Nuprl Lemma : cpsquicksort-quicksort
∀[T:Type]
∀[A:Type]. ∀[cmp:comparison(T)]. ∀L:T List. ∀[k:(T List) ⟶ A]. (cpsquicksort(cmp;L;k) ~ k quicksort(cmp;L))
supposing valueall-type(T)
Proof
Definitions occuring in Statement :
cpsquicksort: cpsquicksort(cmp;L;k),
quicksort: quicksort(cmp;L),
comparison: comparison(T),
list: T List,
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
cpsquicksort: cpsquicksort(cmp;L;k),
quicksort: quicksort(cmp;L),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cons: [a / b],
sq_stable: SqStable(P),
subtract: n - m,
true: True,
callbyvalueall: callbyvalueall,
listp: A List+,
has-value: (a)↓,
has-valueall: has-valueall(a),
comparison: comparison(T),
let: let,
cand: A c∧ B,
l_exists: (∃x∈L. P[x]),
equiv_rel: EquivRel(T;x,y.E[x; y]),
sq_type: SQType(T),
lt_int: i <z j,
assert: ↑b,
refl: Refl(T;x,y.E[x; y])
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
list_wf,
le_wf,
length_wf,
int_seg_wf,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
non_neg_length,
decidable__lt,
lelt_wf,
null_wf3,
subtype_rel_list,
top_wf,
bool_wf,
uiff_transitivity,
equal-wf-T-base,
assert_wf,
eqtt_to_assert,
assert_of_null,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
valueall-type-has-valueall,
hd_wf,
list-cases,
length_of_nil_lemma,
nil_wf,
product_subtype_list,
length_of_cons_lemma,
length_wf_nat,
nat_wf,
not-ge-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel2,
equal_wf,
evalall-reduce,
listp_properties,
not-lt-2,
zero-add,
le-add-cancel,
list-valueall-type,
filter_wf5,
l_member_wf,
lt_int_wf,
eq_int_wf,
length-filter-decreases,
select0,
subtype_base_sq,
int_subtype_base,
comparison-equiv,
select_wf,
itermAdd_wf,
int_term_value_add_lemma,
comparison_wf,
valueall-type_wf,
squash_wf,
true_wf,
iff_weakening_equal,
cpsquicksort_wf,
append_wf,
quicksort_wf,
sorted-by_wf,
permutation_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
functionEquality,
cumulativity,
because_Cache,
productElimination,
unionElimination,
applyEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
hypothesis_subsumption,
dependent_set_memberEquality,
imageElimination,
equalityElimination,
baseClosed,
impliesFunctionality,
promote_hyp,
addEquality,
imageMemberEquality,
minusEquality,
callbyvalueReduce,
setEquality,
instantiate,
hyp_replacement,
productEquality,
universeEquality,
functionExtensionality
Latex:
\mforall{}[T:Type]
\mforall{}[A:Type]. \mforall{}[cmp:comparison(T)].
\mforall{}L:T List. \mforall{}[k:(T List) {}\mrightarrow{} A]. (cpsquicksort(cmp;L;k) \msim{} k quicksort(cmp;L))
supposing valueall-type(T)
Date html generated:
2018_05_21-PM-07_34_57
Last ObjectModification:
2017_07_26-PM-05_09_12
Theory : general
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