Nuprl Lemma : seq-min-upper_wf
∀[k,n:ℕ]. ∀[f:ℕ+ ⟶ ℤ]. (seq-min-upper(k;n;f) ∈ ℕ+)
Proof
Definitions occuring in Statement :
seq-min-upper: seq-min-upper(k;n;f),
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
seq-min-upper: seq-min-upper(k;n;f),
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
prop: ℙ,
int_seg: {i..j-},
nat: ℕ,
le: A ≤ B,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
lelt: i ≤ j < k,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
primrec_wf,
nat_plus_wf,
less_than_wf,
le_int_wf,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
bool_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
assert_of_le_int,
le_wf,
int_seg_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
hypothesisEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
lambdaEquality,
addEquality,
multiplyEquality,
setElimination,
rename,
because_Cache,
applyEquality,
functionExtensionality,
productElimination,
dependent_functionElimination,
unionElimination,
lambdaFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
axiomEquality,
functionEquality
Latex:
\mforall{}[k,n:\mBbbN{}]. \mforall{}[f:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}]. (seq-min-upper(k;n;f) \mmember{} \mBbbN{}\msupplus{})
Date html generated:
2017_10_03-AM-08_43_22
Last ObjectModification:
2017_09_09-AM-08_15_51
Theory : reals
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