Nuprl Lemma : m-not-reg_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[n:ℕ]. ∀[s:ℕn + 1 ⟶ X]. (m-not-reg(d;s;n) ∈ 𝔹)
Proof
Definitions occuring in Statement :
m-not-reg: m-not-reg(d;s;n),
metric: metric(X),
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
m-not-reg: m-not-reg(d;s;n),
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
uimplies: b supposing a,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
true: True,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ,
isl: isl(x)
Lemmas referenced :
m-reg-test_wf,
subtype_rel_function,
int_seg_wf,
int_seg_subtype,
istype-false,
decidable__le,
not-le-2,
condition-implies-le,
minus-add,
istype-void,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-associates,
add-commutes,
le-add-cancel,
subtype_rel_self,
nat_properties,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_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_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
istype-le,
istype-less_than,
btrue_wf,
bfalse_wf,
istype-nat,
metric_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
applyEquality,
because_Cache,
natural_numberEquality,
setElimination,
rename,
hypothesis,
independent_isectElimination,
addEquality,
independent_pairFormation,
lambdaFormation_alt,
unionElimination,
voidElimination,
productElimination,
independent_functionElimination,
isect_memberEquality_alt,
minusEquality,
multiplyEquality,
dependent_set_memberEquality_alt,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
universeIsType,
productIsType,
inhabitedIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
functionIsType,
isectIsTypeImplies,
instantiate,
universeEquality
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[n:\mBbbN{}]. \mforall{}[s:\mBbbN{}n + 1 {}\mrightarrow{} X]. (m-not-reg(d;s;n) \mmember{} \mBbbB{})
Date html generated:
2019_10_30-AM-07_00_30
Last ObjectModification:
2019_10_03-PM-05_55_12
Theory : reals
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