Nuprl Lemma : nonzero-mul-polynom
∀[n:ℕ]. ∀[p,q:polynom(n)].
(poly-zero(n;mul-polynom(n;p;q)) = ff) supposing (poly-zero(n;q) = ff and poly-zero(n;p) = ff)
Proof
Definitions occuring in Statement :
mul-polynom: mul-polynom(n;p;q),
polynom: polynom(n),
poly-zero: poly-zero(n;p),
nat: ℕ,
bfalse: ff,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
mul-polynom: mul-polynom(n;p;q),
has-value: (a)↓,
poly-zero: poly-zero(n;p),
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
uiff: uiff(P;Q),
polynom: polynom(n),
rev_uimplies: rev_uimplies(P;Q),
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cons: [a / b],
polyform-lead-nonzero: polyform-lead-nonzero(n;p),
true: True,
le: A ≤ B,
subtract: n - m,
less_than': less_than'(a;b),
eager-accum: eager-accum(x,a.f[x; a];y;l),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
polyconst: polyconst(n;k),
add-polynom: add-polynom(n;rmz;p;q),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
nil: [],
length: ||as||,
list_ind: list_ind,
has-valueall: has-valueall(a),
squash: ↓T,
colength: colength(L),
less_than: a < b,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
sq_stable: SqStable(P),
polyform: polyform(n)
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
value-type-has-value,
polynom_wf,
value-type-polynom,
polyform_wf,
polyform-value-type,
polyconst_wf,
int-value-type,
poly-zero_wf,
polynom_subtype_polyform,
bool_wf,
bfalse_wf,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
eqff_to_assert,
eq_int_wf,
ifthenelse_wf,
assert_wf,
bnot_wf,
not_wf,
equal-wf-T-base,
eqtt_to_assert,
assert_of_eq_int,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_entire_a,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
equal-wf-base,
le_wf,
istype-assert,
bool_cases,
list-cases,
null_nil_lemma,
product_subtype_list,
null_cons_lemma,
btrue_wf,
btrue_neq_bfalse,
length_of_nil_lemma,
length_of_cons_lemma,
length_wf_nat,
istype-false,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
reduce_hd_cons_lemma,
map_nil_lemma,
map_cons_lemma,
spread_cons_lemma,
subtype_rel-equal,
nat_wf,
base_wf,
uiff_transitivity,
evalall-reduce,
list_wf,
cons_wf,
mul-polynom_wf,
map_wf,
list-valueall-type,
valueall-type-polyform,
valueall-type-has-valueall,
iff_imp_equal_bool,
squash_wf,
true_wf,
length_wf,
map_length_nat,
iff_weakening_equal,
colength-cons-not-zero,
colength_wf_list,
list_ind_cons_lemma,
append_wf,
nil_wf,
void-valueall-type,
length_append,
subtype_rel_list,
top_wf,
length-singleton,
length-append,
length-map,
sq_stable__le,
add-is-int-iff,
false_wf,
istype-top,
less_than_wf,
add-polynom_wf1,
subtract-add-cancel,
add-polynom-length,
imax_ub
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
productElimination,
because_Cache,
unionElimination,
applyEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
hypothesis_subsumption,
callbyvalueReduce,
intEquality,
cumulativity,
Error :equalityIstype,
addEquality,
multiplyEquality,
baseClosed,
lambdaFormation,
equalityElimination,
dependent_pairFormation,
promote_hyp,
impliesFunctionality,
int_eqReduceFalseSq,
sqequalBase,
Error :functionIsType,
minusEquality,
baseApply,
closedConclusion,
sqleReflexivity,
imageElimination,
imageMemberEquality,
universeEquality,
Error :setIsType,
voidEquality,
pointwiseFunctionality,
lessCases,
axiomSqEquality,
Error :inrFormation_alt
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[p,q:polynom(n)].
(poly-zero(n;mul-polynom(n;p;q)) = ff) supposing (poly-zero(n;q) = ff and poly-zero(n;p) = ff)
Date html generated:
2019_06_20-PM-01_53_53
Last ObjectModification:
2019_01_17-PM-04_26_17
Theory : integer!polynomials
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