Nuprl Lemma : l-ordered-is-sorted-by
∀[T:Type]. ∀R:T ⟶ T ⟶ ℙ. ∀L:T List. (l-ordered(T;x,y.R[x;y];L) ⇐⇒ sorted-by(λx,y. R[x;y];L))
Proof
Definitions occuring in Statement :
l-ordered: l-ordered(T;x,y.R[x; y];L),
sorted-by: sorted-by(R;L),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
sorted-by: sorted-by(R;L),
l-ordered: l-ordered(T;x,y.R[x; y];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-},
uimplies: b supposing a,
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,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
rev_implies: P ⇐ Q,
l_before: x before y ∈ l,
sublist: L1 ⊆ L2,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
subtype_rel: A ⊆r B,
ge: i ≥ j ,
nat: ℕ,
select: L[n],
cons: [a / b],
subtract: n - m,
increasing: increasing(f;k)
Lemmas referenced :
select_wf,
int_seg_properties,
length_wf,
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,
l_before_select,
lelt_wf,
int_seg_wf,
all_wf,
l_before_wf,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
non_neg_length,
length_wf_nat,
nat_properties,
equal_wf,
squash_wf,
true_wf,
cons_wf,
nil_wf,
itermAdd_wf,
int_term_value_add_lemma,
iff_weakening_equal,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
setElimination,
rename,
independent_isectElimination,
natural_numberEquality,
hypothesisEquality,
cumulativity,
productElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
imageElimination,
independent_functionElimination,
dependent_set_memberEquality,
functionEquality,
applyEquality,
functionExtensionality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
instantiate,
universeEquality,
addEquality
Latex:
\mforall{}[T:Type]. \mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}. \mforall{}L:T List. (l-ordered(T;x,y.R[x;y];L) \mLeftarrow{}{}\mRightarrow{} sorted-by(\mlambda{}x,y. R[x;y];L))
Date html generated:
2018_05_21-PM-07_38_09
Last ObjectModification:
2017_07_26-PM-05_12_25
Theory : general
Home
Index