Nuprl Lemma : mul-polynom_wf
∀[n:ℕ]. ∀[p,q:polyform(n)]. (mul-polynom(n;p;q) ∈ polyform(n))
Proof
Definitions occuring in Statement :
mul-polynom: mul-polynom(n;p;q),
polyform: polyform(n),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
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],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
mul-polynom: mul-polynom(n;p;q),
polyconst: polyconst(n;k),
subtract: n - m,
has-value: (a)↓,
le: A ≤ B,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
subtype_rel: A ⊆r B,
polyform: polyform(n),
eq_int: (i =z j),
decidable: Dec(P),
nequal: a ≠ b ∈ T ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
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,
less_than_wf,
value-type-has-value,
polyform_wf,
istype-false,
le_wf,
polyform-value-type,
poly-zero_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
subtype_rel_self,
subtract-1-ge-0,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
polyconst_wf,
eq_int_wf,
assert_of_eq_int,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
intformeq_wf,
int_formula_prop_eq_lemma,
int_subtype_base,
neg_assert_of_eq_int,
int-value-type,
subtract_wf,
nat_wf,
nil_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
assert_wf,
bnot_wf,
not_wf,
equal-wf-base,
eager-accum_wf,
list_wf,
list-valueall-type,
valueall-type-polyform,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
uiff_transitivity,
add-polynom_wf1,
btrue_wf,
null_wf,
equal-wf-T-base,
append_wf,
cons_wf,
map_wf,
assert_of_null,
subtype_rel-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
Error :lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
callbyvalueReduce,
sqleReflexivity,
Error :dependent_set_memberEquality_alt,
because_Cache,
unionElimination,
equalityElimination,
productElimination,
Error :equalityIsType1,
promote_hyp,
instantiate,
cumulativity,
multiplyEquality,
applyEquality,
intEquality,
int_eqReduceTrueSq,
Error :equalityIsType2,
baseApply,
closedConclusion,
baseClosed,
int_eqReduceFalseSq,
Error :equalityIsType4,
Error :equalityIsType3
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[p,q:polyform(n)]. (mul-polynom(n;p;q) \mmember{} polyform(n))
Date html generated:
2019_06_20-PM-01_53_01
Last ObjectModification:
2018_10_07-AM-00_23_34
Theory : integer!polynomials
Home
Index