Nuprl Lemma : index-min_wf
∀[zs:ℤ List+]. (index-min(zs) ∈ i:ℕ||zs|| × {x:ℤ| (x = zs[i] ∈ ℤ) ∧ (∀z:ℤ. ((z ∈ zs) ⇒ (x ≤ z)))} )
Proof
Definitions occuring in Statement :
index-min: index-min(zs),
l_member: (x ∈ l),
select: L[n],
listp: A List+,
length: ||as||,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
product: x:A × B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
listp: A List+,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
ge: i ≥ j ,
le: A ≤ B,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
or: P ∨ Q,
cons: [a / b],
less_than': less_than'(a;b),
not: ¬A,
colength: colength(L),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sq_stable: SqStable(P),
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
true: True,
subtype_rel: A ⊆r B,
select: L[n],
top: Top,
index-min: index-min(zs),
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
int_seg: {i..j-},
lelt: i ≤ j < k,
exists: ∃x:A. B[x],
nat_plus: ℕ+,
bool: 𝔹,
unit: Unit,
btrue: tt,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
has-value: (a)↓,
gt: i > j
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
istype-less_than,
list-cases,
product_subtype_list,
colength-cons-not-zero,
istype-nat,
colength_wf_list,
istype-void,
istype-le,
list_wf,
subtract-1-ge-0,
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
spread_cons_lemma,
sq_stable__le,
decidable__equal_int,
subtract_wf,
istype-false,
not-equal-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
minus-minus,
le_weakening2,
listp_wf,
length_of_nil_lemma,
stuck-spread,
istype-base,
less_than_wf,
length_wf,
cons_wf,
length_of_cons_lemma,
list_ind_cons_lemma,
lelt_wf,
l_member_wf,
nil_wf,
select_wf,
length-singleton,
member_singleton,
le_weakening,
non_neg_length,
length_wf_nat,
istype-sqequal,
le_reflexive,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
not-lt-2,
minus-zero,
add-zero,
omega-shadow,
decidable__lt,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
not-le-2,
le_witness_for_triv,
sq_stable__all,
sq_stable__equal,
all_wf,
equal_wf,
sq_stable__and,
list_subtype_base,
add_nat_plus,
int_seg_properties,
decidable__le,
cons_member,
value-type-has-value,
int-value-type,
select_cons_tl_sq2,
int_seg_subtype_nat,
less-iff-le,
mul-distributes,
mul-associates,
mul-commutes,
le-add-cancel-alt,
not-gt-2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
lambdaFormation_alt,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
intWeakElimination,
independent_pairFormation,
productElimination,
imageElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
universeIsType,
lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
intEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
Error :memTop,
equalityIstype,
dependent_set_memberEquality_alt,
because_Cache,
instantiate,
cumulativity,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
addEquality,
minusEquality,
baseApply,
closedConclusion,
applyEquality,
sqequalBase,
isect_memberEquality_alt,
dependent_pairEquality_alt,
productIsType,
equalityIsType4,
functionIsType,
setIsType,
dependent_pairFormation_alt,
multiplyEquality,
equalityElimination,
lessCases,
axiomSqEquality,
isectIsTypeImplies,
functionEquality,
callbyvalueReduce
Latex:
\mforall{}[zs:\mBbbZ{} List\msupplus{}]. (index-min(zs) \mmember{} i:\mBbbN{}||zs|| \mtimes{} \{x:\mBbbZ{}| (x = zs[i]) \mwedge{} (\mforall{}z:\mBbbZ{}. ((z \mmember{} zs) {}\mRightarrow{} (x \mleq{} z)))\} )
Date html generated:
2020_05_19-PM-09_38_08
Last ObjectModification:
2020_01_04-PM-07_59_22
Theory : omega
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