Nuprl Lemma : minus-poly-ringeq
∀r:Rng. ∀p:iPolynomial(). ipolynomial-term(minus-poly(p)) ≡ "-"ipolynomial-term(p)
Proof
Definitions occuring in Statement :
ringeq_int_terms: t1 ≡ t2,
rng: Rng,
minus-poly: minus-poly(p),
ipolynomial-term: ipolynomial-term(p),
iPolynomial: iPolynomial(),
itermMinus: "-"num,
all: ∀x:A. B[x]
Definitions unfolded in proof :
member: t ∈ T,
iPolynomial: iPolynomial(),
all: ∀x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
sq_stable: SqStable(P),
subtype_rel: A ⊆r B,
so_apply: x[s],
less_than: a < b,
prop: ℙ,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
and: P ∧ Q,
lelt: i ≤ j < k,
guard: {T},
uimplies: b supposing a,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
rng: Rng,
uall: ∀[x:A]. B[x],
ringeq_int_terms: t1 ≡ t2,
btrue: tt,
ifthenelse: if b then t else f fi ,
minus-poly: minus-poly(p),
ipolynomial-term: ipolynomial-term(p),
true: True,
le: A ≤ B,
ge: i ≥ j ,
subtract: n - m,
uiff: uiff(P;Q),
int_nzero: ℤ-o,
iMonomial: iMonomial(),
rev_uimplies: rev_uimplies(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
infix_ap: x f y,
minus-monomial: minus-monomial(m)
Lemmas referenced :
rng_wf,
iPolynomial_wf,
squash_wf,
sq_stable__equal,
itermMinus_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
int_seg_properties,
select_wf,
imonomial-less_wf,
iMonomial_wf,
length_wf,
int_seg_wf,
all_wf,
minus-poly_wf,
ipolynomial-term_wf,
ring_term_value_wf,
equal_wf,
rng_car_wf,
sq_stable__all,
false_wf,
int_term_value_add_lemma,
itermAdd_wf,
length_of_cons_lemma,
cons_wf,
length_of_nil_lemma,
nil_wf,
list_wf,
ringeq_int_terms_wf,
list_induction,
rng_minus_zero,
int-to-ring-zero,
ring_term_value_minus_lemma,
ring_term_value_const_lemma,
null_nil_lemma,
map_nil_lemma,
select-cons-tl,
true_wf,
add-subtract-cancel,
lelt_wf,
non_neg_length,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
add-member-int_seg2,
ipolynomial-term-cons-ringeq,
subtype_rel_self,
sorted_wf,
int_nzero_wf,
subtype_rel_product,
imonomial-term_wf,
minus-monomial_wf,
map_cons_lemma,
ringeq_int_terms_functionality,
ringeq_int_terms_transitivity,
itermAdd_functionality_wrt_ringeq,
ringeq_int_terms_weakening,
itermMinus_functionality_wrt_ringeq,
iff_weakening_equal,
int-to-ring_wf,
infix_ap_wf,
int-to-ring-minus,
rng_times_wf,
rng_minus_wf,
rng_plus_wf,
ring_term_value_add_lemma,
imonomial-term-linear-ringeq,
rng_times_over_minus,
rng_minus_over_plus,
rng_plus_comm
Rules used in proof :
hypothesis,
extract_by_obid,
introduction,
cut,
rename,
thin,
setElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
imageElimination,
baseClosed,
hypothesisEquality,
imageMemberEquality,
sqequalRule,
independent_functionElimination,
axiomEquality,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
unionElimination,
dependent_functionElimination,
productElimination,
independent_isectElimination,
natural_numberEquality,
dependent_set_memberEquality,
applyEquality,
functionExtensionality,
because_Cache,
lambdaEquality,
intEquality,
functionEquality,
isectElimination,
promote_hyp,
equalitySymmetry,
equalityTransitivity,
pointwiseFunctionality,
addEquality,
hyp_replacement,
independent_pairEquality,
setEquality,
minusEquality,
universeEquality
Latex:
\mforall{}r:Rng. \mforall{}p:iPolynomial(). ipolynomial-term(minus-poly(p)) \mequiv{} "-"ipolynomial-term(p)
Date html generated:
2018_05_21-PM-03_16_52
Last ObjectModification:
2018_01_25-PM-01_31_02
Theory : rings_1
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