Nuprl Lemma : remove-first-no_repeats-member
∀[T:Type]
∀L:T List. ∀P:{x:T| (x ∈ L)} ⟶ 𝔹. ∀x:T.
(no_repeats(T;L)
⇒ (∀a,b:{x:T| (x ∈ L)} . (((↑(P a)) ∧ (↑(P b))) ⇒ (a = b ∈ T)))
⇒ ((x ∈ remove-first(P;L)) ⇐⇒ (x ∈ L) ∧ (↑¬b(P x))))
Proof
Definitions occuring in Statement :
remove-first: remove-first(P;L),
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
list: T List,
bnot: ¬bb,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s1;s2;s3],
top: Top,
so_lambda: so_lambda3,
remove-first: remove-first(P;L),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uimplies: b supposing a,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
false: False,
not: ¬A,
uiff: uiff(P;Q),
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B
Lemmas referenced :
equal_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
list_wf,
bnot_wf,
assert_wf,
set_wf,
bool_wf,
subtype_rel_dep_function,
remove-first_wf,
l_member_wf,
iff_wf,
all_wf,
no_repeats_wf,
list_induction,
assert_witness,
btrue_neq_bfalse,
btrue_wf,
null_nil_lemma,
nil_wf,
member-implies-null-eq-bfalse,
cons_wf,
no_repeats_cons,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
eqtt_to_assert,
assert_of_bnot,
cons_member,
not_wf,
or_wf,
and_wf,
assert_elim,
not_assert_elim,
subtype_rel_self,
remove-first-member-implies,
bfalse_wf,
istype-assert,
list-subtype,
no_repeats-subtype,
istype-universe,
l_member-settype
Rules used in proof :
cut,
universeEquality,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
independent_functionElimination,
functionExtensionality,
productEquality,
rename,
setElimination,
independent_isectElimination,
setEquality,
applyEquality,
because_Cache,
hypothesis,
cumulativity,
functionEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productElimination,
equalitySymmetry,
equalityTransitivity,
independent_pairFormation,
instantiate,
promote_hyp,
dependent_pairFormation,
equalityElimination,
unionElimination,
andLevelFunctionality,
impliesFunctionality,
addLevel,
inrFormation,
levelHypothesis,
applyLambdaEquality,
dependent_set_memberEquality,
hyp_replacement,
inlFormation,
isect_memberFormation_alt,
lambdaFormation_alt,
dependent_set_memberEquality_alt,
universeIsType,
lambdaEquality_alt,
setIsType,
functionIsType,
productIsType,
equalityIstype,
inhabitedIsType
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. \mforall{}P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}. \mforall{}x:T.
(no\_repeats(T;L)
{}\mRightarrow{} (\mforall{}a,b:\{x:T| (x \mmember{} L)\} . (((\muparrow{}(P a)) \mwedge{} (\muparrow{}(P b))) {}\mRightarrow{} (a = b)))
{}\mRightarrow{} ((x \mmember{} remove-first(P;L)) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} L) \mwedge{} (\muparrow{}\mneg{}\msubb{}(P x))))
Date html generated:
2020_05_19-PM-09_45_37
Last ObjectModification:
2020_01_04-PM-07_59_45
Theory : list_1
Home
Index