Nuprl Lemma : member-merge-int
∀bs,as:ℤ List. ∀x:ℤ. ((x ∈ merge-int(as;bs)) ⇐⇒ (x ∈ as) ∨ (x ∈ bs))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
merge-int: merge-int(as;bs),
list: T List,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
int: ℤ
Definitions unfolded in proof :
merge-int: merge-int(as;bs),
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
iff: P ⇐⇒ Q,
and: P ∧ Q,
prop: ℙ,
or: P ∨ Q,
rev_implies: P ⇐ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
nat: ℕ,
less_than: a < b,
squash: ↓T,
false: False,
guard: {T},
not: ¬A,
decidable: Dec(P),
uiff: uiff(P;Q),
le: A ≤ B,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
list_induction,
all_wf,
list_wf,
iff_wf,
l_member_wf,
reduce_wf,
insert-int_wf,
or_wf,
istype-int,
reduce_nil_lemma,
istype-void,
reduce_cons_lemma,
nil_wf,
null_nil_lemma,
btrue_wf,
stuck-spread,
istype-base,
length_of_nil_lemma,
less_than_transitivity1,
less_than_irreflexivity,
btrue_neq_bfalse,
decidable__equal_int,
istype-false,
not-equal-2,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-swap,
add-commutes,
le-add-cancel,
add-associates,
cons_wf,
member-insert-int,
cons_member
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
intEquality,
Error :lambdaEquality_alt,
closedConclusion,
hypothesis,
because_Cache,
hypothesisEquality,
independent_isectElimination,
Error :inhabitedIsType,
independent_functionElimination,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
rename,
Error :functionIsType,
Error :productIsType,
Error :universeIsType,
Error :unionIsType,
independent_pairFormation,
Error :inlFormation_alt,
unionElimination,
productElimination,
baseClosed,
setElimination,
imageElimination,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
Error :inrFormation_alt,
addEquality,
Error :equalityIstype,
promote_hyp
Latex:
\mforall{}bs,as:\mBbbZ{} List. \mforall{}x:\mBbbZ{}. ((x \mmember{} merge-int(as;bs)) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} as) \mvee{} (x \mmember{} bs))
Date html generated:
2019_06_20-PM-00_42_34
Last ObjectModification:
2019_04_08-PM-03_31_13
Theory : list_0
Home
Index