Nuprl Lemma : quicksort_wf
∀[T:Type]. ∀[cmp:comparison(T)].
∀L:T List. (quicksort(cmp;L) ∈ {srtd:T List| sorted-by(λx,y. (0 ≤ (cmp x y));srtd) ∧ permutation(T;srtd;L)} ) supposin\000Cg valueall-type(T)
Proof
Definitions occuring in Statement :
quicksort: quicksort(cmp;L),
comparison: comparison(T),
permutation: permutation(T;L1;L2),
sorted-by: sorted-by(R;L),
list: T List,
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
quicksort: quicksort(cmp;L),
squash: ↓T,
less_than: a < b,
less_than': less_than'(a;b),
le: A ≤ B,
subtype_rel: A ⊆r B,
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
prop: ℙ,
and: P ∧ Q,
top: Top,
not: ¬A,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
comparison: comparison(T),
true: True,
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
has-valueall: has-valueall(a),
has-value: (a)↓,
listp: A List+,
callbyvalueall: callbyvalueall,
subtract: n - m,
sq_stable: SqStable(P),
cons: [a / b],
refl: Refl(T;x,y.E[x; y]),
assert: ↑b,
lt_int: i <z j,
sq_type: SQType(T),
equiv_rel: EquivRel(T;x,y.E[x; y]),
so_apply: x[s],
so_lambda: λ2x.t[x],
sorted-by: sorted-by(R;L),
trans: Trans(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
nequal: a ≠ b ∈ T ,
band: p ∧b q
Lemmas referenced :
comparison_wf,
valueall-type_wf,
list_wf,
length_wf_nat,
nat_wf,
int_term_value_add_lemma,
itermAdd_wf,
equal_wf,
assert_of_bnot,
eqff_to_assert,
iff_weakening_uiff,
not_wf,
bnot_wf,
iff_transitivity,
assert_of_null,
eqtt_to_assert,
assert_wf,
equal-wf-T-base,
uiff_transitivity,
bool_wf,
top_wf,
subtype_rel_list,
null_wf3,
lelt_wf,
decidable__lt,
non_neg_length,
int_formula_prop_eq_lemma,
intformeq_wf,
false_wf,
int_seg_subtype,
decidable__equal_int,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
int_seg_properties,
int_seg_wf,
length_wf,
le_wf,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
permutation_wf,
l_member_wf,
sorted-by_wf,
permutation-nil,
sorted-by-nil,
sqequal-nil,
eq_int_wf,
lt_int_wf,
filter_wf5,
list-valueall-type,
le-add-cancel,
zero-add,
not-lt-2,
listp_properties,
evalall-reduce,
le-add-cancel2,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-associates,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
condition-implies-le,
sq_stable__le,
not-ge-2,
length_of_cons_lemma,
product_subtype_list,
nil_wf,
length_of_nil_lemma,
list-cases,
hd_wf,
valueall-type-has-valueall,
length-filter-decreases,
comparison-equiv,
int_subtype_base,
subtype_base_sq,
hd_member,
l_exists_iff,
append_wf,
sorted-by-append,
assert_of_eq_int,
set_wf,
select_wf,
filter_type,
all_wf,
l_all_wf2,
l_all_iff,
member_filter,
member-permutation,
assert_of_lt_int,
member_append,
iff_weakening_equal,
true_wf,
squash_wf,
int_term_value_minus_lemma,
itermMinus_wf,
minus-is-int-iff,
permutation_weakening,
append_functionality_wrt_permutation,
permutation_functionality_wrt_permutation,
assert_of_le_int,
equal-wf-base-T,
neg_assert_of_eq_int,
le_int_wf,
permutation-split2,
and_wf,
subtype_rel_sets,
permutation-subtype,
filter-filter,
istype-universe,
iff_imp_equal_bool,
bool_cases,
bool_subtype_base,
band_wf,
btrue_wf,
bfalse_wf,
full-omega-unsat,
istype-int,
istype-le,
equal-wf-base,
assert_of_band,
istype-assert,
istype-less_than
Rules used in proof :
universeEquality,
isect_memberEquality,
because_Cache,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
dependent_functionElimination,
lambdaEquality,
sqequalRule,
hypothesisEquality,
cumulativity,
thin,
isectElimination,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
addEquality,
impliesFunctionality,
baseClosed,
equalityElimination,
imageElimination,
dependent_set_memberEquality,
hypothesis_subsumption,
applyLambdaEquality,
applyEquality,
unionElimination,
productElimination,
independent_functionElimination,
computeAll,
independent_pairFormation,
voidEquality,
voidElimination,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
setEquality,
productEquality,
callbyvalueReduce,
minusEquality,
imageMemberEquality,
promote_hyp,
instantiate,
addLevel,
levelHypothesis,
functionEquality,
impliesLevelFunctionality,
allLevelFunctionality,
allFunctionality,
functionExtensionality,
closedConclusion,
baseApply,
pointwiseFunctionality,
hyp_replacement,
lambdaEquality_alt,
universeIsType,
inhabitedIsType,
functionExtensionality_alt,
lambdaFormation_alt,
approximateComputation,
dependent_pairFormation_alt,
Error :memTop,
equalityIstype,
sqequalBase,
productIsType,
isect_memberEquality_alt,
setIsType
Latex:
\mforall{}[T:Type]. \mforall{}[cmp:comparison(T)].
\mforall{}L:T List
(quicksort(cmp;L) \mmember{} \{srtd:T List| sorted-by(\mlambda{}x,y. (0 \mleq{} (cmp x y));srtd) \mwedge{} permutation(T;srtd;L)\}\000C )
supposing valueall-type(T)
Date html generated:
2020_05_20-AM-08_09_03
Last ObjectModification:
2020_01_08-PM-01_55_15
Theory : general
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