Nuprl Lemma : poly-zero-false
∀n:ℕ. ∀p:polynom(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,
polynom: polynom(n),
poly-zero: poly-zero(n;p),
length: ||as||,
list: T List,
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
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,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than': less_than'(a;b),
le: A ≤ B,
btrue: tt,
ifthenelse: if b then t else f fi ,
subtract: n - m,
eq_int: (i =z j),
poly-zero: poly-zero(n;p),
polynom: polynom(n),
it: ⋅,
nil: [],
null: null(as),
poly-int-val: l@p,
nequal: a ≠ b ∈ T ,
uiff: uiff(P;Q),
cons: [a / b],
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
top: Top,
sq_type: SQType(T),
guard: {T},
bfalse: ff,
bool: 𝔹,
unit: Unit,
assert: ↑b,
true: True,
polyform-lead-nonzero: polyform-lead-nonzero(n;p),
nat_plus: ℕ+,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
select: L[n],
bnot: ¬bb,
sum_aux: sum_aux(k;v;i;x.f[x]),
sum: Σ(f[x] | x < k)
Lemmas referenced :
polynom_wf,
subtract_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
istype-assert,
poly-zero_wf,
polynom_subtype_polyform,
istype-void,
list_wf,
list_subtype_base,
int_subtype_base,
poly-int-val_wf2,
istype-less_than,
primrec-wf2,
iff_wf,
not_wf,
assert_wf,
equal-wf-base,
set_subtype_base,
le_wf,
istype-nat,
subtype_rel_self,
istype-false,
length_wf_nat,
eq_int_wf,
nil_wf,
length_nil,
neg_assert_of_eq_int,
product_subtype_list,
list-cases,
null_nil_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
itermAdd_wf,
intformeq_wf,
decidable__lt,
non_neg_length,
length_wf,
le_weakening2,
length_of_cons_lemma,
bnot_wf,
bool_wf,
null_wf,
bool_cases,
subtype_base_sq,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
uiff_transitivity,
null_cons_lemma,
reduce_hd_cons_lemma,
add_nat_plus,
nat_plus_properties,
add-is-int-iff,
false_wf,
sum-nat,
absval_wf,
int_seg_properties,
select_wf,
int_seg_wf,
add_nat_wf,
nat_properties,
cons_wf,
decidable__equal_int,
spread_cons_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
sum_split_first,
polynom-subtype-list,
exp_wf2,
iff_weakening_equal,
sum_wf,
add-subtract-cancel,
sum_le,
absval_mul,
exp_add,
int_seg_subtype_nat,
select_cons_tl,
exp_wf_nat_plus,
mul_preserves_le,
mul_bounds_1a,
mul-one,
mul-associates,
minus-add,
minus-one-mul,
mul-commutes,
mul-swap,
itermMultiply_wf,
int_term_value_mul_lemma,
add-swap,
add-commutes,
absval_pos,
exp_wf4,
less_than_functionality,
absval_sum,
le_weakening,
sum_scalar_mult,
less_than_wf,
mul_preserves_lt,
exp_step,
less_than_transitivity2,
istype-top,
assert-bnot,
bool_cases_sqequal,
assert_of_lt_int,
lt_int_wf,
sum-unroll,
int_term_value_minus_lemma,
itermMinus_wf,
multiply-is-int-iff,
absval-positive,
istype-base,
stuck-spread,
length_of_nil_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
rename,
setElimination,
sqequalRule,
functionIsType,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
dependent_set_memberEquality_alt,
hypothesisEquality,
natural_numberEquality,
hypothesis,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
productIsType,
applyEquality,
setIsType,
intEquality,
equalityIstype,
inhabitedIsType,
baseApply,
closedConclusion,
baseClosed,
sqequalBase,
equalitySymmetry,
because_Cache,
functionEquality,
productEquality,
setEquality,
equalityTransitivity,
equalityIsType4,
productElimination,
hypothesis_subsumption,
promote_hyp,
isect_memberEquality_alt,
instantiate,
cumulativity,
equalityElimination,
applyLambdaEquality,
pointwiseFunctionality,
imageElimination,
addEquality,
universeEquality,
multiplyEquality,
imageMemberEquality,
minusEquality,
equalityIsType1,
axiomSqEquality,
isect_memberFormation_alt,
lessCases
Latex:
\mforall{}n:\mBbbN{}. \mforall{}p:polynom(n). (\mneg{}\muparrow{}poly-zero(n;p) \mLeftarrow{}{}\mRightarrow{} \mexists{}l:\{l:\mBbbZ{} List| ||l|| = n\} . (\mneg{}(l@p = 0)))
Date html generated:
2020_05_19-PM-09_52_08
Last ObjectModification:
2019_12_31-PM-00_15_19
Theory : integer!polynomials
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