Nuprl Lemma : poly-zero-implies
∀n:ℕ. ∀p:polyform(n). ((↑poly-zero(n;p)) ⇒ (∀l:{l:ℤ List| ||l|| = n ∈ ℤ} . (l@p = 0 ∈ ℤ)))
Proof
Definitions occuring in Statement :
poly-int-val: l@p,
poly-zero: poly-zero(n;p),
polyform: polyform(n),
length: ||as||,
list: T List,
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
polyform: polyform(n),
poly-zero: poly-zero(n;p),
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nat: ℕ,
so_apply: x[s],
or: P ∨ Q,
poly-int-val: l@p,
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
top: Top,
ge: i ≥ j ,
decidable: Dec(P),
le: A ≤ B,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
uiff: uiff(P;Q),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bfalse: ff,
sq_type: SQType(T),
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b,
select: L[n],
nil: [],
it: ⋅,
sum: Σ(f[x] | x < k),
sum_aux: sum_aux(k;v;i;x.f[x])
Lemmas referenced :
set_wf,
list_wf,
equal-wf-base-T,
list_subtype_base,
int_subtype_base,
assert_wf,
poly-zero_wf,
polyform_wf,
nat_wf,
eq_int_wf,
list-cases,
length_of_nil_lemma,
null_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
le_weakening2,
length_wf,
non_neg_length,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
bnot_wf,
not_wf,
equal-wf-T-base,
decidable__le,
add-is-int-iff,
intformnot_wf,
int_formula_prop_not_lemma,
false_wf,
null_cons_lemma,
spread_cons_lemma,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
le_wf,
stuck-spread,
base_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
sqequalRule,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
thin,
intEquality,
lambdaEquality,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
applyEquality,
independent_isectElimination,
setElimination,
rename,
natural_numberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_pairFormation,
int_eqEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
pointwiseFunctionality,
instantiate,
cumulativity,
impliesFunctionality,
dependent_set_memberEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}p:polyform(n). ((\muparrow{}poly-zero(n;p)) {}\mRightarrow{} (\mforall{}l:\{l:\mBbbZ{} List| ||l|| = n\} . (l@p = 0)))
Date html generated:
2017_09_29-PM-06_00_04
Last ObjectModification:
2017_04_26-PM-02_04_49
Theory : integer!polynomials
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