Nuprl Lemma : mul-ipoly-ringeq
∀r:CRng. ∀p,q:iMonomial() List. ipolynomial-term(mul-ipoly(p;q)) ≡ ipolynomial-term(p) (*) ipolynomial-term(q)
Proof
Definitions occuring in Statement :
ringeq_int_terms: t1 ≡ t2,
crng: CRng,
mul-ipoly: mul-ipoly(p;q),
ipolynomial-term: ipolynomial-term(p),
iMonomial: iMonomial(),
itermMultiply: left (*) right,
list: T List,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
mul-ipoly: mul-ipoly(p;q),
uimplies: b supposing a,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
iMonomial: iMonomial(),
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
int_nzero: ℤ-o,
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bfalse: ff,
ipolynomial-term: ipolynomial-term(p),
null: null(as),
nil: [],
it: ⋅,
itermConstant: "const",
ringeq_int_terms: t1 ≡ t2,
crng: CRng,
rng: Rng,
and: P ∧ Q,
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_uimplies: rev_uimplies(P;Q),
true: True,
squash: ↓T,
infix_ap: x f y,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
iMonomial_wf,
list-cases,
valueall-type-has-valueall,
list_wf,
list-valueall-type,
void-valueall-type,
nil_wf,
evalall-reduce,
null_nil_lemma,
product_subtype_list,
product-valueall-type,
int_nzero_wf,
sorted_wf,
subtype_rel_self,
set-valueall-type,
nequal_wf,
int-valueall-type,
cons_wf,
null_cons_lemma,
spread_cons_lemma,
crng_wf,
ring_term_value_const_lemma,
ring_term_value_mul_lemma,
rng_car_wf,
rng_times_zero,
ring_term_value_wf,
ipolynomial-term_wf,
int-to-ring-zero,
null_wf,
bool_wf,
eqtt_to_assert,
assert_of_null,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
equal-wf-T-base,
btrue_wf,
and_wf,
bfalse_wf,
btrue_neq_bfalse,
ipolynomial-term-cons-ringeq,
eager-accum_wf,
mul-mono-poly_wf1,
add-ipoly_wf1,
itermMultiply_wf,
itermAdd_wf,
imonomial-term_wf,
int_term_wf,
ring_term_value_add_lemma,
rng_times_over_plus,
ringeq_int_terms_functionality,
ringeq_int_terms_weakening,
itermMultiply_functionality_wrt_ringeq,
mul-mono-poly-ringeq,
ringeq_int_terms_transitivity,
itermAdd_functionality_wrt_ringeq,
list_induction,
all_wf,
ringeq_int_terms_wf,
list_accum_wf,
list_accum_nil_lemma,
list_accum_cons_lemma,
rng_plus_wf,
squash_wf,
true_wf,
rng_plus_zero,
iff_weakening_equal,
add-ipoly-ringeq,
rng_times_wf,
rng_plus_assoc,
rng_plus_ac_1,
rng_plus_comm,
eager-accum-list_accum
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
unionElimination,
sqequalRule,
voidEquality,
independent_isectElimination,
callbyvalueReduce,
promote_hyp,
hypothesis_subsumption,
productElimination,
lambdaEquality,
setEquality,
intEquality,
because_Cache,
independent_functionElimination,
natural_numberEquality,
isect_memberEquality,
voidElimination,
functionEquality,
setElimination,
rename,
equalitySymmetry,
equalityElimination,
equalityTransitivity,
dependent_pairFormation,
instantiate,
cumulativity,
baseClosed,
dependent_set_memberEquality,
independent_pairFormation,
applyLambdaEquality,
independent_pairEquality,
applyEquality,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}r:CRng. \mforall{}p,q:iMonomial() List.
ipolynomial-term(mul-ipoly(p;q)) \mequiv{} ipolynomial-term(p) (*) ipolynomial-term(q)
Date html generated:
2018_05_21-PM-03_17_17
Last ObjectModification:
2018_05_19-AM-08_08_31
Theory : rings_1
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