Nuprl Lemma : mul-ipoly_wf
∀[p,q:iPolynomial()]. (mul-ipoly(p;q) ∈ iPolynomial())
Proof
Definitions occuring in Statement :
mul-ipoly: mul-ipoly(p;q),
iPolynomial: iPolynomial(),
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
nil: [],
select: L[n],
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
it: ⋅,
unit: Unit,
bool: 𝔹,
less_than: a < b,
nat: ℕ,
exists: ∃x:A. B[x],
true: True,
less_than': less_than'(a;b),
subtype_rel: A ⊆r B,
subtract: n - m,
false: False,
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
decidable: Dec(P),
le: A ≤ B,
uiff: uiff(P;Q),
bfalse: ff,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
top: Top,
cons: [a / b],
btrue: tt,
ifthenelse: if b then t else f fi ,
or: P ∨ Q,
has-valueall: has-valueall(a),
has-value: (a)↓,
callbyvalueall: callbyvalueall,
int_nzero: ℤ-o,
prop: ℙ,
iMonomial: iMonomial(),
so_apply: x[s],
all: ∀x:A. B[x],
guard: {T},
squash: ↓T,
and: P ∧ Q,
lelt: i ≤ j < k,
implies: P ⇒ Q,
sq_stable: SqStable(P),
int_seg: {i..j-},
so_lambda: λ2x.t[x],
iPolynomial: iPolynomial(),
uimplies: b supposing a,
mul-ipoly: mul-ipoly(p;q),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
less_than_irreflexivity,
less_than_transitivity1,
base_wf,
stuck-spread,
length_of_nil_lemma,
mul-mono-poly_wf,
list_accum_wf,
equal-wf-T-base,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert_of_null,
eqtt_to_assert,
bool_wf,
null_wf,
nat_properties,
int_seg_properties,
add-subtract-cancel,
le-add-cancel,
not-lt-2,
decidable__lt,
select-cons-tl,
true_wf,
squash_wf,
add-swap,
lelt_wf,
equal_wf,
int_subtype_base,
le_wf,
set_subtype_base,
nat_wf,
length_wf_nat,
non_neg_length,
length_of_cons_lemma,
le-add-cancel2,
add-zero,
add_functionality_wrt_le,
add-commutes,
zero-add,
minus-one-mul-top,
add-associates,
minus-one-mul,
minus-add,
condition-implies-le,
not-le-2,
false_wf,
subtract_wf,
decidable__le,
cons_wf,
add-member-int_seg2,
nil_wf,
spread_cons_lemma,
null_cons_lemma,
product_subtype_list,
null_nil_lemma,
list-cases,
evalall-reduce,
int-valueall-type,
nequal_wf,
subtype_rel_self,
sorted_wf,
int_nzero_wf,
product-valueall-type,
list-valueall-type,
le_weakening2,
less_than_transitivity2,
sq_stable__le,
select_wf,
imonomial-less_wf,
length_wf,
int_seg_wf,
all_wf,
iMonomial_wf,
list_wf,
set-valueall-type,
iPolynomial_wf,
valueall-type-has-valueall
Rules used in proof :
cumulativity,
instantiate,
equalityElimination,
hyp_replacement,
sqequalIntensionalEquality,
dependent_pairFormation,
applyEquality,
minusEquality,
addEquality,
independent_pairFormation,
dependent_set_memberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
voidEquality,
voidElimination,
isect_memberEquality,
hypothesis_subsumption,
promote_hyp,
unionElimination,
callbyvalueReduce,
lambdaFormation,
intEquality,
setEquality,
dependent_functionElimination,
imageElimination,
baseClosed,
imageMemberEquality,
productElimination,
independent_functionElimination,
rename,
setElimination,
because_Cache,
hypothesisEquality,
natural_numberEquality,
lambdaEquality,
independent_isectElimination,
hypothesis,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[p,q:iPolynomial()]. (mul-ipoly(p;q) \mmember{} iPolynomial())
Date html generated:
2017_04_14-AM-08_59_23
Last ObjectModification:
2017_04_12-PM-05_21_56
Theory : omega
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