Nuprl Lemma : comparison-sort_wf
∀[T:Type]. ∀[cmp:comparison(T)].
∀L:T List. (comparison-sort(cmp;L) ∈ {srtd:T List| sorted-by(λx,y. (0 ≤ (cmp x y));srtd)} ) supposing valueall-type(T)
Proof
Definitions occuring in Statement :
comparison-sort: comparison-sort(cmp;L),
comparison: comparison(T),
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],
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
comparison-sort: comparison-sort(cmp;L),
prop: ℙ,
comparison: comparison(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
sorted-by: sorted-by(R;L),
select: L[n],
nil: [],
it: ⋅,
top: Top,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
sq_stable__le,
insert-no-combine-sorted-by,
insert-no-combine_wf,
int_seg_wf,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
satisfiable-full-omega-tt,
int_seg_properties,
base_wf,
stuck-spread,
length_of_nil_lemma,
nil_wf,
comparison_wf,
valueall-type_wf,
list-valueall-type,
set-valueall-type,
le_wf,
l_member_wf,
sorted-by_wf,
list_wf,
eager-accum_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
setEquality,
because_Cache,
hypothesis,
lambdaEquality,
setElimination,
rename,
natural_numberEquality,
applyEquality,
dependent_functionElimination,
independent_isectElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
universeEquality,
dependent_set_memberEquality,
baseClosed,
voidElimination,
voidEquality,
productElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
imageMemberEquality,
imageElimination
Latex:
\mforall{}[T:Type]. \mforall{}[cmp:comparison(T)].
\mforall{}L:T List. (comparison-sort(cmp;L) \mmember{} \{srtd:T List| sorted-by(\mlambda{}x,y. (0 \mleq{} (cmp x y));srtd)\} ) suppos\000Cing valueall-type(T)
Date html generated:
2016_05_14-PM-02_43_59
Last ObjectModification:
2016_01_15-AM-07_37_55
Theory : list_1
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