-cancel,
add-commutes,
add_functionality_wrt_le,
not-lt-2,
decidable__lt,
mul_nat_plus,
equal-wf-base,
int_formula_prop_less_lemma,
intformless_wf,
nat_plus_properties,
less_than_wf,
subtype_rel_sets,
satisfiable-full-omega-tt,
int_upper_properties,
le_wf,
false_wf,
int_upper_subtype_nat,
int-rmul_functionality,
int-rdiv_functionality,
rnexp0,
uiff_transitivity,
rdiv-zero,
rmul-int,
cosine_wf,
nat_wf,
series-sum-unique
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
isectElimination,
natural_numberEquality,
hypothesis,
independent_functionElimination,
lambdaEquality_alt,
setElimination,
rename,
hypothesisEquality,
minusEquality,
dependent_set_memberEquality_alt,
multiplyEquality,
because_Cache,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
applyEquality,
lambdaFormation_alt,
inhabitedIsType,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
equalityIstype,
promote_hyp,
instantiate,
hypothesis_subsumption,
cumulativity,
intEquality,
closedConclusion,
baseClosed,
sqequalBase,
inrFormation_alt,
imageMemberEquality,
applyLambdaEquality,
setEquality,
computeAll,
voidEquality,
isect_memberEquality,
lambdaEquality,
dependent_pairFormation,
lambdaFormation,
dependent_set_memberEquality
Latex:
cosine(r0) = r1
Date html generated:
2019_10_29-AM-10_35_44
Last ObjectModification:
2019_04_02-AM-09_59_24
Theory : reals
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