Nuprl Lemma : m-regularize_wf_finite
∀[X:Type]. ∀[d:metric(X)]. ∀[b:ℕ]. ∀[s:ℕb ⟶ X]. (m-regularize(d;s) ∈ ℕb ⟶ X)
Proof
Definitions occuring in Statement :
m-regularize: m-regularize(d;s),
metric: metric(X),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
m-regularize: m-regularize(d;s),
has-value: (a)↓,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
sq_stable: SqStable(P),
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
let: let,
sq_type: SQType(T),
guard: {T},
m-not-reg: m-not-reg(d;s;n),
isl: isl(x),
m-reg-test: m-reg-test(d;b;s;x),
int-seg-case: int-seg-case(i;j;d),
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f)
Lemmas referenced :
value-type-has-value,
int_seg_wf,
set-value-type,
lelt_wf,
istype-int,
int-value-type,
first-m-not-reg_wf,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
istype-void,
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,
istype-le,
subtype_rel_function,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
zero-add,
sq_stable__le,
less-iff-le,
add_functionality_wrt_le,
le-add-cancel2,
subtype_rel_self,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-nat,
metric_wf,
istype-universe,
first-m-not-reg-property,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
subtract_wf,
intformless_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
bool_wf,
m-not-reg_wf,
istype-less_than,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__lt,
int_seg_subtype_nat,
bfalse_wf,
it_wf,
unit_wf2,
btrue_neq_bfalse
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
lambdaEquality_alt,
callbyvalueReduce,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
productElimination,
hypothesis,
addEquality,
hypothesisEquality,
natural_numberEquality,
independent_isectElimination,
intEquality,
because_Cache,
dependent_set_memberEquality_alt,
imageElimination,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
applyEquality,
lambdaFormation_alt,
minusEquality,
imageMemberEquality,
baseClosed,
closedConclusion,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
equalityElimination,
equalityIstype,
axiomEquality,
functionIsType,
isectIsTypeImplies,
instantiate,
universeEquality,
cumulativity,
productIsType,
applyLambdaEquality,
sqequalBase,
inrEquality_alt
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[b:\mBbbN{}]. \mforall{}[s:\mBbbN{}b {}\mrightarrow{} X]. (m-regularize(d;s) \mmember{} \mBbbN{}b {}\mrightarrow{} X)
Date html generated:
2019_10_30-AM-07_02_59
Last ObjectModification:
2019_10_03-PM-06_03_54
Theory : reals
Home
Index