Nuprl Lemma : ratLegendre_wf
∀[x:ℤ × ℕ+]. ∀[n:ℕ]. (ratLegendre(n;x) ∈ {y:ℤ × ℕ+| ratreal(y) = Legendre(n;ratreal(x))} )
Proof
Definitions occuring in Statement :
ratLegendre: ratLegendre(n;x),
Legendre: Legendre(n;x),
ratreal: ratreal(r),
req: x = y,
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
product: x:A × B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
false: False,
ratLegendre: ratLegendre(n;x),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
le: A ≤ B,
less_than': less_than'(a;b),
int_upper: {i...},
int_seg: {i..j-},
lelt: i ≤ j < k,
subtract: n - m,
subtype_rel: A ⊆r B,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
true: True,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
nat_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
req_wf,
ratreal_wf,
int-to-real_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
upper_subtype_nat,
istype-false,
nequal-le-implies,
zero-add,
istype-le,
ratLegendre-aux_wf,
int_upper_properties,
intformand_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
decidable__le,
itermAdd_wf,
int_term_value_add_lemma,
Legendre_1_lemma,
req_weakening,
Legendre_0_lemma,
Legendre_wf,
int_seg_subtype_nat,
subtract_wf,
int_seg_properties,
itermSubtract_wf,
int_term_value_subtract_lemma,
istype-nat,
nat_plus_wf,
rdiv_wf,
rless-int,
rless_wf,
req-int-fractions2,
int_subtype_base,
nequal_wf,
decidable__equal_int,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
req_functionality,
ratreal-req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
independent_pairEquality,
natural_numberEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
universeIsType,
inhabitedIsType,
lambdaFormation_alt,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
because_Cache,
hypothesis_subsumption,
independent_pairFormation,
int_eqEquality,
dependent_pairEquality_alt,
addEquality,
productIsType,
setIsType,
applyEquality,
axiomEquality,
isectIsTypeImplies,
closedConclusion,
inrFormation_alt,
imageMemberEquality,
baseClosed,
intEquality,
sqequalBase
Latex:
\mforall{}[x:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}]. \mforall{}[n:\mBbbN{}]. (ratLegendre(n;x) \mmember{} \{y:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}| ratreal(y) = Legendre(n;ratreal(x))\} )
Date html generated:
2019_10_30-AM-11_34_13
Last ObjectModification:
2019_01_10-PM-04_04_28
Theory : reals_2
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