Nuprl Lemma : member-insert-no-combine
∀T:Type. ∀cmp:comparison(T). ∀x,z:T. ∀v:T List. ((z ∈ insert-no-combine(cmp;x;v)) ⇐⇒ (z ∈ [x / v]))
Proof
Definitions occuring in Statement :
insert-no-combine: insert-no-combine(cmp;x;l),
comparison: comparison(T),
l_member: (x ∈ l),
cons: [a / b],
list: T List,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
insert-no-combine: insert-no-combine(cmp;x;l),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
comparison: comparison(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A
Lemmas referenced :
list_induction,
iff_wf,
l_member_wf,
insert-no-combine_wf,
cons_wf,
list_wf,
list_ind_nil_lemma,
list_ind_cons_lemma,
comparison_wf,
nil_wf,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
le_wf,
cons_member,
or_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesisEquality,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
rename,
universeEquality,
independent_pairFormation,
natural_numberEquality,
applyEquality,
setElimination,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
addLevel,
orFunctionality,
inrFormation,
inlFormation
Latex:
\mforall{}T:Type. \mforall{}cmp:comparison(T). \mforall{}x,z:T. \mforall{}v:T List.
((z \mmember{} insert-no-combine(cmp;x;v)) \mLeftarrow{}{}\mRightarrow{} (z \mmember{} [x / v]))
Date html generated:
2017_04_17-AM-08_30_57
Last ObjectModification:
2017_02_27-PM-04_52_03
Theory : list_1
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