Nuprl Lemma : minus-polynom-val
∀[n:ℕ]. ∀[p:polyform(n)]. ∀[l:{l:ℤ List| ||l|| = n ∈ ℤ} ]. (l@minus-polynom(n;p) = (-l@p) ∈ ℤ)
Proof
Definitions occuring in Statement :
minus-polynom: minus-polynom(n;p),
poly-int-val: l@p,
polyform: polyform(n),
length: ||as||,
list: T List,
nat: ℕ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
minus: -n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
guard: {T},
so_apply: x[s],
polyform: polyform(n),
minus-polynom: minus-polynom(n;p),
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
decidable: Dec(P),
or: P ∨ Q,
poly-int-val: l@p,
null: null(as),
nil: [],
it: ⋅,
cons: [a / b],
le: A ≤ B,
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
set_wf,
list_wf,
equal-wf-base,
less_than_transitivity1,
less_than_irreflexivity,
polyform_wf,
le_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_subtype_base,
equal-wf-base-T,
list_subtype_base,
nat_wf,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
le_weakening2,
length_wf,
non_neg_length,
decidable__lt,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
le_weakening,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
minus-polynom_wf,
polyform-value-type,
map-rev-sq-map,
equal-wf-T-base,
colength_wf_list,
map_nil_lemma,
spread_cons_lemma,
set_subtype_base,
decidable__equal_int,
map_cons_lemma,
poly_int_val_nil_cons,
assert_wf,
bnot_wf,
not_wf,
cons_wf,
map_wf,
add-is-int-iff,
false_wf,
poly-int-val_wf,
exp_wf2,
length_wf_nat,
uiff_transitivity,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
squash_wf,
true_wf,
poly_int_val_cons_cons,
minus_functionality_wrt_eq,
iff_weakening_equal,
length-map,
minus-is-int-iff,
itermMultiply_wf,
itermMinus_wf,
int_term_value_mul_lemma,
int_term_value_minus_lemma,
add_functionality_wrt_eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
because_Cache,
dependent_set_memberEquality,
unionElimination,
minusEquality,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
int_eqReduceTrueSq,
instantiate,
cumulativity,
int_eqReduceFalseSq,
applyLambdaEquality,
addEquality,
imageElimination,
pointwiseFunctionality,
multiplyEquality,
impliesFunctionality,
universeEquality,
imageMemberEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[p:polyform(n)]. \mforall{}[l:\{l:\mBbbZ{} List| ||l|| = n\} ]. (l@minus-polynom(n;p) = (-l@p))
Date html generated:
2017_09_29-PM-06_00_37
Last ObjectModification:
2017_04_27-PM-05_04_59
Theory : integer!polynomials
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