Nuprl Lemma : sorted-by-append
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
∀as,bs:T List. (sorted-by(R;as @ bs) ⇐⇒ sorted-by(R;as) ∧ sorted-by(R;bs) ∧ (∀a∈as.(∀b∈bs.R a b)))
Proof
Definitions occuring in Statement :
sorted-by: sorted-by(R;L),
l_all: (∀x∈L.P[x]),
append: as @ bs,
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
sorted-by: sorted-by(R;L),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
int_seg: {i..j-},
prop: ℙ,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
top: Top,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
so_apply: x[s],
rev_implies: P ⇐ Q,
ge: i ≥ j ,
le: A ≤ B,
subtype_rel: A ⊆r B,
subtract: n - m,
cand: A c∧ B,
sq_type: SQType(T),
l_all: (∀x∈L.P[x]),
true: True
Lemmas referenced :
int_seg_wf,
length_wf,
all_wf,
append_wf,
select_wf,
length-append,
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,
add-is-int-iff,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
false_wf,
l_all_wf,
l_member_wf,
list_wf,
non_neg_length,
lelt_wf,
select_append_front,
iff_weakening_equal,
add-member-int_seg2,
le_weakening2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
add-associates,
minus-one-mul,
add-mul-special,
zero-mul,
add-zero,
squash_wf,
le_wf,
less_than_wf,
subtract-is-int-iff,
and_wf,
equal_wf,
subtype_base_sq,
int_subtype_base,
select_append_back,
true_wf,
add-subtract-cancel,
length_append,
subtype_rel_list,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
cumulativity,
because_Cache,
lambdaEquality,
applyEquality,
functionExtensionality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
addEquality,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
productEquality,
setEquality,
functionEquality,
universeEquality,
dependent_set_memberEquality,
independent_functionElimination,
hyp_replacement,
imageMemberEquality,
applyLambdaEquality,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
\mforall{}as,bs:T List.
(sorted-by(R;as @ bs) \mLeftarrow{}{}\mRightarrow{} sorted-by(R;as) \mwedge{} sorted-by(R;bs) \mwedge{} (\mforall{}a\mmember{}as.(\mforall{}b\mmember{}bs.R a b)))
Date html generated:
2017_04_17-AM-07_43_27
Last ObjectModification:
2017_02_27-PM-04_16_51
Theory : list_1
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