Nuprl Lemma : assert-nonneg-monomial
∀m:iMonomial(). ((↑nonneg-monomial(m)) ⇒ (∃m':iMonomial(). ∃k:ℕ+. (mul-monomials(m';m') = mul-monomials(m;<k, []>) ∈ iM\000Conomial())))
Proof
Definitions occuring in Statement :
nonneg-monomial: nonneg-monomial(m),
mul-monomials: mul-monomials(m1;m2),
iMonomial: iMonomial(),
nil: [],
nat_plus: ℕ+,
assert: ↑b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
pair: <a, b>,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iMonomial: iMonomial(),
nonneg-monomial: nonneg-monomial(m),
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_nzero: ℤ-o,
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
bfalse: ff,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
false: False,
lelt: i ≤ j < k,
decidable: Dec(P),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
not: ¬A,
subtract: n - m,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
nat: ℕ,
exists: ∃x:A. B[x],
ge: i ≥ j ,
sq_stable: SqStable(P),
cons: [a / b],
merge-int-accum: merge-int-accum(as;bs),
eager-accum: eager-accum(x,a.f[x; a];y;l),
nil: [],
it: ⋅,
even-int-list: even-int-list(L),
bor: p ∨bq,
bnot: ¬bb,
btrue: tt,
assert: ↑b,
rev_uimplies: rev_uimplies(P;Q),
merge-int: merge-int(as;bs),
label: ...$L... t,
insert-int: insert-int(x;l),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
bool: 𝔹,
unit: Unit,
sorted: sorted(L),
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
mul-monomials: mul-monomials(m1;m2),
has-value: (a)↓
Lemmas referenced :
istype-assert,
nonneg-monomial_wf,
iMonomial_wf,
assert_wf,
le_int_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
band_wf,
btrue_wf,
assert_of_le_int,
even-int-list_wf,
bfalse_wf,
le_wf,
istype-le,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
less_than_transitivity1,
less_than_irreflexivity,
int_seg_wf,
decidable__equal_int,
subtract_wf,
set_subtype_base,
int_subtype_base,
decidable__le,
istype-false,
not-le-2,
less-iff-le,
le_antisymmetry_iff,
condition-implies-le,
minus-add,
istype-void,
minus-minus,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
zero-add,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
decidable__lt,
not-lt-2,
le-add-cancel-alt,
istype-less_than,
subtype_rel_self,
int_seg_properties,
non_neg_length,
length_wf_nat,
istype-sqequal,
istype-int,
le_reflexive,
length_wf,
list_wf,
not-equal-2,
guard_wf,
all_wf,
sorted_wf,
exists_wf,
equal-wf-base,
list_subtype_base,
primrec-wf2,
sq_stable__le,
add-mul-special,
zero-mul,
istype-nat,
nat_properties,
list-cases,
nil_wf,
product_subtype_list,
cons_wf,
null_cons_lemma,
reduce_tl_cons_lemma,
reduce_hd_cons_lemma,
null_nil_lemma,
reduce_tl_nil_lemma,
eq_int_wf,
assert_of_eq_int,
length_of_cons_lemma,
two-mul,
mul-distributes-right,
one-mul,
sorted-cons,
l_all_iff,
l_member_wf,
l_all_wf,
merge-int-accum-sq,
member-merge-int,
reduce_cons_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
merge-int-comm,
iff_weakening_equal,
merge-int_wf,
list_ind_nil_lemma,
l_all_wf_nil,
list_ind_cons_lemma,
l_all_cons,
lt_int_wf,
assert_of_lt_int,
istype-top,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
less_than_wf,
less_than_anti-reflexive,
sq_stable__all,
select_wf,
less_than_transitivity2,
le_weakening2,
le_witness_for_triv,
minus-zero,
nequal_wf,
nat_plus_wf,
mul-monomials_wf,
int_entire_a,
trivial-equal,
value-type-has-value,
int_nzero_wf,
set-value-type,
int-value-type,
list-value-type
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
Error :universeIsType,
natural_numberEquality,
setElimination,
rename,
dependent_functionElimination,
unionElimination,
instantiate,
cumulativity,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
productEquality,
because_Cache,
Error :isect_memberEquality_alt,
Error :productIsType,
independent_pairFormation,
promote_hyp,
voidElimination,
applyEquality,
Error :dependent_set_memberEquality_alt,
imageElimination,
addEquality,
minusEquality,
hypothesis_subsumption,
Error :lambdaEquality_alt,
intEquality,
Error :inhabitedIsType,
Error :dependent_pairFormation_alt,
Error :equalityIstype,
Error :functionIsType,
closedConclusion,
functionEquality,
setEquality,
baseApply,
baseClosed,
Error :setIsType,
imageMemberEquality,
multiplyEquality,
voidEquality,
sqequalBase,
Error :inlFormation_alt,
applyLambdaEquality,
universeEquality,
equalityElimination,
lessCases,
Error :isect_memberFormation_alt,
axiomSqEquality,
Error :isectIsTypeImplies,
Error :functionIsTypeImplies,
independent_pairEquality,
callbyvalueReduce
Latex:
\mforall{}m:iMonomial()
((\muparrow{}nonneg-monomial(m)) {}\mRightarrow{} (\mexists{}m':iMonomial(). \mexists{}k:\mBbbN{}\msupplus{}. (mul-monomials(m';m') = mul-monomials(m;<k, []>\000C))))
Date html generated:
2019_06_20-PM-00_46_00
Last ObjectModification:
2019_04_08-PM-04_00_11
Theory : omega
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