Nuprl Lemma : agree_on_common_filter
∀[T:Type]. ∀P:T ⟶ 𝔹. ∀as,bs:T List. (agree_on_common(T;as;bs) ⇒ agree_on_common(T;filter(P;as);filter(P;bs)))
Proof
Definitions occuring in Statement :
agree_on_common: agree_on_common(T;as;bs),
filter: filter(P;l),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
istype: istype(T),
top: Top,
agree_on_common: agree_on_common(T;as;bs),
list_ind: list_ind,
nil: [],
it: ⋅,
true: True,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
false: False,
not: ¬A,
guard: {T}
Lemmas referenced :
list_induction,
all_wf,
list_wf,
agree_on_common_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
istype-universe,
l_member_wf,
filter_nil_lemma,
istype-void,
nil_wf,
set_wf,
subtype_rel_self,
cons_wf,
ifthenelse_wf,
agree_on_common_nil,
filter_cons_lemma,
list_ind_cons_lemma,
not_wf,
bnot_wf,
assert_wf,
equal-wf-T-base,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
equal_wf,
agree_on_common_cons2,
member_filter,
agree_on_common_cons
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
hypothesis,
functionEquality,
because_Cache,
applyEquality,
setEquality,
setElimination,
rename,
setIsType,
universeIsType,
independent_isectElimination,
inhabitedIsType,
independent_functionElimination,
functionIsType,
dependent_functionElimination,
universeEquality,
isect_memberEquality_alt,
voidElimination,
lambdaEquality,
lambdaFormation,
natural_numberEquality,
productElimination,
voidEquality,
isect_memberEquality,
unionElimination,
baseClosed,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[T:Type]
\mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}as,bs:T List.
(agree\_on\_common(T;as;bs) {}\mRightarrow{} agree\_on\_common(T;filter(P;as);filter(P;bs)))
Date html generated:
2019_10_15-AM-10_53_38
Last ObjectModification:
2018_10_09-AM-10_28_53
Theory : list!
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