Nuprl Lemma : mcauchy_wf
∀[X:Type]. ∀[d:X ⟶ X ⟶ ℝ]. ∀[x:ℕ ⟶ X]. (mcauchy(d;n.x[n]) ∈ ℙ)
Proof
Definitions occuring in Statement :
mcauchy: mcauchy(d;n.x[n]),
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
mcauchy: mcauchy(d;n.x[n]),
mdist: mdist(d;x;y),
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
nat: ℕ,
so_apply: x[s],
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top
Lemmas referenced :
all_wf,
nat_plus_wf,
sq_exists_wf,
nat_wf,
le_wf,
rleq_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_properties,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
istype-nat,
real_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
lambdaEquality_alt,
because_Cache,
functionEquality,
setElimination,
rename,
applyEquality,
hypothesisEquality,
closedConclusion,
natural_numberEquality,
independent_isectElimination,
inrFormation_alt,
dependent_functionElimination,
productElimination,
independent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
inhabitedIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsType,
isectIsTypeImplies,
instantiate,
universeEquality
Latex:
\mforall{}[X:Type]. \mforall{}[d:X {}\mrightarrow{} X {}\mrightarrow{} \mBbbR{}]. \mforall{}[x:\mBbbN{} {}\mrightarrow{} X]. (mcauchy(d;n.x[n]) \mmember{} \mBbbP{})
Date html generated:
2019_10_30-AM-06_37_26
Last ObjectModification:
2019_10_02-AM-10_50_31
Theory : reals
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