Nuprl Lemma : member-listify
∀[T:Type]. ∀m:ℤ. ∀n:{n:ℤ| n ≥ m } . ∀f:{m..n-} ⟶ T. ∀[x:T]. ((x ∈ listify(f;m;n)) ⇐⇒ ∃i:{m..n-}. (x = (f i) ∈ T))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
listify: listify(f;m;n),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
ge: i ≥ j ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
int_seg: {i..j-},
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
listify: listify(f;m;n),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
decidable: Dec(P),
subtype_rel: A ⊆r B,
ge: i ≥ j ,
sq_stable: SqStable(P),
squash: ↓T,
l_member: (x ∈ l),
cand: A c∧ B,
le: A ≤ B,
true: True
Lemmas referenced :
int_seg_wf,
all_wf,
subtract_wf,
iff_wf,
l_member_wf,
listify_wf,
exists_wf,
equal_wf,
set_wf,
less_than_wf,
primrec-wf2,
nat_wf,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
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,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
le_wf,
cons_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__lt,
lelt_wf,
subtype_rel_dep_function,
int_seg_subtype,
cons_member,
itermSubtract_wf,
int_term_value_subtract_lemma,
or_wf,
decidable__equal_int,
int_subtype_base,
intformeq_wf,
int_formula_prop_eq_lemma,
ge_wf,
sq_stable__le,
select_member,
length_wf,
squash_wf,
true_wf,
list_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
universeEquality,
lambdaFormation,
thin,
hypothesisEquality,
functionEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
setElimination,
rename,
hypothesis,
cumulativity,
addEquality,
natural_numberEquality,
intEquality,
because_Cache,
sqequalRule,
lambdaEquality,
functionExtensionality,
applyEquality,
productElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
promote_hyp,
instantiate,
dependent_set_memberEquality,
addLevel,
orFunctionality,
levelHypothesis,
inlFormation,
inrFormation,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[T:Type]
\mforall{}m:\mBbbZ{}. \mforall{}n:\{n:\mBbbZ{}| n \mgeq{} m \} . \mforall{}f:\{m..n\msupminus{}\} {}\mrightarrow{} T.
\mforall{}[x:T]. ((x \mmember{} listify(f;m;n)) \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\{m..n\msupminus{}\}. (x = (f i)))
Date html generated:
2018_05_21-PM-06_53_29
Last ObjectModification:
2017_07_26-PM-04_58_58
Theory : general
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