Nuprl Lemma : mu-ge-bound-property
∀n,m:ℤ. ∀f:{n..m-} ⟶ 𝔹. ((∃m:{n..m-}. (↑(f m))) ⇒ {(↑(f mu-ge(f;n))) ∧ (∀[i:{n..mu-ge(f;n)-}]. (¬↑(f i)))})
Proof
Definitions occuring in Statement :
mu-ge: mu-ge(f;n),
int_seg: {i..j-},
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
nat: ℕ,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
and: P ∧ Q,
guard: {T},
subtype_rel: A ⊆r B,
int_seg: {i..j-},
cand: A c∧ B,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
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),
bnot: ¬bb,
assert: ↑b,
label: ...$L... t,
rev_uimplies: rev_uimplies(P;Q),
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
less_than': less_than'(a;b),
true: True,
has-value: (a)↓
Lemmas referenced :
assert_wf,
int_seg_wf,
bool_wf,
istype-int,
mu-ge-bound,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
istype-void,
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,
assert_witness,
subtract-1-ge-0,
nat_wf,
int_seg_properties,
itermSubtract_wf,
int_term_value_subtract_lemma,
lelt_wf,
le_wf,
subtract_wf,
decidable__lt,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__le,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
assert_functionality_wrt_uiff,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
int_subtype_base,
itermAdd_wf,
int_term_value_add_lemma,
subtype_rel_function,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-associates,
add-commutes,
le-add-cancel,
le_reflexive,
subtype_rel_self,
value-type-has-value,
int-value-type,
iff_weakening_uiff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
sqequalRule,
Error :productIsType,
Error :universeIsType,
because_Cache,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
hypothesisEquality,
hypothesis,
Error :functionIsType,
Error :inhabitedIsType,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
productElimination,
independent_pairEquality,
Error :functionIsTypeImplies,
equalityTransitivity,
equalitySymmetry,
Error :isect_memberFormation_alt,
applyLambdaEquality,
functionExtensionality,
Error :dependent_set_memberEquality_alt,
unionElimination,
equalityElimination,
Error :equalityIsType1,
promote_hyp,
instantiate,
cumulativity,
Error :equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
addEquality,
minusEquality,
multiplyEquality,
intEquality,
callbyvalueReduce
Latex:
\mforall{}n,m:\mBbbZ{}. \mforall{}f:\{n..m\msupminus{}\} {}\mrightarrow{} \mBbbB{}.
((\mexists{}m:\{n..m\msupminus{}\}. (\muparrow{}(f m))) {}\mRightarrow{} \{(\muparrow{}(f mu-ge(f;n))) \mwedge{} (\mforall{}[i:\{n..mu-ge(f;n)\msupminus{}\}]. (\mneg{}\muparrow{}(f i)))\})
Date html generated:
2019_06_20-PM-01_16_54
Last ObjectModification:
2018_10_06-AM-11_21_37
Theory : int_2
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