Nuprl Lemma : mu-ge-property
∀[n:ℤ]. ∀[f:{n...} ⟶ 𝔹]. {(↑(f mu-ge(f;n))) ∧ (∀[i:{n..mu-ge(f;n)-}]. (¬↑(f i)))} supposing ∃m:{n...}. (↑(f m))
Proof
Definitions occuring in Statement :
mu-ge: mu-ge(f;n),
int_upper: {i...},
int_seg: {i..j-},
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
guard: {T},
and: P ∧ Q,
implies: P ⇒ Q,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
int_upper: {i...},
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
mu-ge: mu-ge(f;n),
true: True,
less_than': less_than'(a;b),
le: A ≤ B,
subtract: n - m,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
has-value: (a)↓,
cand: A c∧ B,
nat: ℕ
Lemmas referenced :
assert_witness,
mu-ge_wf,
istype-int_upper,
istype-assert,
bool_wf,
istype-int,
int_seg_wf,
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_subtype_upper,
le_reflexive,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
istype-less_than,
primrec-wf2,
istype-nat,
subtract_wf,
assert_wf,
uall_wf,
all_wf,
exists_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
eqtt_to_assert,
le_wf,
int_upper_wf,
lelt_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
decidable__lt,
int_formula_prop_eq_lemma,
intformeq_wf,
btrue_neq_bfalse,
not_assert_elim,
int_subtype_base,
assert_elim,
decidable__equal_int,
subtype_rel_self,
le-add-cancel,
add-commutes,
add-associates,
add-zero,
zero-mul,
add-mul-special,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
condition-implies-le,
not-le-2,
istype-false,
upper_subtype_upper,
subtype_rel_function,
int-value-type,
value-type-has-value,
set_subtype_base,
int_upper_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
extract_by_obid,
isectElimination,
applyEquality,
hypothesisEquality,
independent_isectElimination,
hypothesis,
independent_functionElimination,
Error :isect_memberEquality_alt,
Error :lambdaEquality_alt,
dependent_functionElimination,
because_Cache,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :isectIsTypeImplies,
Error :productIsType,
Error :functionIsType,
Error :universeIsType,
voidElimination,
Error :lambdaFormation_alt,
addEquality,
natural_numberEquality,
setElimination,
rename,
imageElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
Error :dependent_set_memberEquality_alt,
unionElimination,
Error :setIsType,
productEquality,
functionEquality,
cumulativity,
instantiate,
promote_hyp,
Error :equalityIsType1,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
functionExtensionality,
applyLambdaEquality,
hyp_replacement,
Error :equalityIsType4,
multiplyEquality,
minusEquality,
intEquality,
callbyvalueReduce,
dependent_set_memberEquality,
dependent_pairFormation,
lambdaEquality,
isect_memberEquality,
voidEquality
Latex:
\mforall{}[n:\mBbbZ{}]. \mforall{}[f:\{n...\} {}\mrightarrow{} \mBbbB{}].
\{(\muparrow{}(f mu-ge(f;n))) \mwedge{} (\mforall{}[i:\{n..mu-ge(f;n)\msupminus{}\}]. (\mneg{}\muparrow{}(f i)))\} supposing \mexists{}m:\{n...\}. (\muparrow{}(f m))
Date html generated:
2019_06_20-PM-01_16_43
Last ObjectModification:
2019_03_05-PM-03_39_24
Theory : int_2
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