Nuprl Lemma : minus-poly-equiv
∀p:iPolynomial(). ipolynomial-term(minus-poly(p)) ≡ "-"ipolynomial-term(p)
Proof
Definitions occuring in Statement :
minus-poly: minus-poly(p),
ipolynomial-term: ipolynomial-term(p),
iPolynomial: iPolynomial(),
equiv_int_terms: t1 ≡ t2,
itermMinus: "-"num,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
iPolynomial: iPolynomial(),
member: t ∈ T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
equiv_int_terms: t1 ≡ t2,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
int_seg: {i..j-},
uimplies: b supposing a,
lelt: i ≤ j < k,
and: P ∧ Q,
guard: {T},
so_apply: x[s],
prop: ℙ,
subtype_rel: A ⊆r B,
ipolynomial-term: ipolynomial-term(p),
minus-poly: minus-poly(p),
top: Top,
ifthenelse: if b then t else f fi ,
btrue: tt,
int_term_value: int_term_value(f;t),
itermConstant: "const",
int_term_ind: int_term_ind,
itermMinus: "-"num,
uiff: uiff(P;Q),
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
subtract: n - m,
less_than': less_than'(a;b),
true: True,
exists: ∃x:A. B[x],
nat: ℕ,
less_than: a < b,
iMonomial: iMonomial(),
int_nzero: ℤ-o,
rev_uimplies: rev_uimplies(P;Q),
minus-monomial: minus-monomial(m),
itermAdd: left (+) right
Lemmas referenced :
iPolynomial_wf,
sq_stable__all,
equal_wf,
int_term_value_wf,
ipolynomial-term_wf,
minus-poly_wf,
all_wf,
int_seg_wf,
length_wf,
iMonomial_wf,
imonomial-less_wf,
select_wf,
sq_stable__le,
less_than_transitivity2,
le_weakening2,
itermMinus_wf,
sq_stable__equal,
squash_wf,
list_induction,
equiv_int_terms_wf,
list_wf,
nil_wf,
cons_wf,
map_nil_lemma,
null_nil_lemma,
minus-zero,
add-member-int_seg2,
decidable__le,
subtract_wf,
false_wf,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-associates,
minus-one-mul-top,
zero-add,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel2,
length_of_cons_lemma,
non_neg_length,
length_wf_nat,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
lelt_wf,
add-swap,
int_seg_properties,
nat_properties,
decidable__lt,
true_wf,
select-cons-tl,
not-lt-2,
le-add-cancel,
add-subtract-cancel,
map_cons_lemma,
minus-monomial_wf,
itermAdd_wf,
imonomial-term_wf,
subtype_rel_product,
int_nzero_wf,
sorted_wf,
subtype_rel_self,
equiv_int_terms_functionality,
equiv_int_terms_transitivity,
ipolynomial-term-cons,
itermAdd_functionality,
equiv_int_terms_weakening,
itermMinus_functionality,
mul-associates,
add_functionality_wrt_eq,
imonomial-term-linear,
minus_functionality_wrt_eq,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
hypothesis,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
baseClosed,
imageElimination,
isectElimination,
functionEquality,
intEquality,
lambdaEquality,
functionExtensionality,
applyEquality,
because_Cache,
dependent_set_memberEquality,
natural_numberEquality,
independent_isectElimination,
productElimination,
dependent_functionElimination,
axiomEquality,
voidEquality,
isect_memberEquality,
voidElimination,
independent_pairFormation,
unionElimination,
addEquality,
minusEquality,
dependent_pairFormation,
sqequalIntensionalEquality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
hyp_replacement,
setEquality,
independent_pairEquality,
multiplyEquality,
universeEquality
Latex:
\mforall{}p:iPolynomial(). ipolynomial-term(minus-poly(p)) \mequiv{} "-"ipolynomial-term(p)
Date html generated:
2017_04_14-AM-08_58_46
Last ObjectModification:
2017_02_27-PM-03_41_35
Theory : omega
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