Nuprl Lemma : sorted-by-exists2
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
((∀a,b:T. Dec(a = b ∈ T))
⇒ (∀a,b:T. Dec(R a b))
⇒ Linorder(T;a,b.R a b)
⇒ (∀L:T List. ∃L':T List. (sorted-by(R;L') ∧ no_repeats(T;L') ∧ L ⊆ L' ∧ L' ⊆ L)))
Proof
Definitions occuring in Statement :
sorted-by: sorted-by(R;L),
l_contains: A ⊆ B,
no_repeats: no_repeats(T;l),
list: T List,
linorder: Linorder(T;x,y.R[x; y]),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
and: P ∧ Q,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
guard: {T},
sorted-by: sorted-by(R;L),
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T
Lemmas referenced :
list_wf,
linorder_wf,
all_wf,
decidable_wf,
equal_wf,
dcdr-to-bool_wf,
dcdr-to-bool-equivalence,
assert_wf,
assert_witness,
iff_wf,
exists_wf,
bool_wf,
sorted-by-exists,
linorder_functionality_wrt_iff,
sorted-by_wf,
subtype_rel_dep_function,
l_member_wf,
subtype_rel_self,
set_wf,
no_repeats_wf,
l_contains_wf,
int_seg_wf,
length_wf,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
universeEquality,
rename,
dependent_pairFormation,
independent_pairFormation,
because_Cache,
dependent_functionElimination,
productElimination,
independent_functionElimination,
independent_pairEquality,
axiomEquality,
allFunctionality,
promote_hyp,
productEquality,
independent_isectElimination,
instantiate,
setEquality,
setElimination,
natural_numberEquality,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
imageElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}a,b:T. Dec(a = b))
{}\mRightarrow{} (\mforall{}a,b:T. Dec(R a b))
{}\mRightarrow{} Linorder(T;a,b.R a b)
{}\mRightarrow{} (\mforall{}L:T List. \mexists{}L':T List. (sorted-by(R;L') \mwedge{} no\_repeats(T;L') \mwedge{} L \msubseteq{} L' \mwedge{} L' \msubseteq{} L)))
Date html generated:
2017_04_17-AM-08_33_12
Last ObjectModification:
2017_02_27-PM-04_53_38
Theory : list_1
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