Nuprl Lemma : ipolynomial-nonneg
∀[p:iPolynomial()]. ∀f:ℤ ⟶ ℝ. (r0 ≤ real_term_value(f;ipolynomial-term(p))) supposing ↑nonneg-poly(p)
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
real_term_value: real_term_value(f;t),
int-to-real: r(n),
real: ℝ,
nonneg-poly: nonneg-poly(p),
ipolynomial-term: ipolynomial-term(p),
iPolynomial: iPolynomial(),
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
iPolynomial: iPolynomial(),
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
or: P ∨ Q,
cons: [a / b],
decidable: Dec(P),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
ipolynomial-term: ipolynomial-term(p),
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
req_int_terms: t1 ≡ t2,
iMonomial: iMonomial(),
int_nzero: ℤ-o,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y
Lemmas referenced :
assert-nonneg-poly,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
le_witness_for_triv,
iMonomial_wf,
list-cases,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
list_wf,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
itermSubtract_wf,
itermAdd_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
le_wf,
istype-nat,
real_wf,
istype-assert,
nonneg-poly_wf,
iPolynomial_wf,
null_nil_lemma,
real_term_value_const_lemma,
rleq_weakening_equal,
int-to-real_wf,
l_all_wf2,
nil_wf,
assert_wf,
nonneg-monomial_wf,
subtype_rel_set,
top_wf,
void_wf,
l_member_wf,
l_all_cons,
cons_wf,
ipolynomial-term-cons-req,
real_term_value_add_lemma,
radd_wf,
real_term_value_wf,
imonomial-term_wf,
ipolynomial-term_wf,
rleq_functionality_wrt_implies,
rleq_transitivity,
radd_functionality_wrt_rleq,
rleq_weakening,
req_inversion,
trivial-rleq-radd,
imonomial-nonneg
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
unionElimination,
promote_hyp,
hypothesis_subsumption,
equalityIstype,
because_Cache,
dependent_set_memberEquality_alt,
instantiate,
applyLambdaEquality,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
sqequalBase,
functionIsType,
isectIsTypeImplies,
voidEquality,
functionExtensionality,
setIsType,
independent_pairEquality
Latex:
\mforall{}[p:iPolynomial()]
\mforall{}f:\mBbbZ{} {}\mrightarrow{} \mBbbR{}. (r0 \mleq{} real\_term\_value(f;ipolynomial-term(p))) supposing \muparrow{}nonneg-poly(p)
Date html generated:
2019_10_29-AM-10_08_21
Last ObjectModification:
2019_04_08-PM-05_34_11
Theory : reals
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