Nuprl Lemma : member-update-alist1
∀[A,T:Type].
∀eq:EqDecider(T). ∀x:T. ∀L:(T × A) List. ∀z:A. ∀f:A ⟶ A. ∀y:T.
((y ∈ map(λp.(fst(p));update-alist(eq;L;x;z;v.f[v]))) ⇐⇒ (y ∈ map(λp.(fst(p));L)) ∨ (y = x ∈ T))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
update-alist: update-alist(eq;L;x;z;v.f[v]),
map: map(f;as),
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
pi1: fst(t),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
pi1: fst(t),
so_apply: x[s],
implies: P ⇒ Q,
update-alist: update-alist(eq;L;x;z;v.f[v]),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
prop: ℙ,
deq: EqDecider(T),
iff: P ⇐⇒ Q,
and: P ∧ Q,
or: P ∨ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
eqof: eqof(d),
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
not: ¬A,
guard: {T},
false: False
Lemmas referenced :
list_induction,
all_wf,
iff_wf,
l_member_wf,
map_wf,
update-alist_wf,
or_wf,
equal_wf,
list_wf,
list_ind_nil_lemma,
map_nil_lemma,
map_cons_lemma,
list_ind_cons_lemma,
deq_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
and_wf,
cons_wf,
pi1_wf,
cons_member,
bnot_wf,
not_wf,
eqof_wf,
uiff_transitivity,
eqtt_to_assert,
safe-assert-deq,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
false_wf,
member_singleton,
nil_member,
nil_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
productEquality,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
because_Cache,
productElimination,
applyEquality,
functionExtensionality,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
rename,
universeEquality,
setElimination,
equalityTransitivity,
equalitySymmetry,
baseClosed,
independent_pairFormation,
inlFormation,
addLevel,
hyp_replacement,
dependent_set_memberEquality,
applyLambdaEquality,
levelHypothesis,
independent_pairEquality,
orFunctionality,
promote_hyp,
unionElimination,
inrFormation,
equalityElimination,
independent_isectElimination,
impliesFunctionality,
allFunctionality
Latex:
\mforall{}[A,T:Type].
\mforall{}eq:EqDecider(T). \mforall{}x:T. \mforall{}L:(T \mtimes{} A) List. \mforall{}z:A. \mforall{}f:A {}\mrightarrow{} A. \mforall{}y:T.
((y \mmember{} map(\mlambda{}p.(fst(p));update-alist(eq;L;x;z;v.f[v]))) \mLeftarrow{}{}\mRightarrow{} (y \mmember{} map(\mlambda{}p.(fst(p));L)) \mvee{} (y = x))
Date html generated:
2017_04_14-AM-09_24_59
Last ObjectModification:
2017_02_27-PM-03_59_28
Theory : list_1
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