Nuprl Lemma : listify_wf
∀[T:Type]. ∀[m,n:ℤ]. ∀[f:{m..n-} ⟶ T]. (listify(f;m;n) ∈ T List)
Proof
Definitions occuring in Statement :
listify: listify(f;m;n),
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
gt: i > j,
decidable: Dec(P),
or: P ∨ Q,
true: True,
less_than': less_than'(a;b),
le: A ≤ B,
top: Top,
subtype_rel: A ⊆r B,
subtract: n - m,
lelt: i ≤ j < k,
int_seg: {i..j-},
not: ¬A,
false: False,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
prop: ℙ,
exists: ∃x:A. B[x],
bfalse: ff,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
listify: listify(f;m;n),
int_lower: {...i},
nat: ℕ,
ge: i ≥ j ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T,
nat_plus: ℕ+,
less_than: a < b
Lemmas referenced :
int_seg_wf,
istype-int,
decidable__lt,
lelt_wf,
le-add-cancel,
add-associates,
add_functionality_wrt_le,
less-iff-le,
add-commutes,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
condition-implies-le,
not-le-2,
le_reflexive,
cons_wf,
le_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
nil_wf,
assert_of_le_int,
eqtt_to_assert,
bool_wf,
le_int_wf,
int_lower_wf,
not_wf,
gt_wf,
nat_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
subtract_wf,
decidable__le,
istype-false,
not-ge-2,
zero-add,
istype-void,
minus-minus,
add-zero,
decidable__int_equal,
set_subtype_base,
int_subtype_base,
le_antisymmetry_iff,
not-lt-2,
le-add-cancel-alt,
subtype_rel_self,
int_seg_properties,
not-equal-2,
sq_stable__le,
add-mul-special,
zero-mul,
equal-wf-T-base,
assert_wf,
lt_int_wf,
bnot_wf,
uiff_transitivity,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int,
subtract_nat_wf,
le-add-cancel2,
subtype_rel_function,
int_seg_subtype,
one-mul,
two-mul,
mul-distributes-right,
omega-shadow,
mul-distributes,
mul-swap,
mul-associates,
mul-commutes,
int_lower_properties,
add_nat_wf,
not-gt-2
Rules used in proof :
Error :functionIsType,
Error :universeIsType,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
Error :inhabitedIsType,
universeEquality,
Error :isect_memberFormation_alt,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :isect_memberEquality_alt,
dependent_functionElimination,
unionElimination,
minusEquality,
intEquality,
voidEquality,
isect_memberEquality,
lambdaEquality,
natural_numberEquality,
addEquality,
independent_pairFormation,
dependent_set_memberEquality,
functionExtensionality,
applyEquality,
voidElimination,
independent_functionElimination,
instantiate,
promote_hyp,
dependent_pairFormation,
cumulativity,
independent_isectElimination,
productElimination,
equalityElimination,
lambdaFormation,
Error :lambdaFormation_alt,
setElimination,
rename,
intWeakElimination,
Error :lambdaEquality_alt,
Error :dependent_set_memberEquality_alt,
hypothesis_subsumption,
imageMemberEquality,
baseClosed,
imageElimination,
multiplyEquality,
Error :equalityIsType1
Latex:
\mforall{}[T:Type]. \mforall{}[m,n:\mBbbZ{}]. \mforall{}[f:\{m..n\msupminus{}\} {}\mrightarrow{} T]. (listify(f;m;n) \mmember{} T List)
Date html generated:
2019_06_20-PM-00_38_26
Last ObjectModification:
2018_10_03-PM-03_01_50
Theory : list_0
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