Nuprl Lemma : poly-approx_wf
∀[k:ℕ]. ∀[a:ℕ ⟶ ℝ]. ∀[x:ℝ]. ∀[N:ℕ+]. (poly-approx(a;x;k;N) ∈ ℤ)
Proof
Definitions occuring in Statement :
poly-approx: poly-approx(a;x;k;N),
real: ℝ,
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
poly-approx: poly-approx(a;x;k;N),
has-value: (a)↓,
uimplies: b supposing a,
nat: ℕ,
nat_plus: ℕ+,
real: ℝ,
le: A ≤ B,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
Lemmas referenced :
value-type-has-value,
int-value-type,
mul_nat_plus,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
poly-approx-aux_wf,
nat_wf,
le_wf,
nat_plus_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
nat_plus_wf,
real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
callbyvalueReduce,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
independent_isectElimination,
hypothesis,
multiplyEquality,
natural_numberEquality,
addEquality,
setElimination,
rename,
hypothesisEquality,
because_Cache,
applyEquality,
dependent_set_memberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
independent_functionElimination,
lambdaEquality,
isect_memberEquality,
voidEquality,
minusEquality,
functionExtensionality,
divideEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
baseApply,
closedConclusion,
baseClosed,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[a:\mBbbN{} {}\mrightarrow{} \mBbbR{}]. \mforall{}[x:\mBbbR{}]. \mforall{}[N:\mBbbN{}\msupplus{}]. (poly-approx(a;x;k;N) \mmember{} \mBbbZ{})
Date html generated:
2018_05_22-PM-02_01_30
Last ObjectModification:
2017_10_25-PM-04_15_51
Theory : reals
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