Nuprl Lemma : sorted-by-exists
∀[T:Type]
∀eq,r:T ⟶ T ⟶ 𝔹.
Linorder(T;a,b.↑(r a b))
⇒ (∀L:T List. ∃L':T List. (sorted-by(λx,y. (↑(r x y));L') ∧ no_repeats(T;L') ∧ L ⊆ L' ∧ L' ⊆ L))
supposing ∀a,b:T. (↑(eq a b) ⇐⇒ a = b ∈ T)
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]),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
exists: ∃x:A. B[x],
cand: A c∧ B,
top: Top,
sorted-by: sorted-by(R;L),
select: L[n],
nil: [],
it: ⋅,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
l_contains: A ⊆ B,
or: P ∨ Q,
l_all: (∀x∈L.P[x]),
decidable: Dec(P),
less_than: a < b,
squash: ↓T
Lemmas referenced :
assert_wf,
assert_witness,
equal_wf,
list_induction,
exists_wf,
list_wf,
sorted-by_wf,
l_member_wf,
no_repeats_wf,
l_contains_wf,
linorder_wf,
all_wf,
iff_wf,
bool_wf,
nil_wf,
no_repeats_nil,
l_contains_nil,
length_of_nil_lemma,
stuck-spread,
base_wf,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
intformle_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
int_seg_wf,
insert-by_wf,
cons_wf,
insert-by-sorted-by,
insert-by-no-repeats,
l_all_cons,
member-insert-by,
select_wf,
length_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__lt,
l_all_iff,
cons_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
axiomEquality,
hypothesis,
extract_by_obid,
isectElimination,
applyEquality,
functionExtensionality,
cumulativity,
independent_functionElimination,
rename,
productEquality,
because_Cache,
setElimination,
setEquality,
functionEquality,
universeEquality,
dependent_pairFormation,
independent_pairFormation,
isect_memberEquality,
voidElimination,
voidEquality,
baseClosed,
independent_isectElimination,
natural_numberEquality,
int_eqEquality,
intEquality,
computeAll,
inlFormation,
unionElimination,
imageElimination,
inrFormation
Latex:
\mforall{}[T:Type]
\mforall{}eq,r:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}.
Linorder(T;a,b.\muparrow{}(r a b))
{}\mRightarrow{} (\mforall{}L:T List. \mexists{}L':T List. (sorted-by(\mlambda{}x,y. (\muparrow{}(r x y));L') \mwedge{} no\_repeats(T;L') \mwedge{} L \msubseteq{} L' \mwedge{} L' \msubseteq{} L)\000C)
supposing \mforall{}a,b:T. (\muparrow{}(eq a b) \mLeftarrow{}{}\mRightarrow{} a = b)
Date html generated:
2017_04_17-AM-08_32_54
Last ObjectModification:
2017_02_27-PM-04_53_51
Theory : list_1
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