Nuprl Lemma : cosine-medium_wf
∀[x:{x:ℝ| x ∈ [r(-2), r(2)]} ]. (cosine-medium(x) ∈ {y:ℝ| y = cosine(x)} )
Proof
Definitions occuring in Statement :
cosine-medium: cosine-medium(x),
rccint: [l, u],
i-member: r ∈ I,
cosine: cosine(x),
req: x = y,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
cosine-medium: cosine-medium(x),
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
false: False,
prop: ℙ,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
int-rdiv: (a)/k1,
int-to-real: r(n),
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
real: ℝ,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
rge: x ≥ y,
rgt: x > y,
le: A ≤ B,
nat: ℕ,
sq_stable: SqStable(P),
cand: A c∧ B
Lemmas referenced :
member_rccint_lemma,
istype-void,
rless-case_wf,
int-rdiv_wf,
subtype_base_sq,
int_subtype_base,
istype-int,
nequal_wf,
int-to-real_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,
istype-less_than,
real_wf,
i-member_wf,
rccint_wf,
rabs_wf,
rdiv_wf,
rless-int,
rless_wf,
rmul_preserves_rleq,
rmul_wf,
itermSubtract_wf,
itermMultiply_wf,
rinv_wf2,
itermVar_wf,
nat_plus_properties,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
rleq_functionality,
rabs-of-nonneg,
req_weakening,
int-rdiv-req,
req_transitivity,
rmul-rinv3,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
int-rinv-cancel2,
rmul-int,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening_rless,
rleq-int-fractions2,
istype-false,
rsub_wf,
int-rmul_wf,
rnexp_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
istype-le,
sine-small_wf,
rleq_wf,
rneq-int,
cosine-small_wf,
req_wf,
cosine_wf,
rcos-reduce4,
rcos_wf,
rsin_wf,
sine_wf,
sq_stable__req,
req_functionality,
rcos-is-cosine,
rsub_functionality,
int-rmul_functionality,
rmul_functionality,
rnexp_functionality,
rsin-is-sine,
rmul-rinv,
rcos-reduce2,
rabs-rless-iff,
rless_transitivity2,
rminus_wf,
rleq_weakening,
itermMinus_wf,
rminus_functionality,
rinv-as-rdiv,
real_term_value_minus_lemma,
rless_functionality,
rmul_preserves_rless,
minus-one-mul-top,
req_inversion,
rabs-rleq-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
setElimination,
thin,
rename,
sqequalHypSubstitution,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
hypothesis,
productElimination,
isectElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
lambdaFormation_alt,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
equalityIstype,
baseClosed,
sqequalBase,
universeIsType,
hypothesisEquality,
closedConclusion,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
addEquality,
applyEquality,
because_Cache,
inhabitedIsType,
axiomEquality,
setIsType,
minusEquality,
inrFormation_alt,
int_eqEquality,
imageElimination,
applyLambdaEquality
Latex:
\mforall{}[x:\{x:\mBbbR{}| x \mmember{} [r(-2), r(2)]\} ]. (cosine-medium(x) \mmember{} \{y:\mBbbR{}| y = cosine(x)\} )
Date html generated:
2019_10_30-AM-11_42_52
Last ObjectModification:
2019_02_03-PM-01_18_50
Theory : reals_2
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