Nuprl Lemma : quicksort-int-last
∀[L:ℤ List]. ∀n:ℤ. n ≤ last(quicksort-int(L)) supposing (n ∈ L)
Proof
Definitions occuring in Statement :
quicksort-int: quicksort-int(L),
last: last(L),
l_member: (x ∈ l),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
last: last(L),
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
true: True,
guard: {T},
rev_implies: P ⇐ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q)
Lemmas referenced :
quicksort-int-length,
not_wf,
assert_wf,
subtype_rel_set,
assert_elim,
assert_of_null,
nil_wf,
btrue_neq_bfalse,
member-implies-null-eq-bfalse,
top_wf,
subtype_rel_list,
null_wf3,
equal_wf,
btrue_wf,
null_nil_lemma,
last_wf,
less_than'_wf,
lelt_wf,
quicksort-int-prop1,
int_formula_prop_eq_lemma,
intformeq_wf,
less_than_wf,
decidable__equal_int,
iff_weakening_equal,
decidable__lt,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
permutation_wf,
l_member_wf,
sorted-by_wf,
and_wf,
list_wf,
decidable__le,
nat_properties,
length_wf,
subtract_wf,
quicksort-int_wf,
select_wf,
true_wf,
squash_wf,
le_wf,
quicksort-int-member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
applyEquality,
lambdaEquality,
imageElimination,
isectElimination,
equalityTransitivity,
equalitySymmetry,
intEquality,
because_Cache,
sqequalRule,
natural_numberEquality,
independent_isectElimination,
setElimination,
rename,
setEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageMemberEquality,
baseClosed,
universeEquality,
dependent_set_memberEquality,
independent_pairEquality,
addLevel,
impliesFunctionality,
levelHypothesis,
axiomEquality
Latex:
\mforall{}[L:\mBbbZ{} List]. \mforall{}n:\mBbbZ{}. n \mleq{} last(quicksort-int(L)) supposing (n \mmember{} L)
Date html generated:
2016_05_15-PM-04_30_47
Last ObjectModification:
2016_01_16-AM-11_15_58
Theory : general
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