Nuprl Lemma : first-m-not-reg_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[k:ℕ]. ∀[s:ℕk ⟶ X]. (first-m-not-reg(d;s;k) ∈ ℕk + 1)
Proof
Definitions occuring in Statement :
first-m-not-reg: first-m-not-reg(d;s;k),
metric: metric(X),
int_seg: {i..j-},
nat: ℕ,
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,
first-m-not-reg: first-m-not-reg(d;s;k),
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 ,
all: ∀x:A. B[x],
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
Lemmas referenced :
search_wf,
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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality_alt,
dependent_set_memberEquality_alt,
setElimination,
rename,
hypothesis,
productElimination,
imageElimination,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
applyEquality,
because_Cache,
addEquality,
lambdaFormation_alt,
minusEquality,
imageMemberEquality,
baseClosed,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsType,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
universeEquality
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[k:\mBbbN{}]. \mforall{}[s:\mBbbN{}k {}\mrightarrow{} X]. (first-m-not-reg(d;s;k) \mmember{} \mBbbN{}k + 1)
Date html generated:
2019_10_30-AM-07_01_53
Last ObjectModification:
2019_10_03-PM-05_56_02
Theory : reals
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