Nuprl Lemma : minus-polynom-val
∀[n:ℕ]. ∀[p:polyform(n)]. ∀[l:{l:ℤ List| ||l|| = n ∈ ℤ} ]. (minus-polynom(n;p)@l = (-p@l) ∈ ℤ)
Proof
Definitions occuring in Statement :
minus-polynom: minus-polynom(n;p),
poly-int-val: p@l,
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 :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
colength: colength(L),
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
uiff: uiff(P;Q),
unit: Unit,
bool: 𝔹,
le: A ≤ B,
cons: [a / b],
it: ⋅,
nil: [],
null: null(as),
poly-int-val: p@l,
or: P ∨ Q,
decidable: Dec(P),
btrue: tt,
ifthenelse: if b then t else f fi ,
subtract: n - m,
eq_int: (i =z j),
minus-polynom: minus-polynom(n;p),
polyform: polyform(n),
so_apply: x[s],
guard: {T},
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
top: Top,
all: ∀x:A. B[x],
not: ¬A,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
add_functionality_wrt_eq,
int_term_value_minus_lemma,
int_term_value_mul_lemma,
itermMinus_wf,
itermMultiply_wf,
minus-is-int-iff,
length-map,
iff_weakening_equal,
minus_functionality_wrt_eq,
poly_int_val_cons_cons,
true_wf,
squash_wf,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
uiff_transitivity,
length_wf_nat,
exp_wf2,
poly-int-val_wf,
false_wf,
add-is-int-iff,
minus-polynom_wf,
map_wf,
cons_wf,
not_wf,
bnot_wf,
assert_wf,
poly_int_val_nil_cons,
map_cons_lemma,
decidable__equal_int,
set_subtype_base,
spread_cons_lemma,
map_nil_lemma,
colength_wf_list,
equal-wf-T-base,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
le_weakening,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
itermAdd_wf,
intformeq_wf,
decidable__lt,
non_neg_length,
length_wf,
le_weakening2,
length_of_cons_lemma,
product_subtype_list,
length_of_nil_lemma,
list-cases,
nat_wf,
list_subtype_base,
equal-wf-base-T,
int_subtype_base,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
le_wf,
polyform_wf,
less_than_irreflexivity,
less_than_transitivity1,
equal-wf-base,
list_wf,
set_wf,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties
Rules used in proof :
imageMemberEquality,
universeEquality,
impliesFunctionality,
multiplyEquality,
pointwiseFunctionality,
imageElimination,
addEquality,
applyLambdaEquality,
int_eqReduceFalseSq,
cumulativity,
instantiate,
int_eqReduceTrueSq,
equalityElimination,
equalitySymmetry,
equalityTransitivity,
productElimination,
hypothesis_subsumption,
promote_hyp,
minusEquality,
unionElimination,
dependent_set_memberEquality,
because_Cache,
applyEquality,
baseClosed,
closedConclusion,
baseApply,
axiomEquality,
independent_functionElimination,
computeAll,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_isectElimination,
natural_numberEquality,
lambdaFormation,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[p:polyform(n)]. \mforall{}[l:\{l:\mBbbZ{} List| ||l|| = n\} ]. (minus-polynom(n;p)@l = (-p@l))
Date html generated:
2017_04_20-AM-07_11_50
Last ObjectModification:
2017_04_19-AM-11_05_45
Theory : list_1
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