Nuprl Lemma : real-term-nonneg
∀[t:int_term()]. ∀f:ℤ ⟶ ℝ. (r0 ≤ real_term_value(f;t)) supposing ↑nonneg-poly(int_term_to_ipoly(t))
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),
int_term_to_ipoly: int_term_to_ipoly(t),
int_term: int_term(),
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],
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
req_int_terms: t1 ≡ t2,
guard: {T},
subtype_rel: A ⊆r B,
iPolynomial: iPolynomial()
Lemmas referenced :
ipolynomial-nonneg,
int_term_to_ipoly_wf,
istype-int,
real_wf,
le_witness_for_triv,
istype-assert,
nonneg-poly_wf,
int_term_wf,
real_term_polynomial,
rleq_transitivity,
int-to-real_wf,
real_term_value_wf,
ipolynomial-term_wf,
rleq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
dependent_functionElimination,
functionIsType,
universeIsType,
sqequalRule,
lambdaEquality_alt,
productElimination,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
natural_numberEquality,
applyEquality,
setElimination,
rename,
because_Cache
Latex:
\mforall{}[t:int\_term()]. \mforall{}f:\mBbbZ{} {}\mrightarrow{} \mBbbR{}. (r0 \mleq{} real\_term\_value(f;t)) supposing \muparrow{}nonneg-poly(int\_term\_to\_ipoly(t))
Date html generated:
2019_10_29-AM-10_08_33
Last ObjectModification:
2019_04_08-PM-06_01_06
Theory : reals
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