Nuprl Lemma : Legendre_changes_wf
∀[n:ℕ+]
(Legendre_changes(n) ∈ {L:(ℤ × ℕ+) List| (||L|| = (n - 1) ∈ ℤ) ∧ rat-sign-change-list(λx.ratLegendre(n;x);<-1, 1>;<1, \000C1>;L)} )
Proof
Definitions occuring in Statement :
Legendre_changes: Legendre_changes(n),
ratLegendre: ratLegendre(n;x),
rat-sign-change-list: rat-sign-change-list,
length: ||as||,
list: T List,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
lambda: λx.A[x],
pair: <a, b>,
product: x:A × B[x],
subtract: n - m,
minus: -n,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
Legendre_changes: Legendre_changes(n),
primrec: primrec(n;b;c),
Legendre-sign-changes-ext,
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
nat: ℕ,
prop: ℙ,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
false: False,
sq_exists: ∃x:A [B[x]],
less_than: a < b,
squash: ↓T,
cand: A c∧ B
Lemmas referenced :
Legendre-sign-changes-ext,
subtype_rel_self,
nat_wf,
less_than_wf,
sq_exists_wf,
list_wf,
nat_plus_wf,
equal-wf-base,
list_subtype_base,
product_subtype_base,
istype-int,
int_subtype_base,
set_subtype_base,
le_wf,
rat-sign-change-list_wf,
ratLegendre_wf,
nat_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
nat_plus_properties,
decidable__le,
intformand_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
istype-le,
uimplies_subtype
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalRule,
cut,
isect_memberEquality_alt,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
introduction,
sqequalHypSubstitution,
isectElimination,
functionEquality,
isectEquality,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
productEquality,
intEquality,
lambdaEquality_alt,
baseApply,
closedConclusion,
baseClosed,
independent_isectElimination,
lambdaFormation_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
productIsType,
universeIsType,
independent_pairEquality,
minusEquality,
dependent_set_memberEquality_alt,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
voidElimination,
int_eqEquality,
independent_pairFormation,
because_Cache,
setIsType,
equalityIstype,
sqequalBase,
setEquality,
imageElimination,
productElimination
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]
(Legendre\_changes(n) \mmember{} \{L:(\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}) List|
(||L|| = (n - 1)) \mwedge{} rat-sign-change-list(\mlambda{}x.ratLegendre(n;x);<-1, 1>ə, 1\000C>L)\} )
Date html generated:
2019_10_31-AM-06_21_20
Last ObjectModification:
2019_01_19-PM-04_44_25
Theory : reals_2
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