Nuprl Lemma : assert-poly-zero
∀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],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
assert: ↑b,
ifthenelse: if b then t else f fi ,
iff: P ⇐⇒ Q,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
nat_wf,
polynom_wf,
poly-zero_wf,
polynom_subtype_polyform,
bool_wf,
eqtt_to_assert,
equal_wf,
squash_wf,
true_wf,
poly-zero-implies,
iff_weakening_equal,
set_wf,
list_wf,
equal-wf-base-T,
list_subtype_base,
int_subtype_base,
all_wf,
equal-wf-T-base,
poly-int-val_wf2,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
false_wf,
poly-zero-false,
nat_properties,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
not_wf
Rules used in proof :
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
applyEquality,
sqequalRule,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
independent_pairFormation,
lambdaEquality,
imageElimination,
universeEquality,
intEquality,
dependent_functionElimination,
independent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
because_Cache,
baseApply,
closedConclusion,
setElimination,
rename,
setEquality,
dependent_set_memberEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
isect_memberEquality,
voidEquality,
computeAll
Latex:
\mforall{}n:\mBbbN{}. \mforall{}p:polynom(n). (\muparrow{}poly-zero(n;p) \mLeftarrow{}{}\mRightarrow{} \mforall{}l:\{l:\mBbbZ{} List| ||l|| = n\} . (l@p = 0))
Date html generated:
2017_09_29-PM-06_00_20
Last ObjectModification:
2017_04_26-PM-02_04_53
Theory : integer!polynomials
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