Nuprl Lemma : m-regularize-of-regular
∀[X:Type]. ∀[d:metric(X)]. ∀[s:ℕ ⟶ X]. m-regularize(d;s) = s ∈ (ℕ ⟶ X) supposing m-k-regular(d;2;s)
Proof
Definitions occuring in Statement :
m-regularize: m-regularize(d;s),
m-k-regular: m-k-regular(d;k;s),
metric: metric(X),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
false: False,
prop: ℙ,
m-regularize: m-regularize(d;s),
all: ∀x:A. B[x],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
squash: ↓T,
guard: {T},
rev_implies: P ⇐ Q,
true: True,
sq_type: SQType(T),
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
bfalse: ff
Lemmas referenced :
m-regular-not-m-not-reg,
nat_wf,
m-k-regular_wf,
istype-void,
istype-le,
istype-nat,
metric_wf,
istype-universe,
first-m-not-reg-property,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
subtype_rel_function,
int_seg_wf,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
subtype_base_sq,
int_subtype_base,
equal_wf,
squash_wf,
true_wf,
bool_wf,
bfalse_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
functionExtensionality,
universeIsType,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
sqequalRule,
lambdaFormation_alt,
voidElimination,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsType,
instantiate,
universeEquality,
dependent_functionElimination,
addEquality,
setElimination,
rename,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
applyEquality,
because_Cache,
cumulativity,
intEquality,
productElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[s:\mBbbN{} {}\mrightarrow{} X]. m-regularize(d;s) = s supposing m-k-regular(d;2;s)
Date html generated:
2019_10_30-AM-07_03_41
Last ObjectModification:
2019_10_09-AM-09_07_16
Theory : reals
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