Nuprl Lemma : quicksort-int-single
∀[n:ℤ]. (quicksort-int([n]) ~ [n])
Proof
Definitions occuring in Statement :
quicksort-int: quicksort-int(L),
cons: [a / b],
nil: [],
uall: ∀[x:A]. B[x],
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
quicksort-int: quicksort-int(L),
quicksort: quicksort(cmp;L),
all: ∀x:A. B[x],
top: Top,
ifthenelse: if b then t else f fi ,
bfalse: ff,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
prop: ℙ,
sq_type: SQType(T),
guard: {T},
evalall: evalall(t),
lt_int: i <z j,
nil: [],
it: ⋅,
eq_int: (i =z j),
btrue: tt,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
and: P ∧ Q,
cand: A c∧ B
Lemmas referenced :
permutation_wf,
l_member_wf,
sorted-by_wf,
and_wf,
permutation_weakening,
le_wf,
sorted-by-single,
list_ind_cons_lemma,
list_ind_nil_lemma,
quicksort-int-nil,
nil_wf,
cons_wf,
list-valueall-type,
list_wf,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermConstant_wf,
itermVar_wf,
itermSubtract_wf,
intformeq_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
evalall-reduce,
int-valueall-type,
valueall-type-has-valueall,
filter_nil_lemma,
filter_cons_lemma,
reduce_hd_cons_lemma,
null_cons_lemma,
int_subtype_base,
list_subtype_base,
set_subtype_base,
subtype_base_sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
independent_isectElimination,
sqequalRule,
hypothesis,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
intEquality,
hypothesisEquality,
callbyvalueReduce,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
computeAll,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
sqleReflexivity,
independent_pairFormation,
dependent_set_memberEquality,
setElimination,
rename,
setEquality,
sqequalAxiom
Latex:
\mforall{}[n:\mBbbZ{}]. (quicksort-int([n]) \msim{} [n])
Date html generated:
2016_05_15-PM-04_30_38
Last ObjectModification:
2016_01_16-AM-11_15_17
Theory : general
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