Nuprl Lemma : LegendreSigns_wf
∀[n:ℕ]. ∀[L:(x:ℤ × ℕ+ × {s:{-1..2-}| s = ratsign(ratLegendre(n;x)) ∈ ℤ} ) List].
(LegendreSigns(n;L) ∈ (x:ℤ × ℕ+ × {s:{-1..2-}| s = ratsign(ratLegendre(n;x)) ∈ ℤ} ) List)
Proof
Definitions occuring in Statement :
LegendreSigns: LegendreSigns(n;L),
ratLegendre: ratLegendre(n;x),
ratsign: ratsign(x),
list: T List,
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
product: x:A × B[x],
minus: -n,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
LegendreSigns: LegendreSigns(n;L),
prop: ℙ,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
so_apply: x[s],
all: ∀x:A. B[x],
nat_plus: ℕ+,
nat: ℕ,
so_lambda: λ2x y.t[x; y],
has-value: (a)↓,
so_apply: x[s1;s2]
Lemmas referenced :
interpolate-list_wf,
nat_plus_wf,
int_seg_wf,
equal-wf-base,
product-value-type,
set_subtype_base,
lelt_wf,
istype-int,
int_subtype_base,
product_subtype_base,
less_than_wf,
le_wf,
value-type-has-value,
rat-midpoint_wf,
set-value-type,
int-value-type,
ratsign_wf,
ratLegendre_wf,
list_wf,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
closedConclusion,
intEquality,
hypothesis,
setEquality,
minusEquality,
natural_numberEquality,
because_Cache,
independent_isectElimination,
lambdaEquality_alt,
applyEquality,
setElimination,
rename,
productElimination,
hypothesisEquality,
imageElimination,
baseApply,
baseClosed,
lambdaFormation_alt,
inhabitedIsType,
productIsType,
universeIsType,
callbyvalueReduce,
equalityTransitivity,
equalitySymmetry,
dependent_pairEquality_alt,
dependent_set_memberEquality_alt,
equalityIstype,
sqequalBase,
setIsType,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[L:(x:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{} \mtimes{} \{s:\{-1..2\msupminus{}\}| s = ratsign(ratLegendre(n;x))\} ) List].
(LegendreSigns(n;L) \mmember{} (x:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{} \mtimes{} \{s:\{-1..2\msupminus{}\}| s = ratsign(ratLegendre(n;x))\} ) List)
Date html generated:
2019_10_31-AM-06_21_52
Last ObjectModification:
2019_02_17-PM-11_51_35
Theory : reals_2
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