Nuprl Lemma : polyvar-val
∀[n:ℕ+]. ∀[v:ℤ]. ∀[l:{l:ℤ List| ||l|| = n ∈ ℤ} ]. (l@polyvar(n;v) = if 0 ≤z v ∧b v <z n then l[v] else 0 fi ∈ ℤ)
Proof
Definitions occuring in Statement :
polyvar: polyvar(n;v),
poly-int-val: l@p,
select: L[n],
length: ||as||,
list: T List,
band: p ∧b q,
nat_plus: ℕ+,
le_int: i ≤z j,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat_plus: ℕ+,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
not: ¬A,
false: False,
polyconst: polyconst(n;k),
subtract: n - m,
polyvar: polyvar(n;v),
band: p ∧b q,
btrue: tt,
bfalse: ff,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
lt_int: i <z j,
le_int: i ≤z j,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
decidable: Dec(P),
cons: [a / b],
true: True,
so_apply: x[s1;s2],
top: Top,
so_lambda: λ2x y.t[x; y],
it: ⋅,
nil: [],
select: L[n],
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uiff: uiff(P;Q),
squash: ↓T,
bool: 𝔹,
unit: Unit,
less_than: a < b,
ge: i ≥ j ,
assert: ↑b,
has-value: (a)↓,
nequal: a ≠ b ∈ T ,
cand: A c∧ B
Lemmas referenced :
list_wf,
list_subtype_base,
int_subtype_base,
istype-int,
nat_plus_properties,
set_subtype_base,
equal_wf,
length_wf,
primrec-wf-nat-plus,
equal-wf-base,
less_than_wf,
nat_plus_wf,
subtype_base_sq,
decidable__equal_int,
length_of_cons_lemma,
product_subtype_list,
istype-void,
istype-base,
stuck-spread,
length_of_nil_lemma,
list-cases,
le_wf,
istype-false,
poly_int_val_cons_cons,
polyconst-val,
iff_weakening_equal,
subtype_rel_self,
false_wf,
int_formula_prop_wf,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermAdd_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
add-is-int-iff,
nil_wf,
polyconst_wf,
polyform_wf,
cons_wf,
istype-universe,
true_wf,
squash_wf,
trivial-equal,
exp1,
one-mul,
exp_wf2,
add-zero,
exp0_lemma,
poly_int_val_nil_cons,
itermMultiply_wf,
int_term_value_mul_lemma,
base_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
le_int_wf,
assert_of_le_int,
non_neg_length,
intformle_wf,
intformless_wf,
int_formula_prop_le_lemma,
int_formula_prop_less_lemma,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
decidable__lt,
assert_wf,
iff_weakening_uiff,
istype-top,
istype-le,
add-subtract-cancel,
value-type-has-value,
int-value-type,
istype-less_than,
decidable__le,
polyform-value-type,
subtract_wf,
polyvar_wf,
select_wf,
poly-int-val_wf,
nequal-le-implies,
itermSubtract_wf,
int_term_value_subtract_lemma,
mul-commutes,
add-commutes,
add-comm,
select-cons-tl,
add_functionality_wrt_eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
setIsType,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
equalityIstype,
inhabitedIsType,
hypothesisEquality,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
independent_isectElimination,
sqequalBase,
equalitySymmetry,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
lambdaFormation_alt,
rename,
setElimination,
because_Cache,
isectIsType,
lambdaEquality_alt,
isectEquality,
setEquality,
natural_numberEquality,
int_eqReduceFalseSq,
sqleReflexivity,
callbyvalueReduce,
independent_functionElimination,
cumulativity,
instantiate,
unionElimination,
dependent_functionElimination,
productElimination,
hypothesis_subsumption,
promote_hyp,
equalityTransitivity,
voidElimination,
independent_pairFormation,
dependent_set_memberEquality_alt,
imageMemberEquality,
equalityIsType4,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
pointwiseFunctionality,
universeEquality,
imageElimination,
multiplyEquality,
lambdaEquality,
dependent_set_memberEquality,
lambdaFormation,
dependent_pairFormation,
isect_memberEquality,
voidEquality,
equalityElimination,
lessCases,
isect_memberFormation,
axiomSqEquality,
equalityIsType1,
Error :memTop,
addEquality,
minusEquality,
productEquality
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[v:\mBbbZ{}]. \mforall{}[l:\{l:\mBbbZ{} List| ||l|| = n\} ].
(l@polyvar(n;v) = if 0 \mleq{}z v \mwedge{}\msubb{} v <z n then l[v] else 0 fi )
Date html generated:
2020_05_19-PM-09_52_13
Last ObjectModification:
2019_12_31-PM-00_15_43
Theory : integer!polynomials
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