Nuprl Lemma : mul-polynom-int-val
∀[n:ℕ]. ∀[l:{l:ℤ List| ||l|| = n ∈ ℤ} ]. ∀[p,q:polyform(n)]. (l@mul-polynom(n;p;q) = (l@p * l@q) ∈ ℤ)
Proof
Definitions occuring in Statement :
mul-polynom: mul-polynom(n;p;q),
poly-int-val: l@p,
polyform: polyform(n),
length: ||as||,
list: T List,
nat: ℕ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
multiply: n * m,
int: ℤ,
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)↓,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
polyform: polyform(n),
bnot: ¬bb,
poly-int-val: l@p,
null: null(as),
nil: [],
cons: [a / b],
le: A ≤ B,
assert: ↑b,
less_than': less_than'(a;b),
int_upper: {i...},
nequal: a ≠ b ∈ T ,
eager-accum: eager-accum(x,a.f[x; a];y;l),
colength: colength(L),
less_than: a < b,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
istype: istype(T),
callbyvalueall: callbyvalueall,
has-valueall: has-valueall(a),
respects-equality: respects-equality(S;T),
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
nat_plus: ℕ+,
subtract: n - m,
poly-zero: poly-zero(n;p)
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,
polyform_wf,
list_wf,
list_subtype_base,
le_wf,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
value-type-has-value,
polyform-value-type,
polyconst_wf,
int-value-type,
poly-zero_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
poly-zero-implies,
iff_weakening_equal,
poly-int-val_wf,
itermMultiply_wf,
int_term_value_mul_lemma,
equal-wf-T-base,
bool_wf,
assert_wf,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
eq_int_wf,
equal-wf-base,
eq_int_eq_true,
btrue_wf,
bfalse_wf,
bool_subtype_base,
assert_elim,
btrue_neq_bfalse,
istype-assert,
bool_cases,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
list-cases,
product_subtype_list,
length_of_cons_lemma,
le_weakening2,
length_wf,
non_neg_length,
satisfiable-full-omega-tt,
length_of_nil_lemma,
add-is-int-iff,
false_wf,
subtype_rel-equal,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
upper_subtype_nat,
istype-false,
nequal-le-implies,
zero-add,
int_upper_properties,
cons_wf,
polyconst-val,
exp_wf2,
length_wf_nat,
exp0_lemma,
colength-cons-not-zero,
colength_wf_list,
spread_cons_lemma,
poly_int_val_nil_cons,
valueall-type-has-valueall,
valueall-type-polyform,
evalall-reduce,
add-polynom_wf1,
nat_wf,
base_wf,
null_wf,
assert_of_null,
append_wf,
nil_wf,
map_wf,
mul-polynom_wf,
add_nat_wf,
add-polynom-int-val,
subtype-respects-equality,
null_nil_lemma,
list_ind_nil_lemma,
null_cons_lemma,
list_ind_cons_lemma,
zero-mul,
length-append,
poly_int_val_cons_cons,
exp_step,
add_nat_plus,
add-associates,
add-swap,
add-commutes,
multiply-is-int-iff,
poly_int_val_cons_cons-sq,
map_nil_lemma,
map_cons_lemma,
mul-zero,
map-length,
add_functionality_wrt_eq
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,
Error :setIsType,
intEquality,
Error :equalityIstype,
baseApply,
closedConclusion,
baseClosed,
sqequalBase,
addEquality,
callbyvalueReduce,
cumulativity,
imageElimination,
universeEquality,
imageMemberEquality,
equalityElimination,
Error :equalityIsType1,
int_eqReduceTrueSq,
Error :equalityIsType4,
Error :functionIsType,
int_eqReduceFalseSq,
multiplyEquality,
promote_hyp,
isect_memberEquality,
voidEquality,
dependent_pairFormation,
lambdaEquality,
computeAll,
pointwiseFunctionality,
Error :equalityIsType3,
sqequalIntensionalEquality,
axiomSqEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[l:\{l:\mBbbZ{} List| ||l|| = n\} ]. \mforall{}[p,q:polyform(n)]. (l@mul-polynom(n;p;q) = (l@p * l@q))
Date html generated:
2019_06_20-PM-01_53_23
Last ObjectModification:
2018_11_23-PM-03_14_52
Theory : integer!polynomials
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