Nuprl Lemma : sorted-by-reverse
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀L:T List. (sorted-by(R;L) ⇐⇒ sorted-by(λx,y. (R y x);rev(L)))
Proof
Definitions occuring in Statement :
sorted-by: sorted-by(R;L),
reverse: rev(as),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
sorted-by: sorted-by(R;L),
member: t ∈ T,
int_seg: {i..j-},
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
rev_implies: P ⇐ Q,
top: Top,
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
true: True,
subtract: n - m,
cand: A c∧ B,
sq_type: SQType(T)
Lemmas referenced :
int_seg_wf,
length_wf,
reverse_wf,
sorted-by_wf,
subtype_rel_dep_function,
l_member_wf,
subtype_rel_self,
set_wf,
list_wf,
length-reverse,
lelt_wf,
int_seg_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
subtract_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
select-reverse,
iff_weakening_equal,
all_wf,
squash_wf,
true_wf,
add-associates,
minus-add,
minus-minus,
minus-one-mul,
add-swap,
add-mul-special,
add-commutes,
zero-add,
zero-mul,
add-zero,
select_wf,
le_wf,
less_than_wf,
and_wf,
equal_wf,
subtype_base_sq,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
cumulativity,
applyEquality,
instantiate,
lambdaEquality,
functionEquality,
universeEquality,
setEquality,
independent_isectElimination,
because_Cache,
functionExtensionality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
productElimination,
unionElimination,
imageElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
imageMemberEquality,
baseClosed,
multiplyEquality,
addEquality,
productEquality,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}L:T List. (sorted-by(R;L) \mLeftarrow{}{}\mRightarrow{} sorted-by(\mlambda{}x,y. (R y x);rev(L)))
Date html generated:
2016_10_21-AM-10_11_57
Last ObjectModification:
2016_07_12-AM-05_29_50
Theory : list_1
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