Nuprl Lemma : quicksort-int-prop1
∀[L:ℤ List]. ∀[i:ℕ||L||]. ∀[j:ℕi]. (quicksort-int(L)[j] ≤ quicksort-int(L)[i])
Proof
Definitions occuring in Statement :
quicksort-int: quicksort-int(L),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
le: A ≤ B,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
and: P ∧ Q,
sorted-by: sorted-by(R;L),
le: A ≤ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
less_than: a < b,
sq_stable: SqStable(P)
Lemmas referenced :
length_wf,
int_seg_wf,
less_than'_wf,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
itermConstant_wf,
intformle_wf,
decidable__le,
select_wf,
lelt_wf,
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
int_seg_properties,
sq_stable__le,
permutation_wf,
l_member_wf,
le_wf,
sorted-by_wf,
and_wf,
list_wf,
quicksort-int_wf,
quicksort-int-length
Rules used in proof :
cut,
lemma_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
because_Cache,
applyEquality,
lambdaEquality,
setElimination,
rename,
hypothesis,
sqequalRule,
imageMemberEquality,
baseClosed,
setEquality,
intEquality,
introduction,
imageElimination,
productElimination,
dependent_set_memberEquality,
independent_pairFormation,
natural_numberEquality,
unionElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
isect_memberFormation,
independent_pairEquality,
axiomEquality
Latex:
\mforall{}[L:\mBbbZ{} List]. \mforall{}[i:\mBbbN{}||L||]. \mforall{}[j:\mBbbN{}i]. (quicksort-int(L)[j] \mleq{} quicksort-int(L)[i])
Date html generated:
2016_05_15-PM-04_30_09
Last ObjectModification:
2016_01_16-AM-11_15_38
Theory : general
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