Nuprl Lemma : int_seg_decide_wf
∀[i,j:ℤ]. ∀[F:{i..j-} ⟶ ℙ{u}]. ∀[d:∀k:{i..j-}. Dec(F[k])]. (int_seg_decide(d;i;j) ∈ Dec(∃k:{i..j-}. F[k]))
Proof
Definitions occuring in Statement :
int_seg_decide: int_seg_decide(d;i;j),
int_seg: {i..j-},
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃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,
all: ∀x:A. B[x],
so_apply: x[s],
prop: ℙ,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
and: P ∧ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
exists: ∃x:A. B[x],
le: A ≤ B,
subtract: n - m,
subtype_rel: A ⊆r B,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
sq_stable: SqStable(P),
cand: A c∧ B,
int_upper: {i...},
int_seg_decide: int_seg_decide(d;i;j),
nat_plus: ℕ+
Lemmas referenced :
int_seg_wf,
decidable_wf,
istype-int,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
istype-less_than,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
istype-void,
istype-le,
less-iff-le,
condition-implies-le,
minus-one-mul,
add-associates,
zero-add,
minus-one-mul-top,
add_functionality_wrt_le,
add-swap,
add-commutes,
le-add-cancel,
eqff_to_assert,
int_subtype_base,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
less_than_wf,
iff_weakening_uiff,
assert_of_bnot,
istype-assert,
member-not,
exists_wf,
sq_stable__le,
subtract-1-ge-0,
le_transitivity,
le_reflexive,
decidable__le,
istype-false,
not-le-2,
minus-add,
subtract_wf,
decidable-exists-int_seg-subtype,
le-add-cancel2,
not-lt-2,
istype-nat,
mul-associates,
mul-distributes,
omega-shadow,
mul-distributes-right,
two-mul,
one-mul,
minus-zero,
minus-minus,
add-zero,
zero-mul,
add-mul-special,
false_wf,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :functionIsType,
Error :universeIsType,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
instantiate,
applyEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
universeEquality,
Error :lambdaFormation_alt,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
Error :lambdaEquality_alt,
dependent_functionElimination,
Error :functionIsTypeImplies,
productElimination,
unionElimination,
equalityElimination,
because_Cache,
lessCases,
axiomSqEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination,
functionExtensionality,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
minusEquality,
addEquality,
Error :inlEquality_alt,
Error :dependent_pairEquality_alt,
Error :equalityIsType1,
Error :dependent_pairFormation_alt,
Error :equalityIsType4,
baseApply,
closedConclusion,
promote_hyp,
cumulativity,
Error :inrEquality_alt,
multiplyEquality,
intEquality,
lambdaEquality,
lambdaFormation,
voidEquality,
isect_memberEquality,
dependent_set_memberEquality
Latex:
\mforall{}[i,j:\mBbbZ{}]. \mforall{}[F:\{i..j\msupminus{}\} {}\mrightarrow{} \mBbbP{}\{u\}]. \mforall{}[d:\mforall{}k:\{i..j\msupminus{}\}. Dec(F[k])].
(int\_seg\_decide(d;i;j) \mmember{} Dec(\mexists{}k:\{i..j\msupminus{}\}. F[k]))
Date html generated:
2019_06_20-AM-11_28_08
Last ObjectModification:
2018_10_27-PM-05_54_54
Theory : call!by!value_2
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