Nuprl Lemma : insert-no-combine-sorted-by
∀[T:Type]
∀cmp:comparison(T)
((∀u,x,z:T. ((0 ≤ (cmp x u)) ⇒ (0 ≤ (cmp u z)) ⇒ (0 ≤ (cmp x z))))
⇒ (∀L:T List. ∀x:T. (sorted-by(λx,y. (0 ≤ (cmp x y));L) ⇒ sorted-by(λx,y. (0 ≤ (cmp x y));insert-no-combine(cmp;x\000C;L)))))
Proof
Definitions occuring in Statement :
insert-no-combine: insert-no-combine(cmp;x;l),
comparison: comparison(T),
sorted-by: sorted-by(R;L),
list: T List,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
lambda: λx.A[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
so_apply: x[s],
comparison: comparison(T),
prop: ℙ,
so_lambda: λ2x.t[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
le: A ≤ B,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2],
top: Top,
so_lambda: λ2x y.t[x; y],
it: ⋅,
nil: [],
uimplies: b supposing a,
select: L[n],
sorted-by: sorted-by(R;L),
insert-no-combine: insert-no-combine(cmp;x;l),
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
or: P ∨ Q,
bfalse: ff,
cand: A c∧ B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
unit: Unit,
bool: 𝔹,
squash: ↓T,
less_than: a < b,
decidable: Dec(P),
l_all: (∀x∈L.P[x]),
subtype_rel: A ⊆r B,
true: True,
sq_stable: SqStable(P)
Lemmas referenced :
comparison_wf,
list_wf,
insert-no-combine_wf,
le_wf,
l_member_wf,
sorted-by_wf,
all_wf,
list_induction,
int_seg_wf,
nil_wf,
cons_wf,
select_wf,
less_than'_wf,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
full-omega-unsat,
int_seg_properties,
length_of_cons_lemma,
list_ind_nil_lemma,
base_wf,
stuck-spread,
length_of_nil_lemma,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
sorted-by-cons,
assert_of_le_int,
eqtt_to_assert,
bool_wf,
le_int_wf,
list_ind_cons_lemma,
decidable__lt,
int_formula_prop_not_lemma,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__le,
length_wf,
l_all_cons,
cons_member,
member-insert-no-combine,
l_all_iff,
int_term_value_minus_lemma,
itermMinus_wf,
iff_weakening_equal,
true_wf,
squash_wf,
sq_stable__le
Rules used in proof :
universeEquality,
independent_functionElimination,
dependent_functionElimination,
setEquality,
applyEquality,
natural_numberEquality,
rename,
setElimination,
hypothesis,
because_Cache,
functionEquality,
cumulativity,
lambdaEquality,
sqequalRule,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
independent_pairEquality,
independent_pairFormation,
intEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
productElimination,
voidEquality,
voidElimination,
isect_memberEquality,
independent_isectElimination,
baseClosed,
instantiate,
promote_hyp,
equalityElimination,
unionElimination,
imageElimination,
computeAll,
applyLambdaEquality,
hyp_replacement,
minusEquality,
imageMemberEquality,
functionExtensionality
Latex:
\mforall{}[T:Type]
\mforall{}cmp:comparison(T)
((\mforall{}u,x,z:T. ((0 \mleq{} (cmp x u)) {}\mRightarrow{} (0 \mleq{} (cmp u z)) {}\mRightarrow{} (0 \mleq{} (cmp x z))))
{}\mRightarrow{} (\mforall{}L:T List. \mforall{}x:T.
(sorted-by(\mlambda{}x,y. (0 \mleq{} (cmp x y));L)
{}\mRightarrow{} sorted-by(\mlambda{}x,y. (0 \mleq{} (cmp x y));insert-no-combine(cmp;x;L)))))
Date html generated:
2018_05_21-PM-00_43_57
Last ObjectModification:
2018_05_18-PM-04_18_29
Theory : list_1
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