Nuprl Lemma : mul-polynom_wf2
∀[n:ℕ]. ∀[p,q:polynom(n)]. (mul-polynom(n;p;q) ∈ polynom(n))
Proof
Definitions occuring in Statement :
mul-polynom: mul-polynom(n;p;q),
polynom: polynom(n),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
nat_plus: ℕ+,
le: A ≤ B,
cons: [a / b],
true: True,
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
nil: [],
list_ind: list_ind,
length: ||as||,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
polyform-lead-nonzero: polyform-lead-nonzero(n;p),
so_apply: x[s1;s2],
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
uiff: uiff(P;Q),
it: ⋅,
unit: Unit,
bool: 𝔹,
so_lambda: λ2x y.t[x; y],
guard: {T},
subtype_rel: A ⊆r B,
btrue: tt,
ifthenelse: if b then t else f fi ,
subtract: n - m,
eq_int: (i =z j),
polynom: polynom(n),
or: P ∨ Q,
decidable: Dec(P),
mul-polynom: mul-polynom(n;p;q),
prop: ℙ,
and: P ∧ Q,
top: Top,
all: ∀x:A. B[x],
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,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
iff_imp_equal_bool,
nonzero-mul-polynom,
assert_functionality_wrt_uiff,
assert_of_ff,
nat_plus_properties,
nat_plus_wf,
length_wf_nat,
add_nat_plus,
map_cons_lemma,
map_nil_lemma,
decidable__lt,
top_wf,
length_append,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-swap,
add-associates,
minus-minus,
minus-add,
minus-one-mul-top,
zero-add,
minus-one-mul,
condition-implies-le,
less-iff-le,
not-le-2,
add-commutes,
subtype_rel-equal,
length_cons,
non_neg_length,
length_nil,
null_cons_lemma,
list_ind_cons_lemma,
product_subtype_list,
reduce_hd_cons_lemma,
null_nil_lemma,
list_ind_nil_lemma,
list-cases,
uiff_transitivity,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
assert_of_eq_int,
bool_cases,
map_wf,
false_wf,
int_term_value_add_lemma,
itermAdd_wf,
add-is-int-iff,
length_of_nil_lemma,
length_of_cons_lemma,
length-append,
cons_wf,
append_wf,
subtype_rel_list,
polyform_wf,
hd_wf,
length_wf,
nil_wf,
null_wf,
valueall-type-polynom,
polynom_subtype_polyform,
poly-zero_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
eqtt_to_assert,
bool_wf,
add-polynom_wf,
polyconst_wf,
eager-accum_wf,
int_subtype_base,
equal-wf-base,
not_wf,
bnot_wf,
assert_wf,
eq_int_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
less_than_irreflexivity,
le_weakening,
less_than_transitivity1,
decidable__equal_int,
nat_wf,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
le_wf,
polynom_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 :
addLevel,
applyLambdaEquality,
imageMemberEquality,
minusEquality,
hypothesis_subsumption,
impliesFunctionality,
pointwiseFunctionality,
addEquality,
functionEquality,
imageElimination,
closedConclusion,
baseApply,
cumulativity,
instantiate,
promote_hyp,
productElimination,
equalityElimination,
baseClosed,
int_eqReduceFalseSq,
applyEquality,
int_eqReduceTrueSq,
because_Cache,
multiplyEquality,
unionElimination,
dependent_set_memberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
independent_functionElimination,
computeAll,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_isectElimination,
natural_numberEquality,
lambdaFormation,
intWeakElimination,
sqequalRule,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[p,q:polynom(n)]. (mul-polynom(n;p;q) \mmember{} polynom(n))
Date html generated:
2017_04_20-AM-07_14_57
Last ObjectModification:
2017_04_18-AM-09_45_56
Theory : list_1
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