Nuprl Lemma : first-m-not-reg-property
∀[X:Type]
∀d:metric(X). ∀k:ℕ. ∀s:ℕk ⟶ X.
((first-m-not-reg(d;s;k) = 0 ∈ ℤ ⇐⇒ ∀n:ℕk. m-not-reg(d;s;n) = ff)
∧ let i = first-m-not-reg(d;s;k) - 1 in
(∀n:ℕi. m-not-reg(d;s;n) = ff) ∧ m-not-reg(d;s;i) = tt
supposing 0 < first-m-not-reg(d;s;k))
Proof
Definitions occuring in Statement :
first-m-not-reg: first-m-not-reg(d;s;k),
m-not-reg: m-not-reg(d;s;n),
metric: metric(X),
int_seg: {i..j-},
nat: ℕ,
bfalse: ff,
btrue: tt,
bool: 𝔹,
less_than: a < b,
let: let,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
first-m-not-reg: first-m-not-reg(d;s;k),
member: t ∈ T,
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
let: let,
cand: A c∧ B
Lemmas referenced :
search_property,
m-not-reg_wf,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
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_var_lemma,
int_formula_prop_wf,
istype-le,
subtype_rel_function,
int_seg_wf,
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,
istype-nat,
metric_wf,
istype-universe,
first-m-not-reg_wf,
set_subtype_base,
lelt_wf,
int_subtype_base,
bool_wf,
int_seg_subtype_nat,
bfalse_wf,
eqtt_to_assert,
istype-assert,
intformless_wf,
intformeq_wf,
int_formula_prop_less_lemma,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
decidable__equal_int,
decidable__lt,
assert_elim,
btrue_neq_bfalse,
member-less_than,
istype-less_than,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
itermAdd_wf,
int_term_value_add_lemma,
iff_imp_equal_bool,
btrue_wf,
istype-true
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
lambdaEquality_alt,
isectElimination,
dependent_set_memberEquality_alt,
setElimination,
rename,
hypothesis,
productElimination,
imageElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
applyEquality,
because_Cache,
addEquality,
minusEquality,
imageMemberEquality,
baseClosed,
promote_hyp,
functionIsType,
instantiate,
universeEquality,
equalityIstype,
intEquality,
sqequalBase,
equalitySymmetry,
equalityTransitivity,
inhabitedIsType,
equalityElimination,
cumulativity,
applyLambdaEquality,
productIsType,
independent_pairEquality,
axiomEquality,
functionIsTypeImplies,
isectIsType
Latex:
\mforall{}[X:Type]
\mforall{}d:metric(X). \mforall{}k:\mBbbN{}. \mforall{}s:\mBbbN{}k {}\mrightarrow{} X.
((first-m-not-reg(d;s;k) = 0 \mLeftarrow{}{}\mRightarrow{} \mforall{}n:\mBbbN{}k. m-not-reg(d;s;n) = ff)
\mwedge{} let i = first-m-not-reg(d;s;k) - 1 in
(\mforall{}n:\mBbbN{}i. m-not-reg(d;s;n) = ff) \mwedge{} m-not-reg(d;s;i) = tt
supposing 0 < first-m-not-reg(d;s;k))
Date html generated:
2019_10_30-AM-07_02_16
Last ObjectModification:
2019_10_03-PM-06_01_27
Theory : reals
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