Nuprl Lemma : rcosine_wf
∀[x:ℝ]. (rcosine(x) ∈ {y:ℝ| y = rcos(x)} )
Proof
Definitions occuring in Statement :
rcosine: rcosine(x),
rcos: rcos(x),
req: x = y,
real: ℝ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rcosine: rcosine(x),
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
sq_type: SQType(T),
guard: {T},
false: False,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
top: Top,
cand: A c∧ B,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
rev_uimplies: rev_uimplies(P;Q),
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
int_seg: {i..j-},
sq_stable: SqStable(P)
Lemmas referenced :
reduce-halfpi_wf,
value-type-has-value,
rleq_wf,
rabs_wf,
rsub_wf,
rmul_wf,
int-to-real_wf,
halfpi_wf,
set-value-type,
istype-int,
int-value-type,
nat_wf,
le_wf,
modulus_wf,
subtype_base_sq,
int_subtype_base,
nequal_wf,
2-MachinPi4,
int-rmul_wf,
MachinPi4_wf,
req_wf,
real_wf,
rabs-rleq-iff,
modulus_wf_int_mod,
istype-less_than,
int-subtype-int_mod,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
cosine-medium_wf,
member_rccint_lemma,
istype-void,
squash_wf,
true_wf,
rminus-int,
subtype_rel_self,
iff_weakening_equal,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
rminus_wf,
sine-medium_wf,
rcos_wf,
rleq_functionality,
rabs_functionality,
rsub_functionality,
req_weakening,
rmul_functionality,
req_inversion,
int-rmul-req,
sq_stable__req,
ifthenelse_wf,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
sine_wf,
cosine_wf,
rsin_wf,
req_functionality,
sine_functionality,
int-rmul_functionality,
cosine_functionality,
rsin-is-sine,
rsin-reduce-half-pi,
rcos-is-cosine,
rcos-reduce-half-pi,
rminus-rminus,
rminus_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
inhabitedIsType,
lambdaFormation_alt,
rename,
sqequalRule,
callbyvalueReduce,
setEquality,
intEquality,
natural_numberEquality,
independent_isectElimination,
lambdaEquality_alt,
because_Cache,
setElimination,
dependent_set_memberEquality_alt,
instantiate,
cumulativity,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
equalityIstype,
baseClosed,
sqequalBase,
universeIsType,
applyEquality,
axiomEquality,
productElimination,
closedConclusion,
independent_pairFormation,
imageMemberEquality,
unionElimination,
equalityElimination,
isect_memberEquality_alt,
imageElimination,
universeEquality,
productIsType,
minusEquality,
dependent_pairFormation_alt,
promote_hyp,
approximateComputation
Latex:
\mforall{}[x:\mBbbR{}]. (rcosine(x) \mmember{} \{y:\mBbbR{}| y = rcos(x)\} )
Date html generated:
2019_10_31-AM-06_07_39
Last ObjectModification:
2019_04_03-PM-05_18_27
Theory : reals_2
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