Nuprl Lemma : poly-approx-aux_wf
∀[k:ℕ]. ∀[a:ℕ ⟶ ℝ]. ∀[x:ℝ]. ∀[xM:ℤ]. ∀[M:ℕ+]. ∀[n:ℕ]. (poly-approx-aux(a;x;xM;M;n;k) ∈ ℤ)
Proof
Definitions occuring in Statement :
poly-approx-aux: poly-approx-aux(a;x;xM;M;n;k),
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,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
poly-approx-aux: poly-approx-aux(a;x;xM;M;n;k),
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
subtype_rel: A ⊆r B,
real: ℝ,
nat_plus: ℕ+,
has-value: (a)↓,
guard: {T},
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
nat_wf,
nat_plus_wf,
real_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_plus_properties,
itermAdd_wf,
int_term_value_add_lemma,
le_wf,
value-type-has-value,
int-value-type,
equal_wf,
add_nat_plus,
divide_wf,
absval_wf,
itermMultiply_wf,
int_term_value_mul_lemma,
set-value-type,
eq_int_wf,
bool_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,
int_subtype_base,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-base,
mul_nat_plus,
int_entire_a,
true_wf,
mul_nzero,
subtype_rel_sets,
nequal_wf,
add-is-int-iff,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
unionElimination,
applyEquality,
functionExtensionality,
because_Cache,
dependent_set_memberEquality,
addEquality,
callbyvalueReduce,
multiplyEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
equalityElimination,
productElimination,
promote_hyp,
instantiate,
cumulativity,
divideEquality,
baseApply,
closedConclusion,
addLevel,
setEquality,
pointwiseFunctionality
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[a:\mBbbN{} {}\mrightarrow{} \mBbbR{}]. \mforall{}[x:\mBbbR{}]. \mforall{}[xM:\mBbbZ{}]. \mforall{}[M:\mBbbN{}\msupplus{}]. \mforall{}[n:\mBbbN{}]. (poly-approx-aux(a;x;xM;M;n;k) \mmember{} \mBbbZ{})
Date html generated:
2018_05_22-PM-02_00_56
Last ObjectModification:
2017_10_25-PM-02_12_35
Theory : reals
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