Nuprl Lemma : mul-mono-poly_wf
∀[m:iMonomial()]. ∀[p:iPolynomial()]. (mul-mono-poly(m;p) ∈ iPolynomial())
Proof
Definitions occuring in Statement :
mul-mono-poly: mul-mono-poly(m;p),
iPolynomial: iPolynomial(),
iMonomial: iMonomial(),
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iPolynomial: iPolynomial(),
all: ∀x:A. B[x],
int_seg: {i..j-},
so_lambda: λ2x.t[x],
uimplies: b supposing a,
sq_stable: SqStable(P),
implies: P ⇒ Q,
lelt: i ≤ j < k,
and: P ∧ Q,
squash: ↓T,
guard: {T},
so_apply: x[s],
prop: ℙ,
mul-mono-poly: mul-mono-poly(m;p),
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
false: False,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
le: A ≤ B,
uiff: uiff(P;Q),
subtract: n - m,
nat_plus: ℕ+,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
not: ¬A,
decidable: Dec(P),
or: P ∨ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
iMonomial: iMonomial(),
int_nzero: ℤ-o,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
sq_type: SQType(T),
cons: [a / b],
ge: i ≥ j ,
imonomial-less: imonomial-less(m1;m2),
pi2: snd(t),
imonomial-le: imonomial-le(m1;m2),
mul-monomials: mul-monomials(m1;m2),
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
mul-mono-poly_wf1,
int_seg_wf,
length_wf,
iMonomial_wf,
all_wf,
imonomial-less_wf,
select_wf,
sq_stable__le,
less_than_transitivity2,
le_weakening2,
iPolynomial_wf,
list_induction,
list_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
list_ind_nil_lemma,
length_of_cons_lemma,
list_ind_cons_lemma,
less_than_irreflexivity,
less_than_transitivity1,
cons_wf,
non_neg_length,
length_wf_nat,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
equal_wf,
add-commutes,
less-iff-le,
add_functionality_wrt_le,
subtract_wf,
le_reflexive,
add-associates,
minus-add,
minus-one-mul,
one-mul,
add-swap,
add-mul-special,
two-mul,
mul-distributes-right,
zero-add,
zero-mul,
add-zero,
not-lt-2,
omega-shadow,
less_than_wf,
mul-distributes,
mul-associates,
mul-commutes,
minus-one-mul-top,
int_seg_properties,
nat_properties,
decidable__lt,
add-subtract-cancel,
le-add-cancel,
select-cons-tl,
true_wf,
squash_wf,
lelt_wf,
le-add-cancel2,
condition-implies-le,
not-le-2,
false_wf,
decidable__le,
add-member-int_seg2,
valueall-type-has-valueall,
product-valueall-type,
int_nzero_wf,
sorted_wf,
subtype_rel_self,
set-valueall-type,
nequal_wf,
int-valueall-type,
list-valueall-type,
mul-monomials_wf,
evalall-reduce,
decidable__equal_int,
subtype_base_sq,
select_cons_tl,
iff_weakening_equal,
minus-zero,
not-equal-2,
not-equal-implies-less,
le-add-cancel-alt,
minus-minus,
list-cases,
product_subtype_list,
value-type-has-value,
int-value-type,
list-value-type,
merge-int-accum_wf,
merge-int-accum-sq,
equal-wf-base,
merge-int-one-one,
intlex_wf,
merge-int_wf,
assert_functionality_wrt_uiff,
merge-int-comm,
merge-int-lex,
le_antisymmetry_iff,
imonomial-less-transitive
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
lambdaFormation,
natural_numberEquality,
sqequalRule,
lambdaEquality,
because_Cache,
independent_isectElimination,
independent_functionElimination,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
functionEquality,
voidElimination,
voidEquality,
addEquality,
dependent_pairFormation,
sqequalIntensionalEquality,
applyEquality,
intEquality,
promote_hyp,
multiplyEquality,
minusEquality,
independent_pairFormation,
unionElimination,
hyp_replacement,
setEquality,
callbyvalueReduce,
instantiate,
cumulativity,
universeEquality,
hypothesis_subsumption,
baseApply,
closedConclusion
Latex:
\mforall{}[m:iMonomial()]. \mforall{}[p:iPolynomial()]. (mul-mono-poly(m;p) \mmember{} iPolynomial())
Date html generated:
2017_09_29-PM-05_53_30
Last ObjectModification:
2017_07_26-PM-01_42_47
Theory : omega
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