Nuprl Lemma : l-ordered-decomp
∀[T:Type]. ∀[R:T ⟶ T ⟶ 𝔹]. ∀[x:T].
∀[L:T List]. L = (filter(λy.R[y;x];L) @ filter(λy.(¬bR[y;x]);L)) ∈ (T List) supposing l-ordered(T;x,y.↑R[x;y];L)
supposing Trans(T;x,y.↑R[x;y])
Proof
Definitions occuring in Statement :
l-ordered: l-ordered(T;x,y.R[x; y];L),
filter: filter(P;l),
append: as @ bs,
list: T List,
trans: Trans(T;x,y.E[x; y]),
bnot: ¬bb,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
all: ∀x:A. B[x],
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
iff: P ⇐⇒ Q,
and: P ∧ Q,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
bfalse: ff,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
assert: ↑b,
false: False,
not: ¬A,
trans: Trans(T;x,y.E[x; y])
Lemmas referenced :
list_induction,
isect_wf,
l-ordered_wf,
assert_wf,
equal_wf,
list_wf,
append_wf,
filter_wf5,
l_member_wf,
bnot_wf,
filter_nil_lemma,
list_ind_nil_lemma,
nil_wf,
true_wf,
l-ordered-nil-true,
equal-wf-base,
filter_cons_lemma,
bool_wf,
eqtt_to_assert,
list_ind_cons_lemma,
cons_wf,
iff_weakening_equal,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
filter_is_nil,
l_all_iff,
not_wf,
l-ordered-cons,
all_wf,
trans_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
hypothesis,
lambdaFormation,
setElimination,
rename,
setEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
addLevel,
independent_isectElimination,
productElimination,
natural_numberEquality,
isectEquality,
baseClosed,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
imageElimination,
imageMemberEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
productEquality,
functionEquality,
axiomEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}]. \mforall{}[x:T].
\mforall{}[L:T List]
L = (filter(\mlambda{}y.R[y;x];L) @ filter(\mlambda{}y.(\mneg{}\msubb{}R[y;x]);L)) supposing l-ordered(T;x,y.\muparrow{}R[x;y];L)
supposing Trans(T;x,y.\muparrow{}R[x;y])
Date html generated:
2018_05_21-PM-07_38_36
Last ObjectModification:
2017_07_26-PM-05_12_50
Theory : general
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