Nuprl Lemma : remove-first-member-implies
∀[T:Type]. ∀L:T List. ∀P:{x:T| (x ∈ L)} ⟶ 𝔹. ∀x:T. ((x ∈ remove-first(P;L)) ⇒ (x ∈ L))
Proof
Definitions occuring in Statement :
remove-first: remove-first(P;L),
l_member: (x ∈ l),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
remove-first: remove-first(P;L),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
iff: P ⇐⇒ Q,
and: P ∧ Q,
false: False,
list_ind: list_ind,
nil: [],
it: ⋅,
rev_implies: P ⇐ Q,
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
subtype_rel: A ⊆r B
Lemmas referenced :
list_induction,
all_wf,
l_member_wf,
bool_wf,
remove-first_wf,
list_wf,
list_ind_nil_lemma,
nil_member,
nil_wf,
list_ind_cons_lemma,
cons_wf,
cons_member,
bool_cases,
subtype_base_sq,
bool_subtype_base,
eqtt_to_assert,
equal_wf,
eqff_to_assert,
assert_of_bnot,
subtype_rel_dep_function,
subtype_rel_sets,
subtype_rel_self,
set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
setEquality,
cumulativity,
hypothesis,
functionExtensionality,
applyEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
rename,
because_Cache,
inlFormation,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
instantiate,
independent_isectElimination,
inrFormation,
setElimination,
hyp_replacement,
Error :applyLambdaEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}. \mforall{}x:T. ((x \mmember{} remove-first(P;L)) {}\mRightarrow{} (x \mmember{} L))
Date html generated:
2016_10_21-AM-10_27_23
Last ObjectModification:
2016_07_12-AM-05_39_43
Theory : list_1
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