Nuprl Lemma : select-tuple_wf
∀[L:Type List]. ∀[n:ℕ]. ∀[k:ℤ]. ∀[x:tuple-type(L)]. (x.n ∈ L[n]) supposing n < ||L|| ∧ (k = ||L|| ∈ ℤ)
Proof
Definitions occuring in Statement :
select-tuple: x.n,
tuple-type: tuple-type(L),
select: L[n],
length: ||as||,
list: T List,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
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],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cons: [a / b],
colength: colength(L),
decidable: Dec(P),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
select-tuple: x.n,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
bfalse: ff,
pi1: fst(t),
le: A ≤ B,
bnot: ¬bb,
assert: ↑b,
int_upper: {i...},
cand: A c∧ B,
pi2: snd(t)
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
tuple-type_wf,
length_wf,
equal-wf-base-T,
less_than_transitivity1,
less_than_irreflexivity,
nat_wf,
equal-wf-T-base,
colength_wf_list,
list_wf,
list-cases,
length_of_nil_lemma,
tupletype_nil_lemma,
stuck-spread,
base_wf,
unit_wf2,
equal-wf-base,
int_subtype_base,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
decidable__equal_int,
length_of_cons_lemma,
tupletype_cons_lemma,
ifthenelse_wf,
null_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
null_nil_lemma,
null_cons_lemma,
non_neg_length,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_upper_subtype_nat,
false_wf,
nequal-le-implies,
zero-add,
select-cons-tl,
int_upper_properties,
decidable__lt,
add-is-int-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
instantiate,
universeEquality,
applyEquality,
because_Cache,
unionElimination,
baseClosed,
productElimination,
promote_hyp,
hypothesis_subsumption,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
cumulativity,
imageElimination,
equalityElimination,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}[L:Type List]. \mforall{}[n:\mBbbN{}]. \mforall{}[k:\mBbbZ{}]. \mforall{}[x:tuple-type(L)]. (x.n \mmember{} L[n]) supposing n < ||L|| \mwedge{} (k = ||L||)
Date html generated:
2017_04_17-AM-09_29_35
Last ObjectModification:
2017_02_27-PM-05_30_23
Theory : tuples
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