Nuprl Lemma : ratLegendre-aux_wf
∀[x:ℤ × ℕ+]. ∀[n:ℕ+]. ∀[tr:k:ℕ+n + 1
× {a:ℤ × ℕ+| ratreal(a) = Legendre(k;ratreal(x))}
× {b:ℤ × ℕ+| ratreal(b) = Legendre(k - 1;ratreal(x))} ].
(ratLegendre-aux(n;x;tr) ∈ {y:ℤ × ℕ+| ratreal(y) = Legendre(n;ratreal(x))} )
Proof
Definitions occuring in Statement :
ratLegendre-aux: ratLegendre-aux(n;x;tr),
Legendre: Legendre(n;x),
ratreal: ratreal(r),
req: x = y,
int_seg: {i..j-},
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
product: x:A × B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
ratLegendre-aux: ratLegendre-aux(n;x;tr),
spreadn: spread3,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
has-value: (a)↓,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
Legendre: Legendre(n;x),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
subtract: n - m,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rsub: x - y,
int_seg: {i..j-},
lelt: i ≤ j < k,
pi1: fst(t),
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
nat_plus_wf,
istype-int,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
nat_plus_properties,
decidable__equal_int,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
subtype_base_sq,
int_subtype_base,
req_wf,
ratreal_wf,
Legendre_wf,
decidable__le,
istype-le,
set_subtype_base,
less_than_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtract-1-ge-0,
value-type-has-value,
int-value-type,
int-rdiv_wf,
nequal_wf,
ratadd_wf,
int-rat-mul_wf,
ratmul_wf,
set-value-type,
product-value-type,
rat-nat-div_wf,
decidable__lt,
bool_wf,
bool_subtype_base,
equal_wf,
squash_wf,
true_wf,
istype-universe,
eq_int_eq_false,
bfalse_wf,
subtype_rel_self,
iff_weakening_equal,
add-subtract-cancel,
itermMultiply_wf,
int_term_value_mul_lemma,
add-associates,
add-swap,
add-commutes,
zero-add,
istype-nat,
radd_wf,
int-rmul_wf,
rmul_wf,
rsub_wf,
nat_plus_subtype_nat,
req_functionality,
req_transitivity,
ratreal-rat-nat-div,
int-rdiv_functionality,
ratreal-ratadd,
radd_functionality,
ratreal-int-rat-mul,
int-rmul_functionality,
ratreal-ratmul,
req_weakening,
rminus_wf,
rminus_functionality,
req_inversion,
rmul_functionality,
int-to-real_wf,
rmul_over_rminus,
int-rmul-req,
real_wf,
rminus-int,
pi1_wf_top,
subtract_nat_wf,
int_seg_wf,
int_seg_properties,
subtract-is-int-iff,
false_wf,
lelt_wf,
subtype_rel_set,
nat_wf,
equal-wf-base,
int_seg_subtype_nat,
istype-false
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
universeIsType,
introduction,
extract_by_obid,
hypothesis,
productIsType,
lambdaFormation_alt,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
productElimination,
because_Cache,
unionElimination,
int_eqReduceTrueSq,
instantiate,
cumulativity,
intEquality,
dependent_set_memberEquality_alt,
setIsType,
equalityIstype,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
sqequalBase,
int_eqReduceFalseSq,
callbyvalueReduce,
addEquality,
setEquality,
productEquality,
multiplyEquality,
minusEquality,
dependent_pairEquality_alt,
independent_pairEquality,
imageElimination,
universeEquality,
imageMemberEquality,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp
Latex:
\mforall{}[x:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[tr:k:\mBbbN{}\msupplus{}n + 1
\mtimes{} \{a:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}| ratreal(a) = Legendre(k;ratreal(x))\}
\mtimes{} \{b:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}| ratreal(b) = Legendre(k - 1;ratreal(x))\} ].
(ratLegendre-aux(n;x;tr) \mmember{} \{y:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}| ratreal(y) = Legendre(n;ratreal(x))\} )
Date html generated:
2019_10_30-AM-11_34_07
Last ObjectModification:
2019_01_11-AM-10_30_29
Theory : reals_2
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