Nuprl Lemma : quicksort-int_wf
∀L:ℤ List. (quicksort-int(L) ∈ {srtd:ℤ List| sorted-by(λx,y. (x ≤ y);srtd) ∧ permutation(ℤ;srtd;L)} )
Proof
Definitions occuring in Statement :
quicksort-int: quicksort-int(L),
permutation: permutation(T;L1;L2),
sorted-by: sorted-by(R;L),
list: T List,
le: A ≤ B,
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
lambda: λx.A[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
quicksort-int: quicksort-int(L),
uall: ∀[x:A]. B[x],
member: t ∈ T,
comparison: comparison(T),
and: P ∧ Q,
cand: A c∧ B,
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,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
sorted-by: sorted-by(R;L),
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
le: A ≤ B,
uiff: uiff(P;Q)
Lemmas referenced :
length_wf,
int_seg_wf,
false_wf,
subtract-is-int-iff,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
select_wf,
int_seg_properties,
permutation_wf,
l_member_wf,
sorted-by_wf,
and_wf,
list_wf,
int-valueall-type,
equal-wf-T-base,
all_wf,
le_wf,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
equal_wf,
int_term_value_constant_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
intformand_wf,
int_formula_prop_wf,
int_term_value_minus_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermMinus_wf,
itermVar_wf,
itermSubtract_wf,
intformeq_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
subtract_wf,
quicksort_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
dependent_set_memberEquality,
lambdaEquality,
hypothesisEquality,
hypothesis,
sqequalRule,
dependent_functionElimination,
because_Cache,
unionElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_pairFormation,
productEquality,
applyEquality,
minusEquality,
functionEquality,
baseClosed,
setElimination,
rename,
imageMemberEquality,
setEquality,
introduction,
imageElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion
Latex:
\mforall{}L:\mBbbZ{} List. (quicksort-int(L) \mmember{} \{srtd:\mBbbZ{} List| sorted-by(\mlambda{}x,y. (x \mleq{} y);srtd) \mwedge{} permutation(\mBbbZ{};srtd;L)\} \000C)
Date html generated:
2016_05_15-PM-04_29_46
Last ObjectModification:
2016_01_16-AM-11_14_38
Theory : general
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