Nuprl Lemma : Legendre-sign-changes-ext
∀n:ℕ. ∃L:(ℤ × ℕ+) List [((||L|| = (n - 1) ∈ ℤ) ∧ rat-sign-change-list(λx.ratLegendre(n;x);<-1, 1>;<1, 1>;L))] supposing \000C0 < n
Proof
Definitions occuring in Statement :
ratLegendre: ratLegendre(n;x),
rat-sign-change-list: rat-sign-change-list,
length: ||as||,
list: T List,
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
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 :
member: t ∈ T,
subtract: n - m,
eq_int: (i =z j),
btrue: tt,
it: ⋅,
bfalse: ff,
ifthenelse: if b then t else f fi ,
Legendre_changes: Legendre_changes(n),
let: let,
uall: ∀[x:A]. B[x],
top: Top,
uimplies: b supposing a,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ,
false: False,
sq_type: SQType(T),
guard: {T},
Legendre-sign-changes,
any: any x,
decidable__equal_int,
sq_stable__and,
sq_stable__equal,
decidable__int_equal,
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x.t[x],
so_apply: x[s],
strict4: strict4(F),
and: P ∧ Q,
has-value: (a)↓,
squash: ↓T
Lemmas referenced :
Legendre-sign-changes,
add-associates,
istype-void,
add-inverse,
zero-add-sq,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
itermAdd_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
lifting-strict-int_eq,
strict4-decide,
lifting-strict-decide,
has-value_wf_base,
istype-base,
is-exception_wf,
istype-universe,
sq_stable__and,
sq_stable__equal,
decidable__int_equal
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
isect_memberEquality_alt,
voidElimination,
natural_numberEquality,
because_Cache,
cumulativity,
intEquality,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
minusEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
universeIsType,
hypothesisEquality,
baseClosed,
independent_pairFormation,
lambdaFormation_alt,
callbyvalueCallbyvalue,
callbyvalueReduce,
baseApply,
closedConclusion,
callbyvalueExceptionCases,
inrFormation_alt,
imageMemberEquality,
imageElimination,
exceptionSqequal,
inlFormation_alt
Latex:
\mforall{}n:\mBbbN{}. \mexists{}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))] supposing 0 < n
Date html generated:
2019_10_31-AM-06_21_12
Last ObjectModification:
2019_01_21-PM-03_59_13
Theory : reals_2
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