Nuprl Lemma : l-ordered-filter2
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
∀L:T List. ∀P:{x:T| (x ∈ L)} ⟶ 𝔹. (l-ordered(T;x,y.R[x;y];L) ⇒ l-ordered(T;x,y.R[x;y];filter(P;L)))
Proof
Definitions occuring in Statement :
l-ordered: l-ordered(T;x,y.R[x; y];L),
l_member: (x ∈ l),
filter: filter(P;l),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
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],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_apply: x[s],
top: Top,
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
subtype_rel: A ⊆r B,
guard: {T},
cand: A c∧ B,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False
Lemmas referenced :
list_induction,
all_wf,
l_member_wf,
bool_wf,
l-ordered_wf,
filter_wf5,
list_wf,
filter_nil_lemma,
true_wf,
nil_wf,
l-ordered-nil-true,
filter_cons_lemma,
cons_member,
cons_wf,
eqtt_to_assert,
l-ordered-cons,
subtype_rel_dep_function,
subtype_rel_sets,
equal_wf,
subtype_rel_self,
set_wf,
member_filter_2,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot
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,
because_Cache,
hypothesis,
setElimination,
rename,
applyEquality,
functionExtensionality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
addLevel,
allFunctionality,
impliesFunctionality,
productElimination,
inlFormation,
dependent_set_memberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
inrFormation,
independent_pairFormation,
dependent_pairFormation,
promote_hyp,
instantiate,
productEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
\mforall{}L:T List. \mforall{}P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}.
(l-ordered(T;x,y.R[x;y];L) {}\mRightarrow{} l-ordered(T;x,y.R[x;y];filter(P;L)))
Date html generated:
2018_05_21-PM-07_38_27
Last ObjectModification:
2017_07_26-PM-05_12_42
Theory : general
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