Nuprl Lemma : mul-monomials-ringeq
∀[r:CRng]. ∀[m1,m2:iMonomial()]. imonomial-term(mul-monomials(m1;m2)) ≡ imonomial-term(m1) (*) imonomial-term(m2)
Proof
Definitions occuring in Statement :
ringeq_int_terms: t1 ≡ t2,
crng: CRng,
mul-monomials: mul-monomials(m1;m2),
imonomial-term: imonomial-term(m),
iMonomial: iMonomial(),
itermMultiply: left (*) right,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
iMonomial: iMonomial(),
mul-monomials: mul-monomials(m1;m2),
has-value: (a)↓,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
int_nzero: ℤ-o,
ringeq_int_terms: t1 ≡ t2,
all: ∀x:A. B[x],
top: Top,
crng: CRng,
rng: Rng,
true: True,
squash: ↓T,
prop: ℙ,
infix_ap: x f y,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
merge-int: merge-int(as;bs),
imonomial-term: imonomial-term(m),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
ring_term_value: ring_term_value(f;t),
insert-int: insert-int(x;l),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
itermConstant: "const",
int_term_ind: int_term_ind,
itermMultiply: left (*) right,
itermVar: vvar,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
less_than': less_than'(a;b),
not: ¬A,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b
Lemmas referenced :
value-type-has-value,
int-value-type,
list_wf,
list-value-type,
merge-int-accum_wf,
ring_term_value_mul_lemma,
rng_car_wf,
rng_times_wf,
iMonomial_wf,
crng_wf,
equal_wf,
squash_wf,
true_wf,
imonomial-term-linear-ringeq,
subtype_rel_self,
iff_weakening_equal,
merge-int-accum-sq,
int-to-ring_wf,
ring_term_value_wf,
imonomial-term_wf,
list_induction,
all_wf,
merge-int_wf,
infix_ap_wf,
reduce_nil_lemma,
list_accum_nil_lemma,
ring_term_value_const_lemma,
list_accum_wf,
int_term_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
rng_times_one,
int-to-ring-one,
reduce_cons_lemma,
insert-int_wf,
cons_wf,
list_ind_nil_lemma,
list_accum_cons_lemma,
crng_times_comm,
list_ind_cons_lemma,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
imonomial-cons-ringeq,
crng_times_ac_1,
int-to-ring-mul,
rng_times_assoc
Rules used in proof :
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
productElimination,
thin,
sqequalRule,
callbyvalueReduce,
cut,
introduction,
extract_by_obid,
isectElimination,
intEquality,
independent_isectElimination,
hypothesis,
multiplyEquality,
setElimination,
rename,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
functionEquality,
because_Cache,
natural_numberEquality,
isect_memberFormation,
lambdaEquality,
axiomEquality,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
instantiate,
independent_functionElimination,
independent_pairEquality,
functionExtensionality,
unionElimination,
equalityElimination,
lessCases,
sqequalAxiom,
independent_pairFormation,
dependent_pairFormation,
promote_hyp,
cumulativity,
equalityUniverse,
levelHypothesis
Latex:
\mforall{}[r:CRng]. \mforall{}[m1,m2:iMonomial()].
imonomial-term(mul-monomials(m1;m2)) \mequiv{} imonomial-term(m1) (*) imonomial-term(m2)
Date html generated:
2018_05_21-PM-03_17_03
Last ObjectModification:
2018_05_19-AM-08_08_12
Theory : rings_1
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