Nuprl Lemma : add-poly-lemma1
∀p,q:iMonomial() List. ∀m:iMonomial().
((∀i:ℕ||p||. ∀j:ℕi. imonomial-less(p[j];p[i]))
⇒ (∀i:ℕ||q||. ∀j:ℕi. imonomial-less(q[j];q[i]))
⇒ (0 < ||p|| ⇒ imonomial-less(m;p[0]))
⇒ (0 < ||q|| ⇒ imonomial-less(m;q[0]))
⇒ 0 < ||add-ipoly(p;q)||
⇒ imonomial-less(m;add-ipoly(p;q)[0]))
Proof
Definitions occuring in Statement :
add-ipoly: add-ipoly(p;q),
imonomial-less: imonomial-less(m1;m2),
iMonomial: iMonomial(),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
less_than: a < b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
imonomial-le: imonomial-le(m1;m2),
pi2: snd(t),
imonomial-less: imonomial-less(m1;m2),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
pi1: fst(t),
iMonomial: iMonomial(),
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
unit: Unit,
bool: 𝔹,
or: P ∨ Q,
decidable: Dec(P),
true: True,
nat_plus: ℕ+,
subtract: n - m,
uiff: uiff(P;Q),
nat: ℕ,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
bfalse: ff,
cons: [a / b],
has-value: (a)↓,
less_than: a < b,
btrue: tt,
ifthenelse: if b then t else f fi ,
so_apply: x[s1;s2],
top: Top,
so_lambda: λ2x y.t[x; y],
it: ⋅,
nil: [],
select: L[n],
add-ipoly: Error :add-ipoly,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
so_apply: x[s],
guard: {T},
squash: ↓T,
and: P ∧ Q,
lelt: i ≤ j < k,
sq_stable: SqStable(P),
uimplies: b supposing a,
int_seg: {i..j-},
prop: ℙ,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
int_nzero_wf,
subtype_rel_self,
sorted_wf,
imonomial-less-transitive,
nat_plus_wf,
add_nat_plus,
add-subtract-cancel,
le-add-cancel,
not-equal-2,
select-cons-tl,
true_wf,
squash_wf,
lelt_wf,
le-add-cancel2,
condition-implies-le,
not-le-2,
decidable__le,
add-member-int_seg2,
neg_assert_of_eq_int,
assert_of_eq_int,
eq_int_wf,
int-value-type,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
eqtt_to_assert,
bool_wf,
imonomial-le_wf,
decidable__lt,
nat_properties,
int_seg_properties,
minus-one-mul-top,
mul-commutes,
mul-associates,
mul-distributes,
omega-shadow,
not-lt-2,
add-zero,
zero-mul,
zero-add,
mul-distributes-right,
two-mul,
add-mul-special,
add-swap,
one-mul,
minus-one-mul,
minus-add,
add-associates,
le_reflexive,
subtract_wf,
add_functionality_wrt_le,
less-iff-le,
add-commutes,
equal_wf,
int_subtype_base,
le_wf,
set_subtype_base,
nat_wf,
length_wf_nat,
non_neg_length,
spread_cons_lemma,
null_cons_lemma,
length_of_cons_lemma,
cons_wf,
less_than_irreflexivity,
less_than_transitivity1,
null_nil_lemma,
base_wf,
stuck-spread,
length_of_nil_lemma,
nil_wf,
list-value-type,
value-type-has-value,
add-ipoly_wf1,
false_wf,
less_than_wf,
le_weakening2,
less_than_transitivity2,
sq_stable__le,
select_wf,
imonomial-less_wf,
length_wf,
int_seg_wf,
list_wf,
all_wf,
iMonomial_wf,
list_induction
Rules used in proof :
setEquality,
productEquality,
independent_pairEquality,
hyp_replacement,
int_eqReduceFalseSq,
int_eqReduceTrueSq,
cumulativity,
instantiate,
equalityElimination,
unionElimination,
dependent_set_memberEquality,
minusEquality,
multiplyEquality,
promote_hyp,
equalitySymmetry,
equalityTransitivity,
intEquality,
applyEquality,
sqequalIntensionalEquality,
dependent_pairFormation,
addEquality,
callbyvalueReduce,
voidElimination,
isect_memberEquality,
voidEquality,
independent_pairFormation,
dependent_functionElimination,
imageElimination,
baseClosed,
imageMemberEquality,
productElimination,
independent_functionElimination,
independent_isectElimination,
rename,
setElimination,
hypothesisEquality,
natural_numberEquality,
functionEquality,
because_Cache,
lambdaEquality,
sqequalRule,
hypothesis,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}p,q:iMonomial() List. \mforall{}m:iMonomial().
((\mforall{}i:\mBbbN{}||p||. \mforall{}j:\mBbbN{}i. imonomial-less(p[j];p[i]))
{}\mRightarrow{} (\mforall{}i:\mBbbN{}||q||. \mforall{}j:\mBbbN{}i. imonomial-less(q[j];q[i]))
{}\mRightarrow{} (0 < ||p|| {}\mRightarrow{} imonomial-less(m;p[0]))
{}\mRightarrow{} (0 < ||q|| {}\mRightarrow{} imonomial-less(m;q[0]))
{}\mRightarrow{} 0 < ||add-ipoly(p;q)||
{}\mRightarrow{} imonomial-less(m;add-ipoly(p;q)[0]))
Date html generated:
2017_04_14-AM-08_58_21
Last ObjectModification:
2017_04_03-AM-09_56_54
Theory : omega
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