Nuprl Lemma : select_zip
∀[T1,T2:Type]. ∀[as:T1 List]. ∀[bs:T2 List]. ∀[i:ℕ].
zip(as;bs)[i] = <as[i], bs[i]> ∈ (T1 × T2) supposing i < ||zip(as;bs)||
Proof
Definitions occuring in Statement :
zip: zip(as;bs),
select: L[n],
length: ||as||,
list: T List,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pair: <a, b>,
product: x:A × B[x],
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,
zip: zip(as;bs),
list_ind: list_ind,
nil: [],
it: ⋅,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
select: L[n],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cons: [a / b],
colength: colength(L),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
decidable: Dec(P),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
le: A ≤ B,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
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,
length_wf,
zip_wf,
list_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
nil_wf,
list_ind_nil_lemma,
stuck-spread,
base_wf,
length_of_nil_lemma,
product_subtype_list,
spread_cons_lemma,
equal_wf,
subtype_base_sq,
set_subtype_base,
le_wf,
int_subtype_base,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__equal_int,
cons_wf,
list_ind_cons_lemma,
length_of_cons_lemma,
decidable__lt,
zip_length,
squash_wf,
true_wf,
select_cons_tl,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
productEquality,
cumulativity,
equalityTransitivity,
equalitySymmetry,
because_Cache,
applyEquality,
unionElimination,
baseClosed,
promote_hyp,
hypothesis_subsumption,
productElimination,
instantiate,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
imageElimination,
universeEquality,
independent_pairEquality,
imageMemberEquality
Latex:
\mforall{}[T1,T2:Type]. \mforall{}[as:T1 List]. \mforall{}[bs:T2 List]. \mforall{}[i:\mBbbN{}].
zip(as;bs)[i] = <as[i], bs[i]> supposing i < ||zip(as;bs)||
Date html generated:
2017_04_17-AM-08_54_53
Last ObjectModification:
2017_02_27-PM-05_12_02
Theory : list_1
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