Nuprl Lemma : mu-ge-bound
∀[n,m:ℤ]. ∀[f:{n..m-} ⟶ 𝔹]. mu-ge(f;n) ∈ {n..m-} supposing ∃k:{n..m-}. (↑(f k))
Proof
Definitions occuring in Statement :
mu-ge: mu-ge(f;n),
int_seg: {i..j-},
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
mu-ge: mu-ge(f;n),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
has-value: (a)↓,
subtype_rel: A ⊆r B,
true: True
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
exists_wf,
int_seg_wf,
assert_wf,
bool_wf,
le_wf,
subtract_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
int_seg_properties,
decidable__lt,
lelt_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
value-type-has-value,
int-value-type,
itermAdd_wf,
int_term_value_add_lemma,
subtype_rel_dep_function,
int_seg_subtype,
subtype_rel_self,
int_subtype_base,
assert_elim,
equal-wf-T-base,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
functionExtensionality,
functionEquality,
isect_memberFormation,
productElimination,
because_Cache,
unionElimination,
dependent_set_memberEquality,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
callbyvalueReduce,
addEquality,
applyLambdaEquality
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[f:\{n..m\msupminus{}\} {}\mrightarrow{} \mBbbB{}]. mu-ge(f;n) \mmember{} \{n..m\msupminus{}\} supposing \mexists{}k:\{n..m\msupminus{}\}. (\muparrow{}(f k))
Date html generated:
2017_04_14-AM-09_18_35
Last ObjectModification:
2017_02_27-PM-03_55_05
Theory : int_2
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