Nuprl Lemma : mul-polynom-int-val
∀[n:ℕ]. ∀[l:{l:ℤ List| ||l|| = n ∈ ℤ} ]. ∀[p,q:polyform(n)]. (mul-polynom(n;p;q)@l = (p@l * q@l) ∈ ℤ)
Proof
Definitions occuring in Statement :
mul-polynom: mul-polynom(n;p;q),
poly-int-val: p@l,
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 :
subtract: n - m,
poly-zero: poly-zero(n;p),
primrec: primrec(n;b;c),
exp: i^n,
nat_plus: ℕ+,
append: as @ bs,
has-valueall: has-valueall(a),
has-value: (a)↓,
callbyvalueall: callbyvalueall,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
colength: colength(L),
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
eager-accum: eager-accum(x,a.f[x; a];y;l),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
squash: ↓T,
nequal: a ≠ b ∈ T ,
cons: [a / b],
nil: [],
null: null(as),
poly-int-val: p@l,
mul-polynom: mul-polynom(n;p;q),
less_than: a < b,
int_upper: {i...},
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
polyform: polyform(n),
or: P ∨ Q,
decidable: Dec(P),
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
so_apply: x[s],
int_seg: {i..j-},
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
guard: {T},
prop: ℙ,
and: P ∧ Q,
top: Top,
not: ¬A,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
map-length,
mul-zero,
map_cons_lemma,
map_nil_lemma,
multiply-is-int-iff,
add-subtract-cancel,
exp_step,
poly_int_val_cons_cons,
length-append,
zero-mul,
null_cons_lemma,
add-polynom-int-val,
add_functionality_wrt_eq,
iff_wf,
iff_imp_equal_bool,
add_nat_wf,
uiff_transitivity,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
bool_cases,
mul-polynom_wf,
map_wf,
poly-zero_wf,
nil_wf,
polynom_subtype_polyform,
append_wf,
btrue_neq_bfalse,
not_assert_elim,
null_nil_lemma,
assert_of_null,
null_wf,
btrue_wf,
add-polynom_wf1,
not_wf,
bnot_wf,
assert_wf,
evalall-reduce,
valueall-type-polyform,
valueall-type-has-valueall,
poly_int_val_nil_cons,
list_ind_cons_lemma,
set_subtype_base,
spread_cons_lemma,
exp0_lemma,
list_ind_nil_lemma,
colength_wf_list,
equal-wf-T-base,
int_term_value_mul_lemma,
itermMultiply_wf,
length_wf_nat,
exp_wf2,
polyconst-val,
iff_weakening_equal,
polyconst_wf,
true_wf,
squash_wf,
cons_wf,
poly-int-val_wf,
subtype_rel-equal,
add-is-int-iff,
length_of_nil_lemma,
int_upper_properties,
equal-wf-base,
non_neg_length,
length_wf,
le_weakening2,
length_of_cons_lemma,
product_subtype_list,
list-cases,
nat_wf,
int_subtype_base,
list_subtype_base,
int_term_value_add_lemma,
itermAdd_wf,
lelt_wf,
decidable__lt,
zero-add,
nequal-le-implies,
int_upper_subtype_nat,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
le_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
int_seg_subtype,
decidable__equal_int,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
false_wf,
int_seg_subtype_nat,
int_seg_properties,
int_seg_wf,
equal-wf-base-T,
list_wf,
set_wf,
less_than_irreflexivity,
less_than_transitivity1,
polyform_wf,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties
Rules used in proof :
sqequalAxiom,
levelHypothesis,
equalityUniverse,
sqequalIntensionalEquality,
addLevel,
impliesFunctionality,
callbyvalueReduce,
imageMemberEquality,
universeEquality,
imageElimination,
int_eqReduceFalseSq,
int_eqReduceTrueSq,
pointwiseFunctionality,
multiplyEquality,
addEquality,
cumulativity,
instantiate,
promote_hyp,
equalityElimination,
dependent_set_memberEquality,
hypothesis_subsumption,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
unionElimination,
productElimination,
baseClosed,
closedConclusion,
baseApply,
because_Cache,
applyEquality,
axiomEquality,
independent_functionElimination,
computeAll,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
lambdaFormation,
thin,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[l:\{l:\mBbbZ{} List| ||l|| = n\} ]. \mforall{}[p,q:polyform(n)]. (mul-polynom(n;p;q)@l = (p@l * q@l))
Date html generated:
2017_04_20-AM-07_13_38
Last ObjectModification:
2017_04_17-PM-06_31_12
Theory : list_1
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