Nuprl Lemma : m-regular-not-m-not-reg
∀[X:Type]. ∀[d:metric(X)]. ∀[s:ℕ ⟶ X]. (m-k-regular(d;2;s) ⇒ (∀n:ℕ. m-not-reg(d;s;n) = ff))
Proof
Definitions occuring in Statement :
m-not-reg: m-not-reg(d;s;n),
m-k-regular: m-k-regular(d;k;s),
metric: metric(X),
nat: ℕ,
bfalse: ff,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
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,
implies: P ⇒ Q,
all: ∀x:A. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
false: False,
prop: ℙ,
m-not-reg: m-not-reg(d;s;n),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
or: P ∨ Q,
isl: isl(x),
exists: ∃x:A. B[x],
m-k-regular: m-k-regular(d;k;s),
squash: ↓T,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
iff: P ⇐⇒ Q,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
rev_implies: P ⇐ Q,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
less_than: a < b,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
true: True
Lemmas referenced :
istype-nat,
m-k-regular_wf,
istype-void,
istype-le,
metric_wf,
istype-universe,
m-reg-test_wf,
subtype_rel_function,
nat_wf,
int_seg_wf,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
bfalse_wf,
equal_wf,
squash_wf,
true_wf,
bool_wf,
btrue_wf,
ppcc-problem,
iff_imp_equal_bool,
rless_transitivity1,
radd_wf,
rdiv_wf,
rneq-int,
nat_plus_properties,
int_seg_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
set_subtype_base,
lelt_wf,
int_subtype_base,
intformless_wf,
int_formula_prop_less_lemma,
le_wf,
mdist_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
rless_irreflexivity,
int-to-real_wf,
istype-true,
unit_wf2,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
hypothesis,
extract_by_obid,
universeIsType,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
sqequalRule,
voidElimination,
lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
functionIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
instantiate,
universeEquality,
applyEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
unionElimination,
productElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
imageElimination,
inlEquality_alt,
addEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
baseApply,
closedConclusion,
baseClosed,
intEquality,
sqequalBase,
imageMemberEquality
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[s:\mBbbN{} {}\mrightarrow{} X]. (m-k-regular(d;2;s) {}\mRightarrow{} (\mforall{}n:\mBbbN{}. m-not-reg(d;s;n) = ff))
Date html generated:
2019_10_30-AM-07_00_51
Last ObjectModification:
2019_10_09-AM-09_01_05
Theory : reals
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