Nuprl Lemma : ipolynomial-term-cons-ringeq
∀[r:Rng]. ∀[m:iMonomial()]. ∀[p:iMonomial() List].
ipolynomial-term([m / p]) ≡ imonomial-term(m) (+) ipolynomial-term(p)
Proof
Definitions occuring in Statement :
ringeq_int_terms: t1 ≡ t2,
rng: Rng,
ipolynomial-term: ipolynomial-term(p),
imonomial-term: imonomial-term(m),
iMonomial: iMonomial(),
itermAdd: left (+) right,
cons: [a / b],
list: T List,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
rng: Rng,
ringeq_int_terms: t1 ≡ t2,
cons: [a / b],
btrue: tt,
bfalse: ff,
ifthenelse: if b then t else f fi ,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
top: Top,
ipolynomial-term: ipolynomial-term(p),
or: P ∨ Q,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
int_nzero: ℤ-o,
iMonomial: iMonomial(),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
true: True,
squash: ↓T,
so_apply: x[s],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rng_wf,
list_wf,
rng_car_wf,
list_accum_cons_lemma,
product_subtype_list,
list_accum_nil_lemma,
null_nil_lemma,
spread_cons_lemma,
null_cons_lemma,
list-cases,
iMonomial_wf,
imonomial-term_wf,
ring_term_value_wf,
rng_plus_zero,
int-to-ring-zero,
ring_term_value_const_lemma,
ring_term_value_add_lemma,
iff_weakening_equal,
rng_plus_assoc,
rng_plus_wf,
infix_ap_wf,
true_wf,
squash_wf,
equal_wf,
subtype_rel_self,
sorted_wf,
int_nzero_wf,
subtype_rel_product,
subtype_rel_list,
list_accum_wf,
itermAdd_wf,
ringeq_int_terms_wf,
int_term_wf,
all_wf,
list_induction,
ringeq_int_terms_functionality,
itermAdd_functionality_wrt_ringeq,
ringeq_int_terms_weakening
Rules used in proof :
because_Cache,
rename,
setElimination,
intEquality,
functionEquality,
axiomEquality,
lambdaEquality,
productElimination,
hypothesis_subsumption,
promote_hyp,
voidEquality,
voidElimination,
isect_memberEquality,
sqequalRule,
unionElimination,
hypothesisEquality,
dependent_functionElimination,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_pairEquality,
applyEquality,
functionExtensionality,
equalitySymmetry,
lambdaFormation,
baseClosed,
imageMemberEquality,
natural_numberEquality,
universeEquality,
equalityTransitivity,
imageElimination,
independent_functionElimination,
setEquality,
independent_isectElimination,
productEquality
Latex:
\mforall{}[r:Rng]. \mforall{}[m:iMonomial()]. \mforall{}[p:iMonomial() List].
ipolynomial-term([m / p]) \mequiv{} imonomial-term(m) (+) ipolynomial-term(p)
Date html generated:
2018_05_21-PM-03_16_32
Last ObjectModification:
2018_01_25-PM-02_19_52
Theory : rings_1
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