Nuprl Lemma : Legendre_functionality
∀[n:ℕ]. ∀[x,y:ℝ]. Legendre(n;x) = Legendre(n;y) supposing x = y
Proof
Definitions occuring in Statement :
Legendre: Legendre(n;x),
req: x = y,
real: ℝ,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
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,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
Legendre: Legendre(n;x),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
le: A ≤ B,
less_than': less_than'(a;b),
int_upper: {i...},
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
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,
req_witness,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
Legendre_wf,
req_wf,
real_wf,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
req_weakening,
int-to-real_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
upper_subtype_nat,
istype-false,
nequal-le-implies,
zero-add,
upper_subtype_upper,
int_upper_properties,
subtype_rel_sets_simple,
le_wf,
nequal_wf,
rsub_wf,
int-rmul_wf,
rmul_wf,
int-rdiv_functionality,
rsub_functionality,
req_functionality,
rmul_functionality,
int-rmul_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
productElimination,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
because_Cache,
unionElimination,
applyEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productIsType,
hypothesis_subsumption,
addEquality,
equalityElimination,
equalityIstype,
promote_hyp,
cumulativity,
intEquality,
baseClosed,
sqequalBase,
multiplyEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbR{}]. Legendre(n;x) = Legendre(n;y) supposing x = y
Date html generated:
2019_10_30-AM-11_32_47
Last ObjectModification:
2019_01_01-PM-03_25_43
Theory : reals_2
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