Nuprl Lemma : imonomial-nonneg-lemma
∀[k:ℕ+]. ∀[m,m':iMonomial()].
∀f:ℤ ⟶ ℝ. (r0 ≤ real_term_value(f;imonomial-term(m))) supposing mul-monomials(m';m') = mul-monomials(m;<k, []>) ∈ iMo\000Cnomial()
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
real_term_value: real_term_value(f;t),
int-to-real: r(n),
real: ℝ,
mul-monomials: mul-monomials(m1;m2),
imonomial-term: imonomial-term(m),
iMonomial: iMonomial(),
nil: [],
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
pair: <a, b>,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iMonomial: iMonomial(),
subtype_rel: A ⊆r B,
sorted: sorted(L),
all: ∀x:A. B[x],
select: L[n],
uimplies: b supposing a,
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
nat_plus: ℕ+,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
true: True,
int_nzero: ℤ-o,
req_int_terms: t1 ≡ t2,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
imonomial-term: imonomial-term(m),
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
nat_plus_inc_int_nzero,
nil_wf,
stuck-spread,
istype-base,
istype-void,
length_of_nil_lemma,
int_seg_properties,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
intformnot_wf,
int_formula_prop_not_lemma,
int_seg_wf,
sorted_wf,
real_wf,
le_witness_for_triv,
mul-monomials_wf,
nat_plus_wf,
itermMultiply_wf,
imonomial-term_wf,
mul-monomials-req,
real_term_value_mul_lemma,
req_int_terms_wf,
squash_wf,
true_wf,
int_term_wf,
list_wf,
subtype_rel_product,
int_nzero_wf,
subtype_rel_self,
iff_weakening_equal,
req_int_terms_functionality,
req_int_terms_weakening,
real_term_value_wf,
list_accum_nil_lemma,
real_term_value_const_lemma,
req_wf,
rmul_wf,
int-to-real_wf,
rmul_preserves_rleq,
rless-int,
decidable__lt,
itermSubtract_wf,
rleq_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
req_functionality,
req_weakening,
square-nonneg,
req_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
independent_pairEquality,
hypothesisEquality,
applyEquality,
thin,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
dependent_set_memberEquality_alt,
isectElimination,
intEquality,
lambdaFormation_alt,
baseClosed,
independent_isectElimination,
isect_memberEquality_alt,
voidElimination,
setElimination,
rename,
productElimination,
imageElimination,
dependent_functionElimination,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
independent_pairFormation,
universeIsType,
because_Cache,
unionElimination,
functionIsType,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
equalityIstype,
isectIsTypeImplies,
productIsType,
setEquality,
closedConclusion,
setIsType,
imageMemberEquality,
instantiate,
universeEquality
Latex:
\mforall{}[k:\mBbbN{}\msupplus{}]. \mforall{}[m,m':iMonomial()].
\mforall{}f:\mBbbZ{} {}\mrightarrow{} \mBbbR{}. (r0 \mleq{} real\_term\_value(f;imonomial-term(m)))
supposing mul-monomials(m';m') = mul-monomials(m;<k, []>)
Date html generated:
2019_10_29-AM-10_07_56
Last ObjectModification:
2019_04_08-PM-04_11_39
Theory : reals
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