Nuprl Lemma : l_before-sorted-by
∀[T:Type]
∀L:T List. ∀[R:{x:T| (x ∈ L)} ⟶ {x:T| (x ∈ L)} ⟶ ℙ]. (sorted-by(R;L) ⇒ (∀x,y:T. (x before y ∈ L ⇒ (R x y))))
Proof
Definitions occuring in Statement :
sorted-by: sorted-by(R;L),
l_before: x before y ∈ l,
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
l_before: x before y ∈ l,
sublist: L1 ⊆ L2,
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
nat: ℕ,
guard: {T},
satisfiable_int_formula: satisfiable_int_formula(fmla),
l_member: (x ∈ l),
cand: A c∧ B,
sorted-by: sorted-by(R;L),
increasing: increasing(f;k)
Lemmas referenced :
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
l_before_wf,
sorted-by_wf,
l_member_wf,
list_wf,
list-subtype,
equal_wf,
select_wf,
int_seg_wf,
non_neg_length,
decidable__le,
length_wf_nat,
nat_properties,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
decidable__lt,
length_wf,
intformless_wf,
int_formula_prop_less_lemma,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
isectElimination,
because_Cache,
comment,
cumulativity,
functionExtensionality,
applyEquality,
setEquality,
functionEquality,
universeEquality,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
setElimination,
rename,
independent_isectElimination,
unionElimination,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
computeAll,
productEquality
Latex:
\mforall{}[T:Type]
\mforall{}L:T List
\mforall{}[R:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbP{}]
(sorted-by(R;L) {}\mRightarrow{} (\mforall{}x,y:T. (x before y \mmember{} L {}\mRightarrow{} (R x y))))
Date html generated:
2017_04_17-AM-07_43_14
Last ObjectModification:
2017_02_27-PM-04_16_35
Theory : list_1
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