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 :
decidable: Dec(P),
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
uiff: uiff(P;Q),
bfalse: ff,
it: ⋅,
unit: Unit,
bool: 𝔹,
polyform: polyform(n),
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
le: A ≤ B,
has-value: (a)↓,
btrue: tt,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
polynom: polynom(n),
subtract: n - m,
polyconst: polyconst(n;k),
mul-polynom: mul-polynom(n;p;q),
prop: ℙ,
and: P ∧ Q,
top: Top,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
polyform-lead-nonzero: polyform-lead-nonzero(n;p),
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
less_than: a < b,
squash: ↓T,
cons: [a / b],
nat_plus: ℕ+,
true: True,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y]
Lemmas referenced :
nat_wf,
subtract_wf,
polyconst_wf,
polyform-value-type,
polynom_subtype_polyform,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__le,
value-type-polynom,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
subtract-1-ge-0,
polynom_wf,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
int_subtype_base,
eqff_to_assert,
polyform_wf,
subtype_rel_self,
le_wf,
istype-false,
poly-zero_wf,
int-value-type,
value-type-has-value,
istype-less_than,
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,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
assert_of_eq_int,
eqtt_to_assert,
bool_cases,
istype-assert,
equal-wf-base,
not_wf,
bnot_wf,
assert_wf,
eq_int_wf,
istype-le,
uiff_transitivity,
itermSubtract_wf,
int_term_value_subtract_lemma,
list-cases,
length_of_nil_lemma,
null_nil_lemma,
list_ind_nil_lemma,
nil_wf,
length_wf,
hd_wf,
subtype_rel_list,
product_subtype_list,
length_of_cons_lemma,
reduce_hd_cons_lemma,
null_cons_lemma,
list_ind_cons_lemma,
cons_wf,
append_wf,
polyconst_wf2,
add_nat_plus,
length_wf_nat,
decidable__lt,
length-append,
nat_plus_properties,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf,
map_wf,
map_cons_lemma,
map_nil_lemma,
iff_imp_equal_bool,
nonzero-mul-polynom,
assert_functionality_wrt_uiff,
bfalse_wf,
assert_of_ff,
add-polynom_wf,
eager-accum_wf,
valueall-type-polynom
Rules used in proof :
int_eqReduceFalseSq,
int_eqReduceTrueSq,
Error :equalityIsType1,
multiplyEquality,
cumulativity,
instantiate,
promote_hyp,
baseClosed,
closedConclusion,
baseApply,
Error :equalityIsType2,
productElimination,
equalityElimination,
unionElimination,
applyEquality,
Error :dependent_set_memberEquality_alt,
because_Cache,
intEquality,
sqleReflexivity,
callbyvalueReduce,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
Error :universeIsType,
independent_pairFormation,
voidElimination,
Error :isect_memberEquality_alt,
dependent_functionElimination,
int_eqEquality,
Error :lambdaEquality_alt,
Error :dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
Error :lambdaFormation_alt,
intWeakElimination,
sqequalRule,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
Error :functionIsType,
Error :equalityIsType4,
Error :equalityIstype,
sqequalBase,
imageElimination,
hypothesis_subsumption,
applyLambdaEquality,
pointwiseFunctionality,
addEquality,
imageMemberEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[p,q:polynom(n)]. (mul-polynom(n;p;q) \mmember{} polynom(n))
Date html generated:
2019_06_20-PM-01_54_01
Last ObjectModification:
2019_01_21-PM-10_25_35
Theory : integer!polynomials
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