Nuprl Lemma : addrcos_wf2
∀[x:ℝ]. (addrcos(x) ∈ {f:ℕ+ ⟶ ℤ| f = (x + rcos(x))} )
Proof
Definitions occuring in Statement :
addrcos: addrcos(x),
rcos: rcos(x),
req: x = y,
radd: a + b,
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
req: x = y,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
prop: ℙ,
radd: a + b,
addrcos: addrcos(x),
reg-seq-list-add: reg-seq-list-add(L),
cbv_list_accum: cbv_list_accum(x,a.f[x; a];y;L),
cons: [a / b],
nil: [],
it: ⋅,
rcos: rcos(x),
accelerate: accelerate(k;f),
approx-arg: approx-arg(f;B;x),
has-value: (a)↓,
uimplies: b supposing a,
nat_plus: ℕ+,
real: ℝ,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
sq_type: SQType(T),
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
sq_stable: SqStable(P),
regular-int-seq: k-regular-seq(f),
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
uiff: uiff(P;Q),
le: A ≤ B,
absval: |i|,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
real_wf,
addrcos_wf,
nat_plus_wf,
all_wf,
le_wf,
absval_wf,
subtract_wf,
radd_wf,
rcos_wf,
nat_wf,
value-type-has-value,
int-value-type,
mul_nat_plus,
less_than_wf,
cosine_wf,
int-rdiv_wf,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
nequal_wf,
int-to-real_wf,
subtype_base_sq,
true_wf,
decidable__equal_int,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__lt,
equal_wf,
mul-commutes,
div-mul-cancel,
zero-add,
sq_stable__regular-int-seq,
decidable__le,
intformle_wf,
itermAdd_wf,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
le_functionality,
le_weakening,
req-iff-bdd-diff,
accelerate_wf,
regular-int-seq_wf,
accelerate-bdd-diff,
itermSubtract_wf,
int_term_value_subtract_lemma,
mul_cancel_in_le,
squash_wf,
absval_mul,
iff_weakening_equal,
div_rem_sum,
add-is-int-iff,
multiply-is-int-iff,
false_wf,
int-triangle-inequality,
add_functionality_wrt_le,
int-triangle-inequality2,
le_weakening2,
add_functionality_wrt_lt,
rem_bounds_absval
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
axiomEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
extract_by_obid,
dependent_set_memberEquality,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
lambdaEquality,
applyEquality,
functionExtensionality,
because_Cache,
setElimination,
rename,
natural_numberEquality,
callbyvalueReduce,
sqleReflexivity,
intEquality,
independent_isectElimination,
multiplyEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
addEquality,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
baseApply,
closedConclusion,
divideEquality,
addLevel,
instantiate,
cumulativity,
independent_functionElimination,
unionElimination,
equalityUniverse,
levelHypothesis,
imageElimination,
applyLambdaEquality,
productElimination,
universeEquality,
pointwiseFunctionality,
promote_hyp,
remainderEquality
Latex:
\mforall{}[x:\mBbbR{}]. (addrcos(x) \mmember{} \{f:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}| f = (x + rcos(x))\} )
Date html generated:
2017_10_04-PM-10_22_19
Last ObjectModification:
2017_07_28-AM-08_48_26
Theory : reals_2
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