Nuprl Lemma : quicksort-int-iseg
∀[L,L':ℤ List]. ∀[n:ℤ]. ∀[x:ℤ]. x ≤ n supposing (x ∈ L') supposing L' @ [n] ≤ quicksort-int(L)
Proof
Definitions occuring in Statement :
quicksort-int: quicksort-int(L),
iseg: l1 ≤ l2,
l_member: (x ∈ l),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
iseg: l1 ≤ l2,
int_seg: {i..j-},
nat: ℕ,
lelt: i ≤ j < k,
and: P ∧ Q,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
guard: {T},
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
select: L[n],
cons: [a / b]
Lemmas referenced :
subtract_wf,
length_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
decidable__lt,
le_wf,
less_than_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
lelt_wf,
list_subtype_base,
subtype_base_sq,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
le_witness_for_triv,
l_member_wf,
iseg_wf,
append_wf,
cons_wf,
nil_wf,
quicksort-int_wf,
list_wf,
select_wf,
int_seg_properties,
length-append,
length_of_cons_lemma,
length_of_nil_lemma,
non_neg_length,
itermAdd_wf,
int_term_value_add_lemma,
equal_wf,
istype-universe,
select_append_front,
select_append_back,
quicksort-int-length,
quicksort-int-prop1,
length_nil,
length_cons,
length_append,
subtype_rel_list,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation_alt,
dependent_set_memberEquality_alt,
extract_by_obid,
isectElimination,
intEquality,
hypothesisEquality,
hypothesis,
natural_numberEquality,
setElimination,
rename,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
imageElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
universeIsType,
because_Cache,
productIsType,
equalityIsType4,
inhabitedIsType,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
promote_hyp,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
universeEquality,
hyp_replacement,
applyLambdaEquality,
addEquality,
productEquality,
voidEquality
Latex:
\mforall{}[L,L':\mBbbZ{} List]. \mforall{}[n:\mBbbZ{}]. \mforall{}[x:\mBbbZ{}]. x \mleq{} n supposing (x \mmember{} L') supposing L' @ [n] \mleq{} quicksort-int(L)
Date html generated:
2019_10_15-AM-11_13_26
Last ObjectModification:
2018_10_10-PM-02_08_55
Theory : general
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