Nuprl Lemma : member-zip
∀[A,B:Type]. ∀xs:A List. ∀ys:B List. ∀x:A. ∀y:B. ((<x, y> ∈ zip(xs;ys)) ⇒ {(x ∈ xs) ∧ (y ∈ ys)})
Proof
Definitions occuring in Statement :
zip: zip(as;bs),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
pair: <a, b>,
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
top: Top,
uimplies: b supposing a,
not: ¬A,
false: False,
iff: P ⇐⇒ Q,
or: P ∨ Q,
pi2: snd(t),
subtype_rel: A ⊆r B,
pi1: fst(t),
guard: {T},
cand: A c∧ B,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
all_wf,
list_wf,
l_member_wf,
zip_wf,
guard_wf,
and_wf,
zip_nil_lemma,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
cons_wf,
zip_cons_nil_lemma,
zip_cons_cons_lemma,
cons_member,
pi2_wf,
Error :pi1_wf_top,
subtype_rel_product,
top_wf,
or_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
hypothesis,
functionEquality,
productEquality,
independent_pairEquality,
independent_functionElimination,
rename,
because_Cache,
dependent_functionElimination,
universeEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
unionElimination,
applyEquality,
inlFormation,
independent_pairFormation,
addLevel,
inrFormation
Latex:
\mforall{}[A,B:Type]. \mforall{}xs:A List. \mforall{}ys:B List. \mforall{}x:A. \mforall{}y:B. ((<x, y> \mmember{} zip(xs;ys)) {}\mRightarrow{} \{(x \mmember{} xs) \mwedge{} (y \mmember{} ys)\})
Date html generated:
2020_05_19-PM-09_49_25
Last ObjectModification:
2020_02_03-AM-11_43_39
Theory : list_1
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